THE LIBRARY
OF
THE UNIVERSITY
OF CALIFORNIA
LOS ANGELES
OF- A-..
LIBRARY
LOS ANGELES, CALIF
SEMICENTENNIAL PUBLICATIONS
OF THE
UNIVERSITY OF CALIFORNIA
1868-1918
THE FUNDAMENTAL EQUATIONS
OF DYNAMICS AND ITS MAIN COORDINATE
SYSTEMS VECTORIALLY TREATED
AND ILLUSTRATED FROM
RIGID DYNAMICS
DY
FREDERICK SLATE
UNIVERSITY OF CALIFORNIA PRESS
BERKELEY
1918
THIS BOOK FORMS PART II OF
THE PRINCIPLES OF MECHANICS, PART I,
NEW YORK, THE MACMILLAN COMPANY, 1900
PRESS OF
THE NEW ERA PRINTING COMPANY
LANCASTER, PA.
1918
Sciences Q A
Library
PREFACE
The day has clearly passed when any comprehensive presen-
tation of all dynamics could be compressed and unified within
the compass of one moderate volume of homogeneous plan.
The established connections of dynamical reasoning with other
fields in physics are of increasing number and closeness, as
furnishing for them strongly rooted sequences in their interpre-
tative trains of thought and linking together what would else
have continued to stand separate. And that relation has reacted
powerfully in modern times upon dynamics itself, perpetually
enriching its substance, yet at the same time introducing within
it certain sharpening differences that are stamped upon it by the
type of use for which preparation is being made. These in fact
modify superficially the modes of expression and their tone, and
shift their own emphasis through a range that brings about
what is in effect a subdivision of territory and an acknowledg-
ment of practically diverse interests. It is in response to the
situation which has been thus unfolding, and in conformity with
its drift toward manifold adaptations, that special treatises have
been rendered available whose measure of unquestioned excel-
lence and authority would make superfluous an attempt to
replace any such unit with a marked improvement upon it.
But undoubtedly these differentiations founded in divergencies
and inevitably expressing them in some degree, are entailing a
corresponding need and demand to offset them with a broadening
survey of the common foundation and of the common stock of
resources. And with that end in view another treatment of
dynamics finds a place for itself and holds it for special service.
This will propose to state with catholic inclusiveness the principles
iv Fundamental Equations of Dynamics
that lay out and direct all the main lines of use, and to anticipate
at their common source, as it were, the preferred methods and
forms that are characteristic of various provinces.
On this side also reasonable requirements for the immediate
future have been satisfied up to a definitely recognizable point.
For works on abstract dynamics are at hand to help, whose
number and quality have left no fair opening for renewed exposi-
tion, that could indeed scarcely attain excellence without dupli-
cating them. In the same proportion, however, that their
requisite perspective has grown, until it involves truly panoramic
sweep, its due scope must cease to be secured except from a
distance that expunges most details and spares only landmarks
of the bolder outlines. And under the urgent pressure to con-
dense in order to avoid neglecting and yet not become too
voluminous in summarizing completely, to keep even pace with
widening outlook, this view of dynamics cannot but endure the
attendant risks of abstractness. Because it must lean in building
toward great reliance upon the formal aid of mathematics, per-
force the physical coloring will fade and the bonds with experi-
mental reasoning be loosened. The stated results are pro-
gressively less likely to comprise what is charged with tentative
quality and is held with the candidly provisional acceptance
proper to inductive method.
For a student devoted to physical science though, as the
gifted mathematicians Poincare and Maxwell have been anxiously
insistent that he should be aware, there are lurking elements of
danger in magnifying a bare logical skeleton as a goal, and in
spending best effort upon articulating it. It is a misguidance
apt to control into rigidity thought which can scarcely prove
worthily fruitful unless it is maintained plastic. There is a
plain sense in which dependence upon clarity of demonstration
in terms of mathematical brevity and rigor may operate as a
defect; and that severe pruning which suppresses all but defini-
Introduction v
tive advance may mislead. There is a season for mitigating the
austerity of algebra and daring to become discursive, for relaxing
the ambition that is steadied to attain command of abstract
principles on their highest level and for pausing in reflective
examination of their genesis and their setting. Truly it would
sterilize action to incline thus always; but never to turn aside
from the more arduous pursuit tends to dissipate that atmosphere
for dynamics which has given it life.
At the other extreme are found the practical temperaments,
looking for tools with which to undertake their special tasks, and
largely unmindful of the processes by which those have been
shaped and of the far-reaching equipment in which their func-
tion is but one part, if only a particular routine can be adequately
served or intelligently mastered. And this more empirical frame
of mind that springs from absorption in monopolizing pursuits
can be fostered and strengthened by the sheer difficulties in
external form that are impressed upon abstract dynamics by the
tendencies that have just been referred to, and by the air of
remoteness from things material and mundane which that
treatment, if unconnected, confers. Unless it can be halted,
therefore, a movement toward disintegration which must be
coped with will confront the cultivators of dynamics that
derives a backing also from other circumstances of the present
situation.
The lifting of technical science to a better plane, where the
habitual facing of new problems under the illumination of
theoretical insight is coming to prevail, creates a demand in all
the fundamental sciences that is a modern appeal. It has been
incorporated into fixed plans of preparation for normal careers
in active life to accomplish those things which were formerly
undertaken with dominating inclination by minds self-selected
through their special gifts. There must be, then, in the methods
of presentation and in the execution of them, some recognition
vi Fundamental Equations of Dynamics
of a constituency that is at once larger, less homogeneous, and
more in need of aid. In a restricted sense of the word, there is
a summons to popularize the abstruser sciences, and among them
dynamics, with a design to favor their assimilation by students
at an earlier stage. This will make concessions in view of
hindrances inherent in the subject-matter, and allowance for
faculties of comparison and of analytic judgment not yet ripened
into full command of all resources.
There is some element in the immediate need that is due to
passing a transition and that will be lost in a newly adjusted
order; for it has appeared from manifold experience what
marvels can be wrought by tradition in giving easy currency to
scientific doctrine. Moreover, the obstacles that loomed larger
by mere novelty suffer genuine reduction by more lucid state-
ment. An older generation arrived but gradually at an under-
standing of the principle in conservation of energy, and caught
the advantage and power of absolute measurements first in
glimpses. Yet they have lived to find those unfamiliar ideas
adopted among the smoothly working formulas of unquestioned
truth. So it will not pass the limits of a reasonable anticipation
to forecast how the younger generation of today can move at
ease in their maturity among bold concepts that were obscure
when imperfectly grasped. Nevertheless, as the call now is,
so must the answer be given.
Every aspect of the thoughts here put down is framed in a
personal experience: the profit from quickening perception and
appreciation for the nexus between sharply generalized ideas
and their narrower origins; the benefit of laying stepping-
stones gauged to a student's stride; the reward of implanting
human interest within the routine of an industrial calling; also
the moral gain through confirming intellectual honesty under a
sustained demand for actual comprehension of what one is
challenged to attack among the papers rated as classics, or in
Introduction vii
judging and sifting recent work. Aiding to scent difficulties
first and then to overcome them fits the processes of the average
mind, where the stronger talent can walk self-guided.
The present enterprise was born of the foregoing considerations
in so far as they dictated its material and the ends for which that
was offered in gradual accumulation during many years and
under the influence of contact with students of varied purpose.
It renounces from the outset all claim to be systematically con-
ceived; it is content with a circling return from one point and
another to a core of ideas that are worth reviewing in their
various aspects because they are central. In their nature being
a supplement to standard books that differ in type from each
other, and offering themselves in flexible continuation of an
elementary stage with unsettled achievement, these selected dis-
cussions cannot escape being judged fragmentary by some, redun-
dant by others. But their spirit and their general aim are built
upon ascertained failure to acquire elsewhere a just comprehen-
sion of several matters here made prominent and perhaps in
some degree originally presented.
This kernel of intention in the subject-matter gathered for
these chapters lends to them, it may be claimed legitimately,
something of peculiar appropriateness for the circumstances of
their publication. On the occasion to be celebrated it seems
particularly pertinent that there should be recorded in some
permanent form the working of those influences which our
University has not withheld from her graduates, to nourish in
them a living root of independent thinking and of unflinching
thoroughness without which constructive scholarship cannot
exist.
June, 1917.
CONTENTS
CHAPTER I
PAGE
INTRODUCTORY SUMMARY 1
CHAPTER II
THE FUNDAMENTAL EQUATIONS 21
CHAPTER III
REFERENCE FRAMES: TRANSFER AND INVARIANT SHIFT. . . 76
CHAPTER IV
SOME COORDINATE SYSTEMS 112
NOTES TO CHAPTERS I-IV , 201
INDEX , . . 225
IX
CHAPTER I
INTRODUCTORY SUMMARY
1. Only sciences that have attained a certain ripeness of
strongly rooted development have been found capable of com-
bining a vigorous and progressive activity at their working
frontier for advance with reflective examination of their deeper
foundations and their general method. The activity is aggressive
in devising novel attack upon enlarging material, while reflection
upon what has already become standard must go with recasting
it to meet modified demands. This situation has been promi-
nently realized in the case of dynamics, whose stirrings to self-
criticism have been evermore spurred by the interactions with
mathematics and astronomy, its closer neighbors, at the same
time that its field was broadening to permeate and harmonize the
greater part of physics. A large net gain of helpful stimulus from
common aim must be allowed here, reenforcing the vigor from
rapid growth, though there have been some dangers for dynamics
to avoid, such as becoming infected with the more formal and
abstract spirit of mathematics, or underrating its own basis in
phenomena by acquiescing too generously in philosophy's rating
for empirical science. It is a fitting preliminary to our immediate
purpose to touch upon one or two such reactions between in-
fluences from without and from within; in part because the
inquiries that were provoked, though prolonged through fifty
years or more with acuteness and tenacity, have left practically
unshaken the external forms of quantitative expression, at least.
This is no sign, however, that dj^namics is stationary and stereo-
typed; but only a reassuring fact to beget confidence in the
fabric of the science. The subtle and less obtrusive changes
2 Fundamental Equations of Dynamics
must not be forgotten, that have clarified the concepts and
infused into them added significance by revised interpretation.
Reading the prospects of the imminent future, too, rouses the
expectation that what has been will continue to be, while dy-
namics is adapting itself to a wider scheme of connections and
to a more accurate insight into its own doctrine or theory.
It is indeed an astonishing testimony to the happy strokes of
genius in the founders of mechanics that force, impulse, work,
momentum and kinetic energy still exhaust the primary needs,
though the broader scope of dynamics now covers the chain of
transformations in which mechanical energy is only one link.
And it confirms our belief in the vital and definitive appropriate-
ness of those quantities to find them retained essentially by those
who are trying out another body of principles that might be substi-
tuted entirely or in part for the Newtonian mechanics. Mean-
while the equations of motion have not been superseded, yet
they date from the seventeenth century; the notable advances
due to d'Alembert, Euler and Lagrange in the eighteenth century,
and to Hamilton in 1835, offer still the foundations upon which
we build. But this introduction would outline a one-sided and
misleading picture of mere static stability unless it used its
warrant in bringing to supplementary notice three strands that
have been woven into dynamics more recently, to alter in some
degree its texture and to influence its emphasis. We shall next
attempt to dispose of these in all proper brevity.
2. Under the first label energetics we are called upon to chron-
icle a strong movement that sought to enhance the prestige that
energy in its various forms had already gained by the rapidly
successful campaign about the middle of the nineteenth century. 1
This tendency was an almost inevitable accompaniment of that
dominating relation to physical processes which conservation of
energy as a conceded central principle had justified beyond cavil.
1 See Note 1. Refer to collected notes following Chapter IV.
But the more pronounced utterances about energy overshot the
mark in their zeal, and sought to exalt it in rank as the one
dynamical quantity to which the rest should be held auxiliary,
and upon which they should be based mathematically. Then
the series kinetic energy, momentum, force, mass was to be
unfolded out of its first term by divisions; and violent extremists
were heard, even condemning force as a superfluous concept,
refusing to associate it directly with our muscular sense, or to
recognize it as an alternative point of departure yielding momen-
tum and other quantities by multiplications. Of course deliber-
ate minds looked askance at a professedly universal point of
view that would exclude, save at the cost of an artificial device,
such important elements as constraints that do no work. Com-
mon sense declined to cripple our assault upon problems for
doctrinaire reasons that would bar and mark for disuse certain
highways of approach, but it seized the chance instead to enrich
and strengthen dynamics by wisely adopting the suggestion to
exploit more completely the relations that energy specially
furnishes, and to incorporate them among its resources and
methods. After abating its flare of exuberance, the saner forces
behind the reconstruction that was advocated have been har-
nessed and made contributory to a real advance that grafts new
upon old, and embraces whatever proved advantage attaches to
all reasonable points of view, with the object of reducing finally
their oppositions and fitting them in place within a more compre-
hensive survey.
What is patent to read in the example of energetics should in
prudence be made further to bear fruit; since judging historically,
any new burst of reform spirit will be likely to repeat the main
features of its lesson. An old and thoroughly tested science
especially will less easily break the continuity of its course,
though it is always responsively ready to swerve under every
fresh impulse to amendment by discovery. So the matters
4 Fundamental Equations of Dynamics
offered recently under the caption relativity are surely giving to
dynamics a wider sweep of horizon; but there too, when the
permanent benefit accruing has been sifted out, the residue will
probably prove more moderate than the tone of radical spokes-
men has been implying while the sensation of novelty was
strongest. 1
3. It has been remarked often that Newton's three laws of
motion taken by themselves give a bias toward concentrating
attention upon momentum, and upon force exclusively as its
time-derivative, with a comparative neglect of the counterpart
in work and its relation to force. The restoration of balance
began at once however, and soon the principle of vis viva was
added and recognized as complementary on a level footing to
Newton's second law. The equivalents of what are now known
as the impulse equation and the work equation were established
firmly and put to use. The readjustment thus begun was
continued by steps as their desirableness was felt until with the
ripeness of time it culminated, we may say, in the proposals
that form the nucleus of what we call energetics. It will be
profitable to expand that thought and mention some chief
sources of the need to follow that line, or what gain has been
found in doing so.
In rudimentary shape the idea of conservation of energy had
emerged early; the histories are apt to date it from the method
invented by Huyghens for the treatment of the pendulum.
And so soon as the formal step had been taken in addition, that
set apart under the heading potential energy the work of weight
and of gravitation, because it can be anticipated by advance
calculation exactly and with full security, the invariance of
mechanical energy under the play of these forces- when thus
expressed, or its conservation within these narrower limits,
became a demonstrable corollary of fundamental definitions.
1 See Note 2.
Introductory Summary 5
The discovered inclusion of electric and magnetic attractions or
repulsions under the same differentially applied law of inverse
square that is characteristic of gravitation made natural the
extension of potential energy as a statement of securely antici-
pated work to the field of those actions as well. And a large
group of valuable mathematical consequences was accumulated
which remain classic and which accompany the law of inverse
square wherever it may lead, retaining their validity with only
slight changes of detail.
These developments are controlled to a great extent by the
idea of energy, and they must have built up a general perception
of its power. The invariance of energy was fitted more com-
pletely for use as a principle, wherever its mechanical forms alone
enter which we distinguish as kinetic and potential, when Gauss
had evolved that plan of so-called absolute measure which has
furnished us with the centimeter-gram-second system. He
certainly consolidated into unity all sources of ponderomotive
force in the several fields where a potential had been recognized.
Of course we discriminate between this stage and the conserva-
tion of energy under all its transformations to which the period
of Mayer, Joule and their co workers attained. The earlier
halting-place behind distinct limitations of scope left matters
besides with a formal content only, in the sense that no questions
were raised and squarely faced that looked toward localizing the
latent energy and investigating the possible mechanism by which
a medium might hold it in storage. This formal mathematics
centered on the fact that the work done within a conservative
system and between the same terminal configurations does not
depend upon the particular paths connecting them. It is a
strikingly significant exhibition of that quasi-neutrality that is
now one salient and accepted feature in the procedure of ener-
getics that so much of solid and permanent accomplishment was
possible while certain vital issues were evaded, and without
6 Fundamental Equations of Dynamics
being compelled to register even a tentative decision upon them.
That non-committal attitude towards much else as subsidiary,
provided always that the gains and losses of energy for the
system under consideration can be made to balance, has often
been employed to turn the flank of obstacles and has been in that
respect an element of strength. Or it leaves us in the lurch
weakly, we might say about other occasions where we have
stood in need of some crucial test between alternatives, and have
found but a dumb oracle.
4. The next important advance was then timely and specially
fruitful in giving life and deeper meaning to what had been in
these directions more a superficial form; and at the same time
in moving forward beyond the previous stopping-place to expand
the range of dynamical ideas. 1 It is Maxwell who is credited with
initiating these contributions by treating dynamically new
aspects of electromagnetic phenomena. He took bold and novel
ground by outlining his provisional basis for an electromagnetic
theory of light that converted a colorless temporary vanishing of
energy into a definite and plausible plan for its storage in a
medium. In achieving this change of front he brought three
lines of thought to a convergence-point; for besides the re-
searches of Faraday and those that identified quantitatively the
many transformations of energy, he utilized more fully than
his predecessors had dared the possibilities that the earlier
dynamics had done much toward making ready to his hand.
It is this third element perhaps that marks most strongly for
us the threshold of the new enterprise upon which dynamics will
hereafter be engaged, in whose tasks we can find a union in just
proportion of imaginative speculation with mastery of the
mathematical instruments and with the candid policy of ener-
getics to preserve an open mind and a suspended judgment in
the face of undecided questions.
1 See Note 3.
Introductory Summary 7
Maxwell was a pioneer in prolonging with new purpose the
sequence upon which d'Alembert set out, and which Lagrange
continued, beyond the point at which the latter paused after
recording notable progress. What those earlier men had done
with the discovery of virtual work as a basis for developing
mechanics remained to be restated for dynamics, and adapted
to a more inclusive command of energy transformations. Among
other things this has given us an enlarged interpretation of older
terms. We are ready to view a conservative system as one
whose energy processes are reversible: that is, energy of any
form being put in, it can be restored without loss, in the same
form or in some other. We have learned to group fair analogues
of kinetic and of potential energy for a system thus conservative
according to one defensible test. Potential forms of energy will
be found resilient as the original examples are; that is, they will
exhaust themselves automatically, under the conditions of the
particular combination, unless the corresponding transformation
is prevented actively. But in order to be coordinated with
kinetic energy on the other hand, the passive quality must be in
evidence that requires some decisive intervention for the passage
into other forms. This trend toward assigning wider meaning
to dynamical concepts has given us further generalized force as a
quotient of energy by a change in its correlated coordinate; the
matching of force and coordinate as factors in the product that
is energy being executed on due physical grounds. We have
been led likewise to replace mass by a broader term inertia,
where a quantity is detectable in the phenomena of more general
energy-storage, that stands in essential parallelism with the rela-
tion of mass itself to force and kinetic energy. And the dynami-
cal scheme has been rounded out by allowing to momentum those
privileges of latency and of reappearance in the literal mechanical
form, that were at the outset the monopoly of energy.
5. These comments have been attached to Lagrange's equa-
2
8 Fundamental Equations of Dynamics
tions because Maxwell did in fact make them the vehicle of his
thought; insisting upon sufficient detail to lift the reproach of
indefiniteness, but also by a right inherent in the method passing
over in silence the points where invention had thus far failed. But
it was demonstrated long ago that d'Alembert and Lagrange and
Hamilton have provided us with interconnected lines of approach
to the same goal: except as the element of choice is directed by
convenience Hamilton's principle lends equal favor and support
with Lagrange's equations to the attempt to summarize a com-
prehensive statement in terms of energy. The former however
elects to generalize for all analogous transformations upon a
simple theorem: That potential energy will exhaust itself as
rapidly as imposed constraints allow upon producing kinetic
energy.
Beside the direct intention to indicate some reasons why
dynamics leans increasingly upon energy relations, and borrows
from energetics some modes of attack, these later remarks have
a reverse implication as well. They intimate the belief that
firm hold upon the elementary content of dynamical principles
and intelligent full insight into them are not superseded, nor
yet to be slighted. And the meaning here is not the mere com-
monplace truth that the more modest range satisfies many needs;
or that historically it is the tap-root that has nourished and
sustained the later growth. But recurring to what lies at the
foundation is further the best preparation for the critical dis-
crimination that must be exercised at the advancing frontier,
because it holds the clews of conscious intention by which all
effort there has been guided, and lends effective aid in steering
an undeflected course among a medley of proposals to tolerate in
concepts a figurative shading of their literal acceptation, or to
condone acknowledged fictions on grounds of expediency.
6. The redistribution of emphasis upon which we have been
dwelling has doubtless exercised the most penetrating influence
Introductory Summary 9
to alter the complexion of mechanics as Newton left it, and
therefore we have put it first. But there has been a second
movement whose modifying effect as dynamics has grown must
not be neglected, and which also like the leavening with energetics
has been spread over a considerable period, though our report
of its outcome can be compressed into a brief space. 1 This
exhibited itself in a searching and protracted discussion on the
relativeness of velocity and acceleration that did its part in con-
tributing to clearness by removing ambiguity from a group of
terms and carrying through a completer analysis of their bearings.
The main concern here was not so much with the baldly kine-
matical side of the question; since it is plain that the final truth
in that sense lies very near the surface. But the endeavor was
quite specially shaped by the ambition to contrive at least soundly
consistent expression for all dynamical processes that shall be
recognized in physics; perhaps with some reach toward an ideal
of universal and ultimate validity. The entire relativeness of
those motions, which furnish leading factors of importance in
decisions upon working values of dynamical quantities, is now a
standard item in the opening chapters of dynamics as a corol-
lary to choice of reference elements by agreement.
The acquirement of this point of view has therefore excluded
all search for truly absolute motion and canceled the unqualified
significance of the phrase which dates as far back as Newton.
Since it seems flatly contradictory to unshackled relativeness,
an impression may be created at first hearing that here for once
the older thought has been overturned and radically revised.
Yet the case is not so weak as it sounds, nor do we see, when
we look below the surface, that any foundations have been
affected vitally. We may be comforted to observe only another
striking instance where a great mind did not everywhere and
straightway hit upon most felicitous terms to describe how it
1 See Note 4.
10 Fundamental Equations of Dynamics
dealt with powerful nascent conceptions. Newton seems to call
motions absolute if they dovetailed easily with the spacious
frame of physical action that his discovery of gravitation was
beginning to build; and himself engrossed in a swift recon-
naissance through the new region, he left later invention to
amend his notation. But it is chiefly the philosophical conno-
tations of his word absolute and not its unfitness in physics that
have made it the center of futile controversy. Thus the idea that
the older writers really had in mind when they spoke of absolute
motion was scarcely different from one that continues to hold
its ground and compels us still to separate two lines of inquiry.
Because beyond the settlement of kinematical equivalences that
is direct and simple since it is unhampered by any physical
considerations, the questions of real difficulty remain unsettled
to confront us. They have had a certain elusive character by
involving a complicated and tentative estimate that must
balance on the largest scale and through the whole range of
physics net gain against loss in simplicity. What common back-
ground, as it were, of reference-elements is decipherable upon
which the interplay of forces and of energies shall stand in
simplest and most consistently detailed relief?
In consequence it has not been displaced as a tenet of orthodox
dynamical doctrine that standards by which to judge of the
energy, momentum and force that ought to appear in its accounts
will not stand on a par if adopted at random, however inter-
changeable they have proved in passing upon rest, velocity and
acceleration by the mathematical criteria in the more indifferent
domain of kinematics. Dynamics has never hesitated to stig-
matize apparent forces, for example, as spurious or fictitious in
relation to its general procedure, and to revise its lists of rejec-
tions on due grounds derived from advance in knowledge and
in method. The definitive resolution of uncertainties that affect
reasonable decision for the questions here implied is still awaited ;
11
of necessity that objective is not attainable conclusively while
the surveys in the several provinces of physics remain both
fragmentary and disconnected. Though it has been claimed
indeed that secure foothold was being gained through reliance
upon a reference to stellar arrangement in removing excres-
cences that showed by the light of its corrective tests.
7. The growing practice to designate that reference as ultimate,
however, has not excluded a proper admission that its lines of
specification were to be improved by whatever greater precision
new discovery and analysis of it reveal definitely to be progress.
And it is fairly probable that majority opinion was looking
entirely in that direction for fresh landmarks until other prospects
were opened with vigor in recent years. These depend upon a
certain increase in freedom to retain functional forms when the
time-variable is added to the coordinates and included in the
group of quantities that are involved in the readjustment when a
change of base in the reference is undertaken. This far-reaching
proposal derived its original suggestion from optical phenomena
peculiar to electromagnetism and in one sense exceptional; yet
since it is the crux of this situation that a decision of universal
application is sought, any unreconciled indications of alternative
must be reckoned with, whereby two plans for attaining the
maximum simplicity that is desired become divergent. The
competitive schemes of ultimate reference cannot be weighed
decisively before the ramifications of both have been traced
everywhere in that detail which can afford a satisfactory con-
clusion through their final comparison. And for that the time
does not seem ripe; especially as each thus far falls short of
established universal quality by seeming to leave some combina-
tions unreduced, or abnormal to its plan. It is therefore reassur-
ing to our logical sense to note how the practically available
devices of proximate reference persist and are neutral, save in
the formulation of the limits upon which their steps of increasing
12 Fundamental Equations of Dynamics
precision may be declared to converge. For that their own
framework is by spontaneous intention approximate can be
conceded without discussion.
8. The contrast upon which we have been remarking, between
an indecision toward many-phased equivalences and the evolution
of preference among them is then one characteristic of the trans-
ition from kinematics to dynamics; that is, from a range fixed by
mathematical conformity to a selection narrowed by physical
meanings. We can proclaim a forward step in that direction when
the allowable mathematical range has been plausibly delimited, as
with the transverse wave of optics from Fresnel's wave-surface in
crystals to recent descriptive spectroscopy; but it is the crown of
attainment to master insight into the causes of the effects ob-
served, or into their sources, or into their explanations, in whatever
chosen terms the phrase may stand. This persistent effort to
identify physical sequences with a mechanism, to link a series
of phenomena by means of a mechanical interpretation, has
absorbed its full quota of sanguine activity since Newton scored
his early partial success with the propagation of sound. The
record shows in the main that the harvest of reward for these
attempts has continued into this later era, slackening somewhat
of course by exhaustion of the material. Yet there has been,
too, a baffling of the imagination in its task of dissecting the
complicated workings of energy in less traceable manifestations
by traveling on parallels to direct sense-experience. And again
optics illustrates; but now is shown a kind of failure, both with
the abandoned types of its theory and in its electromagnetic
alliance.
Every move in bestowing thus upon dynamics the control of a
larger domain has been healthy growth, keeping pace with
progress in other directions; and always sufficiently safeguarded
against speculative vagueness by bonds with the method of its
beginnings. Wherever mechanical energy in ponderable masses
Introductory Summary 13
exhibits itself in the actual chain of transformations, it gives a
touchstone through the measurable quantities, like pondero-
motive force, by which to try the conceived series for its validity
or consistency.
There are assumed successions, however, in which mechanical
energy is not directly in evidence though equivalents of it appear
in amounts known by using the change-ratios. Suppose we trans-
late the given facts or quantities and introduce mechanical energy
fictitiously. We have been prone to incline our judgment of the
original case according to the analogies of its artificial substitute,
and accordingly to accept the assumptions of the former or to
speak skeptically of its paradoxes. But in the puzzling region
that we have just mentioned there may be written a hidden
caution about the cogency of such transferred conclusions. The
absence of mechanical energy from the transformations that do
occur, as we are ready to suppose for light during transmission,
or for a free electron with inertia and without mass but traversing
an electromagnetic field, may be a contributory circumstance in
precluding a mechanical model and in leaving us thus far in the
twilight of kinematics, wrecked on obstacles of seeming internal
contradiction. And to the extent to which this indicated possi-
bility is entertained, the leverage of these unreduced phenomena
will be diminished, to guide or to modify dynamical thought that
discusses ponderable materials.
9. The third gain that we must bring forward is the improved
formulation of dynamics by replacing the cartesian expansions
with vector analysis, whenever general discussions and theorems
are taken in hand, or indeed everywhere unless we are barred
by the needs of detailed calculation to which the vector notation
is not so well adapted. The direct influence here is confined to
external forms, it is true; yet indirectly an undeniable effect
will always be exerted to favor continuity in the presentation of
reasoning, and to preserve with fewer breaks an intelligent
14 Fundamental Equations of Dynamics
orientation during extended developments. These advantages
are felt already, and they will accrue perpetually as a natural
accompaniment of increased compactness in stating relations
and of accentuating resultants first, only passing on to their
partial aspects where necessary. We should all lend our aid to
banish the obscurities and the disguises inseparable from the
older system of equation-triplets. The subdivision of the newer
analysis that is known distinctively as vector algebra is stand-
ardized fairly to the point of rendering great help in dynamics,
and adjustments to this specific use are perfecting. As regards
the vector operators like gradient, curl and divergence, they are
as yet far from establishment in full effectiveness, by unforced
extension of their original relation to field-actions and abatement
of its comparative abstruseness.
10. This introduction will distort the truth of its own words
and convey an unbalanced false impression, unless our reading
of it can be depended upon to counterpoise the omissions that have
trimmed it to these succinct proportions. So it is well to make
room at this point for a few sentences that bear upon maintaining a
real perspective against the tendency of extreme compression.
And first it must be realized that the personal careers of a small
group of geniuses do not constitute scientific history. To men-
tion one great man and to picture him advancing with long sure
strides implies with scarce an exception a whole accompanying
period.active with sporadic anticipations of some larger swing; an
epoch of transition busy with foreshadowings of a new alignment.
One's own thought should always supply this current of perhaps
unrecorded preparation for an impetus that has given enduring
reputation to its standard-bearer. The moulding of dynamics
therefore is not the merit of its master-builders alone; we must
not ignore those who had an inconspicuous share in establishing
and in perpetuating its governing traditions.
Then secondly it may prove misleading to speak exclusively
Introductory Summary 15
of changes and innovations, though some temporary aim compels
that. So we should return to the thesis of our opening para-
graphs and allow them a corrective weight: That the large
body of principles acquired early for dynamics and since un-
questioned has steadied its course. It has been capable of
assimilating the material that we have chosen to mention more
explicitly without sacrifice of comparative power to treat for
example the mechanics of solids and fluids. The considerations
derived within that older territory must hold their place in what
now follows.
11. It will be helpful in the direction of forestalling verbal
quibbles and of clearing the ground otherwise if we enter next
upon an explanation of the usage that we shall adopt for a few
convenient terms; and also proceed to indicate the general
attitude chosen in which to approach mathematical physics, of
which dynamics forms one part. It may be well to premise
once for all that no such personal choice covers a mistaken en-
deavor to close a question that is regarded reasonably as open,
and to silence dissenting opinion. But there is often a practical
necessity for taking a definite position, where adherence to one
view colors exposition; and thus it should be candidly an-
nounced, although the occasion is not appropriate for extended
argument.
In accordance with the unavoidable compulsion to take up
piecemeal the phenomena and the processes given by observation
and experiment in the physical world, any particular problem
of dynamics is obliged to concern itself with a solution obtained
under recognized limitations. These exhibit themselves on one
side in setting a boundary to the region within which the course
of events shall be investigated. If we distinguish within such a
boundary a part enclosed that is ponderable and a part that is
imponderable, we shall apply those terms on a plain etymological
basis; so that the ponderable contents have weight as evidenced
16 Fundamental Equations of Dynamics
by the balance and are subject to gravitation, while the im-
ponderable contents are not thus detectable. We shall speak
of the former also as masses or as bodies. The latter if not
alluded to as free space are called the ether, or the medium,
meaning the medium for the transmission of light and other
electromagnetic action. It is assumed that the ether-medium
has not mass in the sense just specified; but this does not deny
to it the more inclusive quality of inertia in certain connections.
A distinction need not be always upheld between mechanics and
dynamics; but where this is done the second name has the broader
scope, in that it may bring both masses and medium under
consideration, which comprise then a dynamical system rather
than a mechanical one. By contrast the older branch, me-
chanics, attempts only to deal with masses grouped into one
body, or into a system of bodies. We shall conceive a body to
fill its volume continuously and therefore to be adapted in so
far to expressing by means of an integral its total, either of mass
or of any quantity that is a function of the mass-distribution.
The conception behind the phrase system of bodies is somewhat
flexible; it may denote a discrete arrangement of bodies, whose
mass and the like are then given as a sum of a finite number of
terms, of which usage the astronomical view of our sun and its
planets grouped as bodies in the solar system affords a typical
instance; but it is applied also to a closely articulated assemblage
of bodies like a machine, under suppositions that might or might
not naturally justify integration. The opposition between body
and system of bodies is retained and does some service though
it is not tenable under stricter scrutiny, and cannot be radical so
long as physical theory actually analyzes all accessible bodies
into fine-grained systems for the purposes of molecular and
atomic dynamics. On the other hand the contrast between
systems of bodies and dynamical systems loses somewhat in
significance where the interspaces are assumed to be void and
Introductory Summary 17
the ether-medium is ignored; an abstraction common every-
where but in electro-magnetism; and the epithet, dynamical,
then points only towards inclusion of all transformations of
energy that remains associated with masses.
12. The tangled complexity in phenomena as they occur
however compels our official accounts of them to be given piece-
meal in other respects than by isolation of the region that lies
within an assigned boundary. What is further to be done may
be denominated variously; but it runs toward idealizing condi-
tions, both by selecting certain elements as most important for
study of their quantitative consequences and by a restatement
of these that consciously relaxes somewhat precision of corre-
spondence with the facts. It is evident how the two sources of
distortion are likely to conspire in simplifying the mathematics;
since neglecting weaker influences puts aside their smaller effects
as mere modifying terms of a main result. To prune difficulties
by this procedure as a preliminary to formulation and discus-
sion is in some sort a contrivance of approximation, conceding
the lack of desirable full power in our mathematical machinery.
That several determining reasons blend in it can perhaps be
recognized, though that is a subtle question upon which we shall
not touch; but what has practical weight is to separate two uses
of approximation, if such omission be accepted as one of them,
at the same time granting that both are drawn upon partly
because mathematics limps. 1
To put the case briefly, sometimes we lay down a rule strictly
but approximate to the results of it; which is a purely mathe-
matical operation, utilizing for example a convergent series as
we do when calculating the correction for amplitude in the period
of a weight pendulum. Or again the assumed rule itself is known
to be approximate, as is the fact when we call the pendulum
rigid and the local weight-field uniform and constant. A further
1 See Note 5.
18 Fundamental Equations of Dynamics
distinction is that the first type relates to obstacles which may
be overcome entirely by device, as in reducing finally some
obstinate integral, but which lie off the track of advances in
physics. In the instance just quoted the correction for ampli-
tude will remain untouched, because an angle and its sine will
never be equal. But with approximations of the second or
physical type it is otherwise; we cannot make a body more accu-
rately rigid by taking thought, nor can we bestow upon the
field-vector (g) any quality of constancy that it lacks; so they
progress by changing their rule. If provisional and marking
imperfect knowledge while we await amendments of magnitude
not yet ascertained, they move toward refinement of precision
parallel to the advancing front of experimental research, as the
law of Van der Waals about gases is seen to improve upon that
of Boyle. Yet no supreme obligation is felt to make such changes
everywhere; permanent and voluntary renunciations of achiev-
able accuracy are frequent, too; we shall probably continue in
many connections to discuss rigid solids and ideal fluids, not-
withstanding the volume of fruitful investigations in elasticity,
in viscosity and elsewhere, whose data are now at our disposal.
13. All these points are self-evident at first contact, and yet
it is advisable to name them, in order to put aside what is inci-
dental and focus attention upon the intrinsic structure of our
equations, which leaves them inevitably approximate as an
accepted limitation due to idealized or simplified statement.
Clothing this thought in a figure, let us say that the principles of
physics crystallize from the data of discovery into the concepts
that have been shaped by invention to express them, but not
without revealing traces of constraint and distortion that are
not subdued and made quite to vanish under repeated attempts
at adjustment. Historical inquiry has brought to light some
remarkable interdependences here, and furnished a list of ex-
amples how discovery has stimulated the invention of concepts
Introductory Summary 19
to match, and how on the other hand a stroke of inspiration in
devising a well-adapted concept has smoothed the path to dis-
covery of principle. Nevertheless the intimate psychology of
such reciprocity is one of those deep secrets that have been
securely guarded, and it need not concern us; we reach the kernel
of the matter for the present connection when we insist upon the
framework of dynamics as built of invented concepts and add
one or two corollaries of that central idea.
In the first place, in order to proceed by mathematical reason-
ing from specified assumptions, the margin of ambiguity in the
terms that are used must be cut down as much as is feasible.
A controversy about Newton's third law; whether or not it
applies to a source of light, could be settled easily under our
agreement that the ether-medium is not a body (corpus). And
the emancipation from corroborative tests in the free realm of
concepts is some compensation for the trouble of defining. It
has been laborious to disentangle the mean solar second as a
uniform standard of time; but the fluxion-time (t) of Newton in
its quality of independent variable must be equicrescent. So in
the concept of unaccelerated translation there is no place for
differences of velocity anywhere or at any time; and values
specified to be simultaneous cannot be affected by uncertain
deviations from that assumption; and for the conceptual iso-
tropic solid under Hooke's Law the stress-strain relation is
rigorously linear. Likewise, if according to the tenets of rela-
tivity the light-speed in free space and relative to the source is
always the same, we go on unflinching to work out the conse-
quences; and any such assumption with its demonstrable deduc-
tions will be entertained with candor, so long as its contacts with
observed facts given by correct mathematics do not fail either
as plausible physics. However, from the side of these perpetual
tests there is sleepless critical judgment upon all our mental
devices, to continue, to revise or to reject them. In other respects
20 Fundamental Equations of Dynamics
the schemes may be plastic to shift the point in precision at which
they halt, and we are reasonably tolerant also of conventional
fictions.
This brings to a close the short preface of such verbal comment
as may provide a setting in which to frame the equations that
follow, and at the same time assist in some respects to receive
more appreciatively their meaning by bringing to view what
underlies them.
CHAPTER II
THE FUNDAMENTAL EQUATIONS
14. Any standard exposition of dynamics ; though it may not
attempt a comprehensive and most general treatment of the
methods and principles, will introduce into its resources for
carrying on the discussion the six quantities : Force, Momentum,
Kinetic Energy, Power, Force-moment and Moment of Momen-
tum. The terms in detail that are required for the specification
of these, and a certain group of propositions into which they
enter, are so fundamental that they become practically in-
dispensable in establishing the necessary developments. The
units that their function as measured quantities demands are
supplied according to the centimeter-gram-second system with so
nearly universal adoption that we can regard it as having dis-
placed all competitors, everywhere except in some technical
applications where special needs prevail; so that we shall con-
sider no alternative plan of measurement.
Since the six quantities named are not independent of each
other, but are connected by a number of cross-relations that we
can assume to be familiar in their elementary announcement, it
is clear that the way lies open to select for a starting-point a
certain set as primary, the others then falling into their own
place as derived or even auxiliary quantities. It is also plain,
as a mere matter of logical arrangement, that any particular
selection of a primary set will not be unique, with a monopoly
of that title to be put first; and this leaves the exercise of prefer-
ence to be governed ultimately by reasons drawn from the
subject-matter. Not only is it possible to make beginnings
from more points than one in presenting the six quantities on a
21
22 Fundamental Equations of Dynamics
definite basis, and in exhibiting the links among them, but it is the
truth that beginnings have been made differently and defended
vigorously. We have already alluded to one such period of
polemic through which dynamics has passed. It is a necessity
however to choose a procedure by some one line of advance;
but let it be understood that we do this with no excessive claim
for its preponderant advantage or convenience, and explicitly
without prejudice to the validity of some other sequence that
may be preferred.
15. In the light of this last remark we shall make our start
by picking out for first mention a group of three quantities:
Momentum, Kinetic Energy and Moment of Momentum. With-
out anticipating a more specific analysis of them, it is evident on
the surface that they all apply in designating an instantaneous
state depending on velocities, and that momentum is the core
of the three; entering as free vector, as localized vector, and as
factor in a scalar product. And further it can be noted at once,
without presuming more than a first acquaintance with me-
chanics, that the remaining three quantities constituting a second
group can be described in symmetrical relation to the first three
as their time-rates. Then force is made central; and it in turn
appears as a free vector, as a localized vector, and as a factor in a
scalar product. We take the first step accordingly by laying
down for application to any body or to any system of bodies the
three defining equations :
Total momentum = 2 / m vdm = Q ; (I)
Total kinetic energy = 2 / m ^(v-vdm) = E; (II)
Total moment of momentum = S / m (r x vdm) = H. (Ill)
These indicate in each case, with notation that, is so nearly
standard as to carry its own explanation, the result of a mass-
summation extended to contributions from all the mass included
in the system at the epoch, under the terms of some agreement
The Fundamental Equations 23
covering the particular matter in hand, and isolating in thought
temporarily, for purposes of study and discussion, the phenomena
in a limited region. In conformity with a previous explanation
in section 11 any assumed continuous distributions of mass are
included under the integrals, whose further summation indicated
by (S) may be necessary when a system of bodies, discrete or
contiguous, is to be considered. It deserves to be emphasized
perhaps that these are defining equalities merely; so that (Q)
and (H) and (E) only denote aggregate values associated with
the system at the epoch, and so to speak observable in it; neither
side of the equalities conveys any implication about external
sources, or causes by whose action these aggregates may have
originated, or which may be operative at that epoch to bring
about changes affecting them.
16. Because the variables (r) and (v) occur in the quantities
with which we are now dealing, if for no deeper reason, it is
implied that a definite system of reference has been fixed upon
as an essential preliminary to actual attachment of values to
momentum, kinetic energy and moment of momentum. For
the ordinary routine which is likely to involve recasting vector
statements into semi-cartesian equivalents, or the inverse opera-
tion of arriving at the former by means of the latter, the requisite
elements for the reference are obtained by selecting an origin
from which to measure distances and axes for orienting directions.
Unless special exception be explicitly noted we shall follow the
prevalent usage of taking axes of reference that are orthogonal
and in the cycle of the right-handed screw; and shall for con-
venience conduct the main discussion on this permanent back-
ground, reserving any substitution of equivalents for occasions
where that has some peculiar fitness. The reference-frame that
has been agreed upon, it must not be forgotten, is in the essence
of it conceptually fixed while the agreement to use it continues in
force, because it has been singled out as the unique standard in
3
24 Fundamental Equations of Dynamics
relation* to which we specify or trace what can be called the
configurations (r) and the motions (v).
As an antecedent condition of algebraic evaluation for our
three fundamental quantities in a given system at any epoch,
the choice of some reference-frame then is necessary; but it is
likewise evident that any one choice that may be made is equally
sufficient in respect to removing mathematical indeterminateness.
And consequently it will be found true that much can be done
in advancing a satisfactory exposition of dynamical principles to
the point where we stand at the threshold of calculations that
rest on a basis of observed phenomena, without going beyond the
potential assumption of that reference-frame that must be faced
finally, in order to complete the necessary and sufficient condi-
tion for the definiteness of the physical specifications. In other
words, a considerable proportion of the usual developments in
dynamics can be provided ready-made to this extent, and yet
fitting the measure of any reference-frame that is particularly
indicated as appropriate by a physical combination or by a line
of argument. 1
These considerations are adapted to bring to the front also
the idea that quantities like the three with which we are con-
cerned at tnis moment can be evaluated for two or more different
reference-frames, perhaps with the object of reviewing their
comparative merit, especially in being adjusted to the preferences
of consistent physical views (see section 7) . It follows naturally
therefore that provision must be made quantitatively for trans-
fers of base from one reference-frame to another, either in progress
toward ultimate reference, as in abandoning a frame fixed
relatively to the rotating earth, or as a device of ingenuity in
order to reach certain ends simply. The material of Chapter in
in large part bears upon questions of that nature.
17. The range of the mass-summations that are stipulated in
1 See Note 6.
The Fundamental Equations 25
>
the expressions with which we are dealing can vary with time
for several reasons that can be operative separately or con-
currently. It is compatible with many conditions about bound-
ary-surface that material may be added or lost, as is the case
when gas is pumped into a tank or out of it, or when unit volume
of an elastic solid gains or loses by compression or extension.
Or it may fit the circumstances best to mark off a boundary that
changes with time, as when we take up mechanical problems
like those of a growing raindrop or a falling avalanche. The
values of (Q, E, H) are accommodated to any complication of
such conditions, with the single caution that the total mass shall
then be delimited as an instantaneous state at the epoch.
We go on to assume, however, in connection with any transfers
of reference that we are called upon to execute, that mass remains
unaffected thereby in its differential elements and in its total,
being guided by the absence of experimental evidence that mass,
in our adopted use of the word, needs to be made dependent upon
position or velocity. Assembling these suppositions, we see that
mass will play its part in the equations as a pure scalar and
positive constant, except as accretions or losses of recognizable
portions may be a feature of the treatment. And consequently
equation (II) can be made algebraic at once, since the vector
factors are codirectional, and be given the form
E = 2 /mCiVdm), (1)
although the original model should be preserved besides, as a
point of departure for parallelisms that will show themselves later.
18. Return now to examine the two remaining equations, in
order to extract some additional particulars of their meaning.
In the first the total momentum appears as a vector sum, so
built up that its constituents are usually described as free vectors.
This term is seen to justify combining the dispersed elements to
one resultant, on reflecting that the predicated freedom of such
26 Fundamental Equations of Dynamics
vectors lies wholly in the non-effect of mere shift to another base-
point; and that this renders legitimate the indefinite repetition of
the parallelogram construction for intersecting vectors until all
the differential elements have been absorbed, into the total aggre-
gate. But this incidental and as it were graphical convenience
must not lead us to neglect the fact that we are nevertheless
retaining the idea of momentum as a distributed vector, and con-
tinuing to associate each element of it locally with some element
of mass. However formed its total belongs to the system as a
whole; and it can be localized, as it sometimes is at the center
of mass of the system, only by virtue of a convention or an
equivalence. 1
We can call the total momentum a free vector, of course; but
its freedom does not quite consist in an indifference about its
base-point; more nearly it expresses the inherent contradiction
there would be in localizing anywhere what in fact is still con-
ceived to pervade the mass of the system. At several points we
shall discover how the service of vectors in physics makes desir-
able some addition to the formal mathematical handling of them.
It will not be overlooked, finally, how the above analysis of com-
position enlarges upon the addition qf parallel forces to constitute
.a total, through the similar properties of an algebraic and a geo-
metric sum; the latter reduces to its resultant by complete
cancellation in a plane perpendicular to the resultant.
19. In the third equation each local element of momentum
has the attribute of a localized vector through definite assign-
ment to the extremity of its radius-vector. It is not apparent
that the vector product in which it is a factor is thereby deter-
mined to be unequivocally localized; but here again physical
considerations enter that are extraneous to the mathematics; the
practice tacitly followed localizes the several elements of moment
of momentum, not at the differential masses to which they in
1 See Note 7.
The Fundamental Equations 27
one sense belong, but at the origin in acknowledgment of their
intimate connection with rotations about axes there, and of the
origin's importance in determining the lever-arms when the
mass-arrangement is given. Each differential moment of
momentum thus located being perpendicular to the plane of its
(r) and (v) of that epoch, is evidently also normal to the plane
containing consecutive positions of the radius-vector; that is,
(dH) is colinear with (dy), if the latter denotes the resultant
element of angle- vector that (r) is then describing; and on this
we can found a transformation that is worth noting. If (ds) is
the element of path for (dm),
d Y = ^ 2 (r x ds); Y = pfrx T); dH = T (r 2 dm); (2)
and the last equation reproduces differentially the type of an
elementary and partial relation among moment of momentum,
moment of inertia and angular velocity for a rigid solid. Only
(Y) is here individually determined in magnitude and in direction
for each (dm); no common angular velocity and collective
moment of inertia are assigned, as they are in the case of a
rigid solid, but with disturbance in general of the colinearity
shown by (dH) and (Y) into a divergence of the resultant vectors
for angular velocity and moment of momentum.
20. The three equations of section 15 are simplified remarkably
whenever the condition prevails that the velocity (v) has a
common value throughout the system that is in question. This
state of affairs is designated as translation of the system; it may
persist during a finite interval of time, or it may appear only
instantaneouslj 7 ', and in either case naturally it entails a corre-
sponding quality in the simplifications. When the condition of
translation persists the common velocity (v) need not be con-
stant; but the simultaneous velocities everywhere must be
equal. The resulting forms applying to translation are then
seen to be for a total mass (m),
28 Fundamental Equations of Dynamics
Q = v2 / m dm =' mv; (3)
E = i(vv)Z/ m dm = >v 2 ; (4)
H = (S / m rdm) x v = r x mv. (5)
The last equation introduces the familiar mean vector (f) which
locates the center of mass of the system through the mass-
average of the individual radius-vectors (r) according to' the
defining equation
mr = S / m rdm. (6)
The last group of equations contains the suggestion from which
has been worked out a notion that has had some vogue and
convenience in dynamics: that of an equivalent or representative
particle to which are attributed negligible dimensions but also
the total mass, momentum and kinetic energy of the system.
Equations (3, 4, 5) show that such a fictitious particle at the
position of the center of mass of the system would replace the
latter in respect of (Q, E, H) while translation >continues. And
since it is their ratio to other lengths that settles whether
dimensions are physically negligible, the absurdity that there
would be in concentrating momentum and energy into a mathe-
matical point is sensibly mitigated.
21. Even when the condition is not met that simultaneous
velocities shall be equal everywhere, a constituent translation
can be carved out artificially from the actual totals (Q, E, H)
at the epoch and for the system. Let every local velocity (v)
be split into two components in conformity with the relation
v = c + u, (7)
in which (c) is assigned at will, but taken everywhere equal, and
(u) denotes whatever remains of (v). Then substitution in the
fundamental equations of section 15 will segregate the totals
into a part that corresponds to translation and a supplement.
Among the indefinite number of possibilities, we select one
The Fundamental Equations 29
particularly fruitful plan for illustration. Let (v) be the mass-
average of velocities determined by the condition
mv = 2 / m vdm. (8)
Then if
v = v + u. 2 / m udm = necessarily. (9)
But we have also, in consequence of equation (9),
E = |S / m (v + u) (v + u)dm = |mv 2 + ^2 / m u 2 dm. (10)
And further,
H = 2 / m [r x (v + u)dm] = (2 / m rdm) x v + 2 / m (r x udm). (11)
In order to reduce the last term place r = r -f- r', so that (r')
like (u) is departure of the local value from the mean. Then
finally
H = (r x vm) + 2 / m (r' x udm), (12)
in which the segregation according to mean values and de-
partures from them is complete.
Taking equation (8) in conjunction with the first terms on the
right-hand of equations (10) and (12). the idea of a particle at the
position of the center of mass reappears, having the total mass (m) ,
the total momentum (mv = Q), and the kinetic energy (f mv 2 ).
But whereas equations (3, 4, 5) covered the data completely, this
contrived and partial translation with the mean velocity (v)
leaves residual amounts of kinetic energy and moment of momen-
tum; and these are due to departures from the mean values of (r)
and (v), as the last terms in equations (10) and (12) indicate
plainly, which items, as is also evident, have no resultant influ-
ence on the momentum. It is clear that this plan of partition is
adapted to accurate use; but it proves to have some advantages
too as the basis of an approximation, where the residual terms
are in small ratio to the translation-quantities and can be ne-
glected in comparison with the latter. The so-called simple
pendulum affords one instance.
30 Fundamental Equations of Dynamics
22. The recognition of elements of momentum as localized
vectors brings in an additional detail of their physical specifica-
tion; so this alone could be alleged as one valid reason for con-
ceding to moment of momentum its place in the general founda-
tion of dynamics. But we are now in a position to realize another
advantage of which that third equation gives us control. Mean
values have admitted elements of strength in smoothing out
accidental or systematic differences in a series of data, and in
enabling us to convert an integral into a product of finite factors.
Yet this acceptable aid may be offset in part by such elimination
on the whole of departures from the mean as is shown in
S / m r'dm = 0; S / m udm = 0.
Now first inspection of equation (12) shows how it serves to
retrieve by means of the vector products the divergencies that
would be lost from sight in the mean values, and thus to piece
out the support in that direction which equation (10) accom-
plishes through its scalar products, wherever we have an interest
to gauge effects of divergence that are cumulative and not
self-cancelling.
23. Before passing on to another topic it is worth taking occa-
sion to remark that the values for the totals of momentum,
kinetic energy and moment of momentum can be adjusted
without difficulty to expression as summations extended over a
volume; for in terms of the local density (5) and element of
volume (dV) the mass-element there is expressible by
dm = 5dV.
This density will be rated always a pure scalar on account of
its correlation with mass, and both density and volume are best
standardized in dynamics as positive factors in the positive
product that is mass, though it is not advisable to brush aside
too lightly the combinations that the character of volume as a
The Fundamental Equations 31
pseudo-scalar permits. Since the value of the density is zero
throughout the volume that is left unoccupied by the supposed
distribution of mass, the inclusion of such portions into a summa-
tion throughout the whole region within the assumed boundary
is without influence upon the result and can be indicated formally
without error. To declare a density zero is the equivalent of
excluding a volume from a mass-summation.
. Hence the need of a double notation (2) and ( / m ) will dis-
appear, if the continuous volume can be paired with a density
also effectively continuous, by any of the plausible devices that
evade abrupt changes at passage from volume with which mass
is associated to volume from which mass is said to be absent. 1
With these words of explanation the alternative forms that follow
are interpretable at once:
Q = / vol y(5dV); (13)
E = |/ vo ,v-v(6dV); (14)
H = / vol .(rxv(5dV)). (15)
Let us add for its bearing upon the lines of treatment when
mass is variable that then both (6) and (dV) are susceptible of
change. And also recall how there will always be that out-
standing question about mean values in comparison with diver-
gence from them, of which we spoke above, whenever we face the
contradictory demands of mathematical continuity and of open
molecular structure, in order to reconcile them adequately for
instance, in the concept of a homogeneous body with a value of
density that is common to all its parts.
24. We shall next approach the remaining group of funda-
mental quantities that we have enumerated already as three:
Force, Power, and Force-moment. The first object must be to
set forth in satisfying clearness and completeness their relations
1 See Note 8.
32 Fundamental Equations of Dynamics
to the previous group of three, in order to proceed then securely
with reading the lesson how the interlinked and consolidated
set of six quantities provides all requisite solid and efficacious
support, both for the current general reasoning of dynamics
and for its specialized lines of employment.
We began to follow the track over ground that has become
well-trodden since Newton's day, when we laid down a meaning
for the phrase total momentum of a system of bodies and the symbol
(Q) representing it which in effect only renames the intention of
the historic words " Quantity of motion." We also continue the
tradition that has been perpetuated ever since Newton's second
law launched its beginnings for approval, by fixing attention in
its turn upon the rate of change in the momentum, in its differ-
ential elements and in its total, and regard that as delivering to
us the clews, that we shall later follow up, to the forces brought
to bear upon the system of bodies that is under investigation,
with which forces we must undertake to reckon. The gist of
that law has not yielded perceptibly under all the proposals to
improve upon it, though we may be rewording it more flexibly
under widening appeal to experience. Its drift makes the claim
that changes in (Q) are not spontaneous; that when they are
identified to occur there is reason to be alert and detect why,
with gain for physical science in prospect by success.
25. The first move toward formulation could scarcely be
simpler; it is to indicate the time-derivative of equation (I) and
write
Q=^[Z/ m vdm}. (IV)
Yet the mere mathematics of execution blocks the way with
distinctions to be made, unless we are resolved to carry an over-
weight of hampering generality. For it is common knowledge
that the masses are often comprised in such a summation on a
justified footing that they are in every respect independent of
The Fundamental Equations 33
time; and consequently it is then legitimate to differentiate
behind the signs of summation in equation (IV). But forms
alter as the mass included is in any way a function of time;
they will differ besides if only the total mass changes by loss or
gain of elements, or only the elements change leaving the total
constant, or if both sorts of dependence upon time are permissible
under the scheme of treatment. The first supposition of complete
mass-constancy underlies the dynamics of rigid solids and is a
stock condition in much dynamics of fluids as well. And because
it prevails most naturally to that extent, it is perhaps fair to
select this mass-constancy as standard; especially when depar-
tures from it are likely soon to be cut off from the stream of
systematic development by running into specializing restrictions
and a narrowly applicable result. 1
However opinion may stand on that matter, it seems certain
that no aspect should be allowed to escape us finally that belongs
to the full scope of mathematical possibility attaching to the
indicated time-derivative of (Q). An}' contribution to the
changes in momentum may mature a suggestion about force-
action and gain physical meaning. Therefore the tendency seems
unfortunate to borrow the terms of Newton's second law, for
its professedly general statement, from the special though widely
prevalent case which throws all the change in momentum upon
the velocity-factor. To speak of force as universally measured
by the product of mass and acceleration is misleading if the
habit blinds us to the fuller scope of the second law, and atrophies
at all our capacity to use it.
26. In order that the derivative of an expression may be
formed for use, certain conditions of continuity must not be
violated, as we know; but when a derivative is to be made
representative of a sequence of states, mathematical physics has
available a repertory of resources in constructing this requisite
1 See Note 9.
34 Fundamental Equations of Dynamics
continuity of duration and distribution. Examples are plentiful
among the classic methods of attack, how variously the proper
degree of identifiable quality is assigned to a succession of states,
that links the individual terms into a continuous series. Rankine's
device for studying a sound-wave in air is a travelling dynamics
that keeps abreast of the propagation; Euler's hydrodynamical
equations stand permanently at the same element of volume, and
record for successive portions of liquid that stream by; and many
processes where material passes steadily through a machine are
most tractable in similar fashion. We shall not insist further
then upon this point, except to say that advanced stages of the
subject are less apt to rely upon straightforward sameness and
constancy in the masses specified for summation under the term
body or system of bodies. With the reserves of that cautious pre-
amble, we can afford to qualify the case of mass-constancy and
literal sameness as standard in a limited sense, and exploit some
of the consequences flowing from that assumption.
27. On the grounds now announced explicitly the indicated
operation of equation (IV) yields
Q = 2 / m vdm = S / m dR = R. (16)
As a symbol, therefore, (dR) is defined to mean the local resultant
force at each differential mass for which there is evidence through
the local acceleration; and accordingly (R) denotes the vector
sum of such elements of force when the whole system of bodies
is included. This total force is in nature a dispersed aggregate
like the total momentum, and the line of comment under that
heading applies here with a few changes, which however are
obvious enough to absolve us from repeating it.
28. Before we carry the discussion into farther detail it seems
best to bring equations (II) and (III) to this same level by
putting down their time-derivatives, observing consistently
there also the imposed limitation to complete mass-constancy,
The Fundamental Equations 35
but remembering always that vve halted exactly on that line and
postponed until due notice shall be given the further step in
restriction that will introduce a rigidly unchanging arrange-
ment or configuration of all the mass-elements. Writing first the
general defining equation as preface,
r-f.
we can then make the application to the specialized conditions
that gives
= S/ m (v-dR). (17)
This indicates at each element of mass a local manifestation of
power that is measured by the scalar product of the force-element
and velocity this scalar product being of course not merely
formal, since (v) and (dR) are not in general colinear. It has
been called also, perhaps with equal appropriateness, the activity
of the force.
29. In this preliminary consideration there remains only the
time-derivative of equation (III) . And we shall preserve a help-
ful symmetry of statement by giving its place here also to the
general defining equation, and following it as we have done
previously with its present special value. Then
M = H; (VI)
and
M = S / m ^ (r x vdm) = 2 / m (r x dR); (18)
the reduction of the expansion to one of its two terms being the
evident consequence of the identity of (f) and (v). The last
equation demonstrates within the limits set for it that the time-
derivative of the total moment of momentum measures the total
force-moment of the local elements of force that are calculated
36 Fundamental Equations of Dynamics
according to equation (16). As a postscript to equation (18)
repeat with the necessary modifications what was inserted in
section 19, about equation (III), and at the end of section 22.
The example of a force-couple will come to mind at once, where
the pair of its forces is self-cancelling from the free-vector aggre-
gate of force, and it devolves upon the localized force-vector of a
moment to restore for consideration the important effects of
couples.
Observe also the peculiar prominence of the radius-vector in
vector algebra. Where the cartesian habit is to bring both
moment of momentum and force-moment into direct and ex-
clusive relation to a line or axis, vector methods substitute rela-
tion to the origin, which is a point. Upon examination, however,
the difference partly vanishes, because the vector reference to a
point is only a superficial feature. We have explained in con-
nection with equation (2), how a resultant axis is tacitly added.
The element (dM) is similarly a maximum or resultant, the
factors in (r x dR) being given, the effective fraction of the
moment for other axes through the origin being obtainable by
projecting (dM). 1
30. Equations (16, 17, 18) bear on their face and for their
particular setting sufficient reasons for interpreting (Q) in terms
of those forces (dR); (P) or (dE/dt) in terms of the activity of
those same forces (v-dR); and (M) or (H) in terms of their
force-moments (r x dR). There seems to be neither confusion
nor danger imminent if we extend the names thus rooted in
commonplace experience to the (at least mathematically) more
complicated possibilities of equations (IV), (V), (VI). We can
be bold to identify (Q) always as some force (R); (dE/dt) as a
power (P); and (H) always as a force-moment (M); if we have
made ourselves safely aware how terms in any completed mathe-
matical expansion may remain non-significant physically until
1 See Note 10.
The Fundamental Equations 37
discovery confirms them. We have alluded before to the fact
that dj r namics does not altogether shrink from a figurative tinge
in extensions of terms first assigned literally, if essentials of
correspondence are adequately preserved. But notice particu-
larly that the verbal adoptions proposed above cannot of them-
selves assure the occurrence of the duplicate adjustments among
equations (16), (17), and (18). To forces whose sum is (Q) will
correspond activities that we may denote by (v-dQ), and mo-
ments of type (r x dQ). But we must not conclude in advance
that the former group will in their sum match (dE/dt) ; nor that
the latter group will match exactly (H); though both equiva-
lencies hold under the condition of mass-constancy. And for
discrepancies there will be no general corrective formula; they
must be newly weighed wherever they may appear.
31. Let us next turn back to the ideas of translation and
equivalent particle of which we spoke in sections 20 and 21,
and continue them in the light of equations (16, 17, 18). In the
first place note that the mean velocity (v) as previously specified
by equation (8) becomes now identical with the velocity of the
center of mass, because the time-derivative of equation (6) takes
the form
mf = 2 / m fdm = mv. (19)
Secondly the conditions justify for the next time-derivative,
mv = 2 / m vdm, (20)
showing that the center of mass has the mass-average of accelera-
tions. Hence a particle having the total mass (m) of the system
and retaining always its position (f) at the center of mass would
show at every epoch the total momentum (Q) ; and its accelera-
tion would determine the value (R) of the total force in equation
(16) through the product (mv).
But if the first terms in the second members of equations (10)
and (12) and the derivatives of those terms with regard to time
38 Fundamental Equations of Dynamics
be now considered, with the new meaning for (v), it is seen that
the specified particle at the center of mass duly represents all
the dynamical quantities for the system, except those parts which
depend upon departures (r') from the mean vector (f) and upon
departures (u) from the mean velocity (v). Hence such an
artificial or fictitious translation with the center of mass runs like a
plain thread through all the equations for the actual system, and
reproduces accurately their six dynamical quantities when we
simply superpose upon it the additional kinetic energy, moment
of momentum, power and force-moment whose source is in
the deviations from mean values. It is a self-evident corollary
that in a real or pure translation the particle at the center of mass
represents the system without corrections, since the local accelera-
tions must be of common value while translation continues, in
order that simultaneous velocities may remain equal. This
keeps each velocity (u) permanently at zero.
32. It will be instructive to enforce without delay the differ-
ences from parallelism with the preceding details that appear at
several points, in the simplest combinations where it becomes
natural to regard the total mass as variable with time. Let us
then take up for consideration a body in translation, or equiva-
lently a representative particle, denoting by (m) and (v) the
instantaneous values of its mass and velocity. For the momen-
tum and the kinetic energy at the epoch we still have
Q = mv; E = f (mv-v). (21)
If we stand by the agreement that (Q) shall be force and embody
it in the time-derivative of the first equation, we shall write
Q = R = mv +^v. (22)
When mass is constant, resultant force and resultant acceleration
have the same direction, as we can read in equation (16). But in
The Fundamental Equations 39
striking contrast with that consequence, equation (22) shows
that its (R) does not in general coincide with either velocity or
acceleration.
Proceeding next to examine the power, and continuing to
specify it as the derivative of (E) we find
==P = ^(mv-v + mvv) +- -JT-(V-V)
1 dm
Comparison with equation (22) brings out the relation
dm
(23)
/ dm \
R-v = I mv +~J7" V )' v = m *
= mv v -\- - v v
dE 1 dm
And once more a variation from the previous model is impressed
upon us; the power (P) is thrown out of equivalence with the
activity or working-rate of the force (R), thus realizing the
suggested contingency of section 30. The time-integral of the
last equation assumes the form
/ti
(R-v)dt = [Eft + | (dmvv), (25)
and expresses on its face the conclusion that the total work of
the force (R) for the interval is not accumulated in the change
shown by the kinetic energy. What the form and the fate are
of the energy summed in the last integral remains as a physical
question for further study; it may, for instance, cease to be
available, or it may be stored reversibly ready to appear again
by transformation.
If instead of dealing with the resultant (R) we proceed by the"
standard resolution into tangential and normal parts, these are
4
40 Fundamental Equations of Dynamics
R(t) = -JJTV + mv (t) ; R (n) = mv (n) ; (26)
and if we should maintain that measure of force which is ex-
pressible as the product of mass and its acceleration, the inferences
from the above equations would lead through the quotients of
force by its acceleration to different estimates of the mass in-
volved. From the first equation we obtain as a ratio of tensors
R(t) dm dv
-^ = - a -v + m, since ^,^; (27)
.
and from the second equation
^ = m. (28)
V(n)
33. The last value agrees with our initial supposition, and is
to that extent the true mass ; and the value given by the first quo-
tient in equation (27) has been distinguished as effective mass
since the motion of a submerged body through a liquid suggested
the term. We are aware how that idea has been borrowed and
systematized in connection with the dynamics of electrons; and
it is, therefore, of interest to verify that the difference between
longitudinal mass and transverse mass originally introduced there,
though now perhaps in course of abandonment, is quantitatively
identifiable with the term (vdm/dv) according to the assumed
relations for electrons of dependence of mass upon speed.
The effect when we are conscious of the whole situation must
be to make evident how much turns upon attributing the entire
force (R) to the mass (m), because a force diminished by the
amount of the last term in equation (22) would reestablish con-
formity with the type of equation (16) as
Qt
And this is not mere mathematical ingenuity, for in the hydro-
(R - AR) = mv; AR = v. (29)
Qt
The Fundamental Equations 41
dynamical conditions at least we know that the excess of effective
mass over the weighed mass is only a disguised neglect of back-
ward force upon the advancing body due to displacement of the
liquid. So that while groping among phenomena that are less
understood, our attention should keep equal hold upon both
alternatives of statement until experimental analysis decides
finally between them. It is in some degree a question of words
whether all of the force (R) falls within a specified boundary.
34. The formal changes that have been pointed out, and their
possible reconcilement with a larger group of facts through a
second physical view, are important enough to justify this
immediate effort to fix attention upon them. The path is beset
with similar ambiguities whenever the details attendant upon
transformations of the subtler forms of energy are sought.
Therefore it is vital to pursue the thought of the section referred
to, and to perceive with conviction even in this simplest example
offered, how the bare assertion that a time rate for mass will be
introduced for better embodiment of the data leaves the dy-
namics still impracticably vague for decision. We could not
pass upon the physical validity and sufficiency of the force (R)
assigned by equation (22) without fuller insight into suppositions.
The instinctive control of the mathematics by repeated references
to the physics is so well worth strengthening that we shall dwell
upon one other side of the instance before us, though for sug-
gestion only and not with any elaborate intention of exhausting
it.
35. If a stream of water flows steadily in straight stream lines
and with equal velocity everywhere, there is no loophole for
acceleration, neither of an individual particle nor in passage
systematically from one to another. Yet under an arbitrary
agreement to include more and more water in the stipulated
boundary the total momentum would gain an assigned time rate
and the (R) of equation (22) a value
42 Fundamental Equations of Dynamics
dm
R = ^v. (30)
This is plainly illusory and void of dynamical meaning. We
must cut off change of mass by mere lapse of time; this is one
wording of the conclusion. But on the other hand conceive the
mass (m) to grow continuously by picking up from rest differ-
ential accretions, somewhat as a raindrop may increase by
condensation upon its surface, and equation (30) traces a phys-
ical process.
Investigation of this as a physical action confirms equation (30)
quantitatively for a proper surface distribution of the elementary
impacts, as force called for if the slowing of speed is to be com-
pensated that would be consequent upon redistribution of the
same total momentum through a continuously increasing mass.
Thus much of force being allotted to keeping the velocity of the
growing system constant, only the margin above this part would
be registered in the acceleration. Moreover the way is then
opened to interpret the last term in equation (25) by adapting
specially the usual expression for kinetic energy converted at
impact into other forms. Quoting, in a notation that will be
understood at sight, we write that loss in the form
L = 1 - 62
Applying this to the conditions of inelastic central impact
(e = 0); with the ratio (mi/m 2 ) negligible, as (dm/m) is; and
when the relative speed (vi v 2 ) is (v) ; we find
L = ^dmv 2 . (32)
And this wastage of kinetic energy finds due representation
through the integral in question.
The essential condition, however, about (L) is a conversion of
kinetic energy; and as remarked already that conversion might
The Fundamental Equations 43
just as well be reversible. It is, therefore, again suggestive and
perhaps even significant, that the sharing of energy between two
forms indicated in the second member of equation (25) can be
seen to correspond quantitatively with the partition of energy
between the electric and the magnetic field of an electron as
authoritatively calculated according to the assumed rate of
change in its mass with speed. Of course this verifies or proves
nothing, except the possibility in this direction as in others of
constructing a mechanical process that is quantitatively adjusted
to other and different processes where energy is converted. 1
36. The six chosen quantities have been made definite by
means of defining equations, which are truly designated as funda-
mental to the degree that the quantities involved possess that
quality. With these identities we have been content to occupy
ourselves mainly thus far, and confine discussion to phenomena
observed or observable in a system of bodies, and to be described
in terms of the masses, their radius-vectors, and two derivatives
of the latter. With data of this type a range of inferences can
be drawn, quantitatively determinate, too, up to a certain point,
regarding the physical influences under which the system will fur-
nish those data. Any assumed local distribution of mass, velocity
and acceleration demands calculable aggregates of force, momen-
tum and the rest, which the equations can be taken to specify.
But nowhere along this line of thought is the further question
mentioned, about how the influences shall be provided and
brought to bear in producing what we see and measure, or what
is visible and measurable in the system that is under observation.
Not that the relations prove finally to be so one-sided as the
sequence of our mathematics would suggest, according to which
it happens that first mention is given to (Q, E, H), and they are
made primary in the sense that the group (R, P, M) then follow
by differentiation.
1 See Note 11.
44 Fundamental Equations of Dynamics
Yet the latter group would precede more naturally if the
object were to reach the first group by integration; and this
inverse order is revealed to be also a normal alternative. That
procedure erects into data the physical influences like Force,
Power and Force-moment to which the system is externally or
internally subjected, and makes attack in the direction of pre-
dicting the response of the system in detail. The unconstrained
tendency of this line of approach is then to set forth the supple-
mentary idea that the accumulations of Momentum, Kinetic
Energy and Moment of Momentum in the system of bodies are
to be read as integrated consequences of the influences first
specified. 1
The formal change is inconsiderable, though the spirit of it
guides three of our announced identities into full-fledged equa-
tions either of whose members is calculable in terms of the other.
By usual title, these are the Equations of Motion, Work and Im-
pulse that are an important part of dynamical equipment and
that will next engage our attention. Since deciphering and list-
ing the operative physical conditions comes now to- the front, the
weighing of arguments converges upon making the list of forces
that is sought exhaustive, and upon weeding out illusory items
from it. It must be apparent how that search and critical
revision are bound up with inquiries like the suggestions of the
previous section.
,37. Dynamical analysis of results in its field has everywhere
made tenable and corroborated the thesis that momentum and
kinetic energy are traceable as fluxes. This is understood to
imply that each local increase of those quantities will be found
balanced against some other local decrease, either manifest in
the quantity as such, or finally detectable under certain disguises
of transformation. In application to a system of bodies, this
means identifying a process of exchange dependent upon what is
i See Note 12.
The Fundamental Equations 45
in some sort external to it, and sometimes located to occur over
the whole boundary or over limited areas of it, or sometimes
recognized to permeate the whole volume or limited regions of it.
Under the conditions that go with change in total mass by the
passage of material through the stipulated boundary, the mass
thus gained or lost may just carry its momentum and kinetic
energy out or in, without any complicating interactions.
If, however, we exclude and put aside sach processes of pure
convection by confining ourselves to complete mass-constancy,
there is evidence that changes in the total kinetic energy and the
total momentum of a system of bodies are accompanied uni-
versally by exhibitions of force at the seat of the transfer. And
this remains equally true whether a transformation between
other recognizable forms and the mechanical quantities denoted
by (Q) and (E) is taking place there or not. The possible ex-
changes between kinetic energy and other types, and the change-
ratios corresponding to them are a commonplace of modern
physics; as also we know how refined measurement has attested
the forces upon bodies at transformations like that into light-
energy. The settled anticipations in those respects have become
even strong enough to look confidently upon occasional failure
as only postponed success. The more recent proposal is to in-
clude momentum as well as kinetic energy within the scope of
these ideas and concede for both alike a conversion into less
directly sensible modifications, with force exerted upon bodies
of the system or by them as a symptom of the transformation.
And there seems to be no cogent reason why this should not hold
its ground.
38. The quantitative formulation of these two transfers by
flux in relation to what we shall call the transfer-forces tem-
porarily and for the purpose of present emphasis because they
are symptomatic of such action, presents to us the familiar
equations of impulse and work which shall be first written, with
the usual mass-constancy supposed, in the forms
46 Fundamental Equations of Dynamics
Q - Qo = 2 I dR'dt (The Equation of Impulse); (33)
t/O
E - Eo = 2 I dR'-ds' (The Equation of Work). (34)
Jo
They are intended to express total change from (Q ) to (Q)
during any time-interval (0, t), and total change from (E ) to
(E) during any simultaneous displacements (0, s') at the points
of application of the transfer-forces (dR'). The integrations
then cover the summation of effect over time or distance for each
differential force (dR'); and the symbol (S), though open to
mathematical criticism as a crude notation, is doubtless suf-
ficiently indicative of a purpose to include the aggregate of all
such forces at every area and volume where the transfers may
be proceeding. We must make also the necessary discrimination
between the forces denoted by (dR') and those symbolized by
(dR) in equation (16), that are localized by association with
elements of mass and not by participation in some transfer
process, and that express themselves through the local accelera-
tions manifested within the material of the system of bodies,
while the forces (dR') can be determined wholly or to an im-
portant degree by data extraneous to the system.
It should be remarked next how one summation prescribed
by the second member of equation (33) can be executed without
further knowledge or specification, since the one time-interval
applies in common to all force elements (dR') that are making-
simultaneous contributions toward the total change (Q Q ).
Hence if the vector sum of these forces in whatever distribution
they occur be written (R'), the explained sense of this addition
standing entirely in parallel with the comment attached to (R)
in equation (16), we see that
Q - Qo = f R'dt.
Jo
(35)
The Fundamental Equations 47
A corresponding general reduction of equation (34) would first
require equal vector displacements (ds) at all points of appli-
cation throughout the group of (dR'), a condition that need not
be satisfied.
A second essential difference between the equations of impulse
and work is that the former includes indifferently every force
(dR'), in that some duration of its action is a universal charac-
teristic. But in order that a force (dR') may be effective in
work, not only must there be displacement at the point where
it acts upon the system, but that displacement must not be
perpendicular to the line of the force. Either of these conditions
may be at variance with the facts. It is a convenient usage to
distinguish transfer-forces as constraints when they do no work;
which signifies also when their work is negligible, of course.
39. Both (R') and (R) are vector sums and have been exposed
in their formation similarly to cancellation, but there is no pre-
supposed relation of correspondence in detail between the two
groups that would coordinate the occurrence and the extent of
such spontaneous or automatic disappearances from the two
final totals. If however we begin by confining comparison to
those totals as such, that is yielded through the correlation of
two statements which are now before us. Form the time-deriva-
tive of equation (35), replacing (Q) by its defined general equiva-
lent from equation (I) and repeating its conditional derivative
from equation (16). The consequence to be read is
L'dt = R'; (36)
and the relation between the extreme members of the equality
is contingent only upon the validity of equation (35). This
would carry the equality unconditioned otherwise of (R') and
(R) if (Q = R) can be introduced as a defining general equation.
It gives latitude enough for the present line of thought to accept
(R) as first quoted.
48 Fundamental Equations of Dynamics
On its surface the last equation offers the meaning that the
forces applied to the system under the rubric (R') are competent
to furnish exactly the total of force exhibited through the con-
stituents of (R). And the same leading idea dictates the other
verbal formula: The forces (dR) are an emergence of the group
(dR') after a transmission and a local redistribution. But neither
reading is a truism, as the world has realized since d'Alembert
first made the truth evident; for equation (36) does no more
than convey one fruitful aspect of d'Alembert's principle which
declares equality for the impressed forces (R') and the effective
forces (R), which names sanctioned by general usage we shall
now adopt, and standardize the relation as the equation of motion
under the form
Z(dR') = 2 / m vdm. [The Equation of Motion.] (37)
In the first member the sign (2) recurs to the intention explained
for equations (33, 34); and the particular basis of the second
member has been made part of the record.
It is already clear that we have now come to deal with an
equation by whose aid can be calculated either what total of
impressed force is adequate to produce designated accelerations
in given masses or what distributions of accelerations through-
out a mass are compatible with a known group of impressed
forces as their consequence. But the predicated equality is
restricted to the totals and contains that element of indeter-
minateness which affects every resultant, in so far as it is an
unchanging representative of many interchangeable sets of com-
ponents. And in any properly guarded terms that are equiva-
lent to the statement made above, the acknowledged deduction
from the equality is in its chief aspect a conclusion about the
acceleration at the center of mass of the system when (R') is
known, or a foreknowledge of what (R') must be somehow built
up if that center of mass is to be accelerated according to a
known rule.
The Fundamental Equations 49
40. If there were complete physical independence among the
masses of a system, or. in the current phrase, if there were no
connections and constraints active between them to hamper
mutually the freedom of their individual motions, impressed
forces would make their effects felt only locally where they were
brought to bear. And then for each such subdivision of the
total mass as was thus affected equation (37) would apply, and
an impulse equation would follow. Observe however that the
question of minuteness in the subdivision enters, and that
practically halt will be made with some undivided unit, assigning
to it a common value of acceleration; so that the center of mass
idea reappears in this shape ultimately, and duly proportioned
to the scale of force-distribution symbolized by (dR')
In actual fact there are found to be connections among the
parts of a system of bodies, whose local influence deflects the
acceleration from being purely the response to the local quota
of (dR'). In other words, the masses of the system can exercise
upon each other a group of forces internally, which must be re-
garded as superposed upon the impressed forces before the account
of locally active force is to be held complete. To be sure this
reduces to the now almost instinctive perception that external
and internal are relative in use, and that an action may be
impressed from outside upon a part which is exercised internally
in respect to a larger whole. But like many other simple thoughts
it was once announced for the first time.
Now certain forces being impressed, and with whatever
internal connections interposed that the system is capable of
exercising, the net outcome is an observable group of effective
forces. It is therefore common sense to conclude that this
net effect could be entirely nullified, in respect to the accelera-
tions produced locally, by a second group of impressed forces
applied also locally, and everywhere equal and opposite to the
local value given by (dR). In virtue of equation (37) moreover
50 Fundamental Equations of Dynamics
it becomes apparent that the supposititious second group of im-
pressed forces would always amount in their aggregate to ( R') .
Hence two auxiliary conclusions can be stated: First and nega-
tively, that the superposed internal connections do not on the
whole modify the original net sum (R'); and the second is
positive, to the effect that the office of internal connections in
these relations is to transmit and make effective where they
would otherwise not be felt in the system, the distribution of
impressed forces (dR').
The internal connections can be described legitimately as them-
selves in equilibrium; they are the lost forces of d'Alembert.
And the really applied group (dR') would be in equilibrium also
with our second group of locally impressed forces. But this
compensation is a supposition contrary to fact; the resultant
(R') is unbalanced force to use the ordinary phrase. These
details of interpretation are requisite exposition of the formally
insignificant change that writes instead of equation (37)
S(dR') - 2 / m tdm = 0;
2(dR'-Ss') - S/ m (vdm)-6s = 0;
as a formulation of d'Alembert's principle. The second form in-
volves the so-called virtual velocities (6s', 6s), which term is fairly
misleading; for these symbols designate any displacements con-
sistent with preserving the internal connections intact, and capa-
ble of occurring simultaneously; one group at the driving points
of (dR') and the other locally at each (dm). Obviously either
form aims to express that fictitious equilibrium which is derivable
from the real conditions. Because the second form is cast into
terms of work, it seems to call for the remark that the founda-
tion upon which all of this is reared lies nevertheless in the im-
pulse equation, and the development might be called an expan-
sion of consequences under Newton's third law; there is no vital
bearing upon the actual energy relations definitively established
The Fundamental Equations 51
by it. What remains to be said in the latter respect we shall
next consider.
41. The first and familiar fact is that the kinetic energy of
a system of bodies can be affected by interactions that are usually
styled internal: quotable instances being gravitational attraction
between sun and earth, and the effects of resilience upon distorted
elastic bodies. Therefore some deliberate caution must be
observed in delimiting the terms external and internal in rela-
tion to impressed forces, if equation (34) is to cover the total
change in kinetic energy and yet make no dislocation from the
impulse equation. It will be noticed that the critical instances
are connected with transformations of energy; and of energy
that one mode of speech would describe as internal to the system. 1
We can put force exercised upon a body by action of the ether-
medium into the other category, since that medium is by explicit
supposition external to our conception of body.
The case of gravitation is resolved by the consideration that
the conversion of its potential energy into the kinetic form is
attended with exercise of equal and opposite forces upon two
bodies, according to inference from observation. If both bodies
are included in the system, these forces cancel each other and do
not disturb previous conclusions; and if one body is outside the
system's boundary, its action appears among the (dR') A
parallel statement can be drawn up for elastic deformations;
but there is a remnant of combinations that are more obscure,
like the transformations of molecular and atomic energies that
can also affect kinetic energy, and that are by common usage
attributed to the system as an internal endowment. Our
ignorance of their more intimate nature however does not seem a
barrier; we can still look upon every change in a system's kinetic
energy as accompanied by impressed forces (dR/), whether these
are exerted in self-compensated pairs and removed thus from
1 See Note 13.
52 Fundamental Equations of Dynamics
influence upon the impulse equation, or whether there are un-
balanced elements that affect the total momentum in addition
to changing the kinetic energy. To this extent all impressed
forces can be called external, though there may be hesitation
about classing as external or internal the particular type of
energy that is under transformation to or from the kinetic form.
The corollary may be added, that so long as equal and opposite
elements of force are also colinear, their moments for any origin
are self-cancelling; otherwise they constitute couples.
With the attempt to formulate correct equations of motion,
the difficulties of physical dynamics may be said to begin, when
it is required to make the list of impressed forces what we have
spoken of as exhaustive and freed from illusions. Outside the
range of rather direct perceptions, we grapple with uncertainties
under conditions of imperfect knowledge with hypothetical
forces, intangible energies, figurative masses. Dynamics that
was ready to renounce criticism of provisional equations of
motion would be over-sanguine. Conversions of energy into the
one distinctively mechanical form that we call kinetic are perhaps
closest to direct inquiry into attendant circumstances; and
though it would be overcautious to construct on that base only,
it seems probable that dissecting there first is the clew to larger
success, and that equations (33, 34, 36) are landmarks on that
road.
In practice, the bare statement of d'Alembert's principle as
given by any one of the three forms indicated is supplemented
with some record of the particular connections that overcomes
the difficulty of specifying every individual local acceleration,
and reduces the number of indispensable data within manageable
limits. The forces of the connections are thus described in-
directly through the geometrical equations of condition; and this
method is more effective than the more direct one, because the
magnitude of the constraining forces will in general depend upon
The Fundamental Equations 53
the speeds, though the kinematical analysis of the linkages
remains unaltered. It is this thought that introduced Lagrange's
use of indeterminate multipliers. 1
None of these devices though qualifies the character of d'Alem-
bert's equality in asserting a quantitative equivalence between
a net total of external agency (impressed forces) and the response
to it on the part of a system of bodies, as expressed in the states
of motion that the effective forces summarize. The physical
thought attaching to the equation of motion will be clearer
when cause and effect are kept apart, and will tend toward
obscurity or confusion when a shuffling of terms from one member
to the other, as a mathematical device or for other reasons, has
impaired this desirable homogeneousness.
42. One large section of dynamics is devoted to working out
its principles in their application to rigid solids. As these are
specified, they carry to an extreme limit a scheme of inter-
connections among their constituent parts that provides an
ideal. of internal structure which knows no rupture nor even
distortion, but which provides inexorably all necessary con-
straining connections. Like other such concepts its considera-
tion yields results which are not only valuable in themselves,
but which also furnish a point of departure for the introduction
of conditions that approach their standards closely enough to
be taken account of by means of small corrective terms. Beside
repeating that frequent and useful relation of a concept to actual
data, the study of rigid dynamics has some more special reasons
to support it, of which one is discoverable in the trend of the-
oretical views about the constitution of all systems of bodies.
The boldest analysis of molar and molecular and atomic units,
as a substratum for the increasing number of energy-forms that
we associate with them and give passage through them, has not
broken away entirely from utilizing rigid solids of smaller scale
'- Sec Note 14.
54 Fundamental Equations of Dynamics
and their dynamics. This gives the prevailing tone in attacking
the atomic nucleus and its atmosphere of electrons even, with
only such mental reactions to modify the trust in the details of
the reasoning as have a wholesome influence to maintain the flexi-
bility that is scientific and make our dynamics more nearly
universal in what it embraces. 1 In this sense the kinematical
phase, through which so many of these matters evolve, remains
uncompleted or we may dub it empirical until dynamics can
serve it with reasoned argument.
In the second place, however, any rule of constancy is likely
to have an advantage of particular kind over the multifarious
rules of variation in correlation with which it is unique. This
goes beyond the formal gain in abolishing some mathematical
complications, though that, too, frees our minds to entertain the
salient ideas with fuller concentration. Like our previous
assumption of constancy in mass, this added supposition of
permanent internal arrangement puts off particularizing among
rules of change, and enables us to carry forward through instruc-
tive developments the task of bringing some general principles
more nakedly to discussion. This grows cumbrous or impossible
where conclusions are subject to many contingent decisions.
43. It bears rather closely upon these suggestions that we can
make one good entry upon the particular inquiries about rigid
solids by resuming and continuing the line of thought that
paused at equation (20). In that section some glimpses were
secured of a superposition by means of which a serviceable sketch
can be drawn of a dynamical outline for certain systems of bodies.
Or otherwise stated, the actual totals of the important quantities
are grouped round the concept of a representative particle, leav-
ing only specified remainders for further consideration. Let us
now separate from such a system one body that we shall suppose
rigid and having continuous mass-distribution, and deduce for
1 See Note 15.
The Fundamental Equations 55
it, with increased finality of detail, the special consequences that
seem valuable for our purpose. It is clear that the center of
mass of this body will retain all the functions already assigned
to the representative particle, and also that it must now in
addition, because the body is rigid, fall into an unchanging
configuration that makes constant in length all such vectors as
(r') of equation (12). And it follows too from the conception of
rigidity that the internal connections are excluded from net
effect upon the sequences of conversion that change the body's
kinetic energy. They are reduced in their final influence to the
office of transmitting and distributing the consequences of con-
versions and constraints that have been effected otherwise than
by any machinery of readjustments, named or unnamed, of in-
ternal arrangement. The intended meaning is not essentially
varied, though it has been rendered less explicit perhaps, when
it is said that the impressed forces can here only displace the body
as a whole, or that the internal connections can do no work.
44. Now it is the elementary characteristic of translation that
it does apply to the body as a whole and affect it uniformly
throughout in all kinematical respects. Our next natural step,
therefore, is to examine the remaining possibility that is con-
sistent with the constant length of every (r'), and that therefore
restricts the locus of each mass-element to some sphere 'that is
centered on the center of mass. If we accept for this type of
motion as a whole the term rotation, there still remain some
particulars to establish definitely; and of these the first will be
the general value of the velocity denoted by (u) in equation (9),
for which one fitting name is the local velocity relative to the center
of mass. It is evidently identical with the local velocity (v) of
each (dm) if (v) is zero, or if the center of mass is the origin of
reference. With control of the value for (u) we can ultimately
take up the evaluation of the terms that contain (u) or depend
upon it, knowing in advance that these can appear in (E, H,
P, M) but not in (Q, R).
5
56 Fundamental Equations of Dynamics
45. In order to approach the matter conveniently let (C')
denote the center of mass, and locate orthogonal axes there that
are lines of the body: that is, they move with the body and retain
their positions in it. The unit-vectors of those axes shall be
(i', j', k') in the standard right-handed cycle. Then using the
word temporarily in an untechnical sense, any rotation relative
to (C') will in general change all the angles that (i', j', k') make
with the reference-axes. Consider first differential changes of
orientation (a, (5, 5) matching the order of the unit- vectors.
Then (a) as an angle- vector is normal to the plane of the con-
secutive positions of (i'); similarly for ((3) and (j'), and for (5)
and (k'). The corresponding linear displacements on unit
sphere around (C') are given as products of perpendicular factors
by
di / =oxi / ; dj' = &xj'; dk' = 5 x k'. (39)
The vector products are not affected, and hence these equalities
are not disturbed, if we introduce three arbitrary elements of
angular displacement; (V) in the line of (i') into the first, (y')
in the line of (j') into the second, and (v') in the line of (k') into
the third, writing
di'= (a+*')xi'; dj' = (S+')xj';
dk' = (5 + v') x k'.
But because the axis-set must remain orthogonal in the rigid
body, the elements of angular displacement in the line of the
third axis must always be equal for the two other axes at the
same stage. This renders possible the adjustments of particular
values that make equations (40) simultaneous:
v = o(k') = (S(k');
with the consequence that equations (40) are satisfied in the
forms
The Fundamental Equations 57
di' = d r xi'; dj' = d Y xj'; dk' = d Y xk';
(42)
d Y = 3. + y + v.
The occurrence of the vector (d Y ) as a common factor in all three
equations, combined with its determination by projections on
axes arbitrarily chosen and with the fact that simultaneous
linear displacements at points in the same radius-vector must
be proportional to distances from (C'), shows that at each epoch
and for every (r') of constant length,
dr' = df x r'; r' = u = to x r'; to = Y . (43)
Here (to) denotes the rotation-vector for either body or axis-set,
of course, since they are supposed to turn together. It follows
without further question that if a rigid solid moves so that all
its radius-vectors (r) measured from any reference-origin remain
of constant length, the simultaneous velocities (v) of all mass-
elements conform to the relation
v = a) x r. (44)
Any such motion as a whole is described as a pure rotation with
angular velocity (CD), for which vector the origin is conventionally
the base-point.
46. The vector (to) is usually termed the angular velocity of
the body at the epoch, the phrase being made reasonable by the
appearance of (to) as a factor common to all radius-vectors in
equations like (43) or (44). But both the procedure by which
this angular velocity was determined and its appearance in a
vector product show plainly that its resultant value is not
effective to produce changes of direction in all radius-vectors. 1
This common factor has been seen to include three elements
that become superfluous each for one axis, as not influencing
1 See Note 16.
58 Fundamental Equations of Dynamics
angular displacement of it, nor the corresponding linear displace-
ment of points in it. The rotation-vector is thus open to inter-
pretation as a maximum value, useful in giving through its pro-
jection upon the normal to any plane at its base-point the part
effective to bring about a complete angular displacement oc-
curring in that plane. If we identify (<>) with the line of a rota-
tion-axis, permanent or instantaneous, these explanations are
consistent with the elementary ideas of spin about the rotation-
axis and linear velocity given by the product of rate of spin and
distance from the axis.
47. The preceding identification of a rotation-vector connects
its considerations with departures from configurations of (i'j'k')
that are themselves subject to self-produced change, in so far
as they move with the body; and this might conceivably modify
the result. But if that loop-hole seems to exist it is closed when
we detect the same vector (dy) in direct terms of its projections
upon the reference-axes oriented by (ijk) permanently. And
it is, further, worth while to do that, because these projections
are uniquely advantageous in preparing for algebraic additions
to .express any resultant angular displacement according to the
relation
y = /dy = i/d7 (i) + j/d7 (j) + k/d7 (k) , (45)
the tensors that are integrated being those of the projections of
each (dy) upon the axes of (i, j, k). The confirmation sought
depends upon satisfying the relations,
(46)
i/k'-k.
Ordinary routine verifies that equations (46) fulfil identically
the necessary conditions:
The Fundamental Equations 59
di' = df x i' = i(i'( k )d7(j) - i' (j) d7 (k) )
dj' = dy x j' = etc.
dk' = df x k' = etc.
It is not without interest to notice in detail how algebraic cancel-
lations now preserve the obligatory independence of (3t) in the
results for (di') ; of (y) in those for (dj') ; and of (v) in those for
(dk'). This second development is more circuitous, because
the permanently orthogonal condition, due to rigidity, pertains
intimately to (i', j', k'), the coincidence of results by both attacks
being a special instance under a general theorem that will be
proved subsequently (see section 85). The equal corroboration of
equation (44) is a plain inference, and hence, wherever a rotation-
vector covers the local velocities of a rigid body, or the body is
in pure rotation about a fixed point, the summed projections are
invariant :
to(i) + <0(j) + fa>(k) = to(i') + 0)(j)' + G)(k') = C>. (48)
Substitute in equation (44), use the standard relation for common
origin,
r = x + y + z = x' + y' + z', (49)
and omit products of colinear factors. This yields
v = to ( i) x (y + z) + o>(j) x (z + x) + <a ( k) x (x + y)
(50)
= (i ', x (y' + z') + <o (j ') x (z' + x') + (k ', x (x' + y'), J
and is the foundation for a standard rule: Linear velocities in a
rotating rigid body are given correctly by superposing those due
to separate partial' rotations, either about the reference-axes or
about the positions at the epoch of any three lines of the body
intersecting orthogonally at the origin.
60 Fundamental Equations of Dynamics
48. In the present connection however we are dealing with
a rotation relative to (C') as superposed upon the concept
of a representative particle and supplementing the latter, with a
proved equivalence of translation and rotation thus combined
in replacing the most general group of velocities in our rigid
body. On incorporating these recent restatements into equa-
tions (10) and (12), they take on the more special forms that
we can now exhibit. Denote the last terms in the two equations
by (E R ) and (H R ), which we shall call briefly the kinetic energy
and the moment of momentum relative to the center of mass.
Then for the one body of continuous mass
H R = / m (r' x udm) = / m (r' x (w x r')dm)
= / m (o>(r'-r') -r'(to-r'))dm; (51)
ER = | /mU-udm = I / m (w x r'.) (to x r')dm
- I / m ((cO 2 - (o>T') 2 )dm = i(-Ha); (52)
the final reduction of (E R ) being readily verifiable, when we
remember that (<>) is common to all elements in these mass-
summations.
49. Next we continue into equations (17) and (18) the same
plan of partition between representative particle and supple-
mentary term. Direct substitution there according to the rela-
tions previously used,
v = v + u; r = f + r'; (53)
gives
P=^ = v.R + / ra u-dR; (54)
M = H = (f x R) + / m (r' x dR). . (55)
We may remind ourselves that the first terms in the final
members of both these equations are in harmony with the time-
derivatives of corresponding terms in equations (10) and (12)
The Fundamental Equations 61
if we bear in mind equation (20) ; and they show how the particle
can be relied upon still to present these contributions to power
and to force-moment as based upon its artificial translation with
the center of mass. Denote the additional power and force-
moment by (P R ) and (M R ); then from equations (54, 55),
PR = /m(w xr')-dR = o>./ m (r' x dR) = w-M R ; (56)
(57)
We shall compare these statements with the consequences of
equations (51, 52), which give for their derivatives
^(E R ) =i(6.H B + .H H ); (58)
H R = -^ / m (r' x udm) = / m (r' x lidm); (59)
because (u) and (?) are identical. Further, since differentiation
of equation (9) shows
v = v + u, (60)
a natural name for the last term is the local acceleration relative
to the center of mass, which would indicate also a local force-
element (udm) differing from (vdm) that is (dR) and thereby
breaking the equality of (H R ) and (M R ). But since
/ m r'dm = 0, (/ m r'dm) x v = / m (r' x vdm) = 0; (61)
and this term can be added without error to equation (59),
giving
H R = / m (r' x (v + u))dm = / m (r' x dR) = M R . (62)
Evidently the value in equation (61) could reversely be sub-
tracted without error from equation (57). The interchange-
ableness of these forms should not be lost sight of.
50. A similar concordance of equations (56, 58), though it is
62 Fundamental Equations of Dynamics
not superficially evident, follows at once on showing a right to
add the third member in the equality
w
-M R = o>-H R = <b-H R , (63)
whose first and second members are now known to be equal.
The required proof is got by differentiating equation (51), where
we find
H R = / m {d>(r'.r') - U(G>T') - r'(-r')}dm, (64)
whose scalar product with (G>) is, omitting everywhere scalar
products of perpendicular factors,
o>-H R = / m {(fa>-fa>)(r'-r') - (fa>-r')(oVr'))}dm
= / m (b-(o>(r'T') - r'(o>-r'))dm = <b-H R .
The vector (d>) which is the time-derivative of the rotation-
vector (w) is named the vector of angular acceleration. Of course
it provides for both changes of direction (or of axis) in the rota-
tion, and for changes in its magnitude (or spin); and (<b) must
be of common application at any epoch to all mass-elements,
because that is true for (to).
51. With the support of equations (51, 52, 56, 58), we have
given .consideration to all four quantities that need specifying,
for the rotation that is the remainder over and above the fic-
titiously segregated translation, since the representative particle
as it has been determined engages the totals of force and momen-
tum. And having brought the discussion to this point, in terms
connected with the effective forces whose resultant is (R), it
remains to make that transition to impressed forces with equal
resultant (R'), which we have learned to associate with d'Alem-
bert's name. Under the conditions explained for rigid bodies,
certain sources of impressed force are not to be permitted, but
the total work done must appear in the energies of translation
and rotation. Let us then next summarize how matters stand
The Fundamental Equations 63
with the six dynamical quantities, in the two groups that we
have recognized.
I. Translation:
1. Force (R' = R) at (f).
2. Momentum (Q = mv) at (f).
3. Energy (E T = |mv 2 ).
4. Moment of Momentum (H T = f x mv) ; consistent
with (2).
5. Power (P T = R'-v = (d/dt)(E T )); consistent with (1)
and (3).
6. Force-moment (M T = f x R' = H T ) ; consistent with
(1) and (4).
II. Rotation:
1. Force = 0; consistent with couples expressing self-
compensating elements in (R').
2. Momentum = always; consistent with impulse of zero
force.
3. Energy (E R = u-H R ).
4. Moment of Momentum (H R ); consistent with zero
momentum.
5. Power (P R = o>-M R = (d/dt)(E R )); consistent with (1)
and (3).
6. Force-moment (M R = H R ); consistent with (1), (3)
and (5).
52. The review of these details impresses the fact that the
above conventional separation accomplishes complete inde-
pendence for two such constituents of the actual data, in the sense
that the course of events can be duly expressed for each group,
with indifference to the presence or absence of the other, by a self-
contained use of the general dynamical scheme. This secures
the full simplicity attendant on pure superposition, by shrewdly
exploiting center of mass for its average properties, and kinetic
energy with moment of momentum for their salvage of what the
64 Fundamental Equations of Dynamics
mean values sacrifice, utilizing also a form of Poinsot's allowance
through a couple for off-center action of a force. The idea is
successful, besides, in concentrating into the rotation elements
where the form and the mass-distribution of the body complicate
the data with differences; and this frees the translation for giving
expression to broad traits of similarity.
The rudiments of the steps now taken are perceivable in equa-
tions (10) and (12), where it is plain that an internal energy like
(E R ) could belong to radial pulsations of mass-elements about
(C'), either alone or added to spin as a whole; but development
is checked until (u) is particularized in its value and distribution.
It is plain, however, that adaptation to many combinations is
feasible, whose general feature is non-appearance in translational
energy of full equivalent for the total work done. Failing
definite knowledge that forbids, a rotation can be devised as one
possible means of absorbing a quota of kinetic energy, and as
one guide to conjecture among the facts of an observed diversion
of energy from a translation. It is scarcely necessary to insist
that the equivalence of any such devices is restricted to those
particulars according to which their lines were laid down; the
particle plus a rotation is an equivalent for the general motion
of a rigid body only in the six respects enumerated. 1
53. At equation (44) the idea was introduced that pure rota-
tion of a rigid body about a reference-origin, instead of the center
of mass, is describable in corresponding terms on substituting
(r) for (r') and (v) for (u). The intrinsic difference lies in the
necessity that a reference-origin is a fixed point, whereas the
possible velocity of the center of mass runs like a thread through
all our recent discussion. Let us realize that the main results
now added can be similarly extended, and put down as applicable
to pure rotation about the reference-origin these parallels specif-
ically to equations (51, 52, 56, 62, 65):
1 See Note 17.
The Fundamental Equations 65
H = / m (<o(r-r) - r( U T))dm;l (66)
E = |(a>-H); Total quantities (67)
P = to M ; - for pure rota- (68)
H = M; tion. (69)
<b-H = <o H = (o-M. (70)
Since in this case supposed, the center of mass need not coincide
with the origin, the alternative choices will be open to treat the
body as exhibiting rotation alone, or as affected with translation
and with a rotation besides. But translation cannot bring in
change of direction for lines of the body, hence both views of
the rotation must agree in their rotation-vectors permanently.
And because the center of mass cannot change its position relative
to its rigid body, a relation distinctive of pure rotation must be
v = to x f. (71)
The comparative directness and convenience of the two methods
will be decided according to circumstances. One method ex-
cludes from (M) any forces really acting through the origin;
the other can omit from (M R ) any forces acting through (C')
54. We proceed with the requisite analysis of rotation, by
examining the specialized values of local accelerations and some
consequences of them, conscious always in the light of what has
just been said, that the conclusions will be available for twofold
use. One is more important, doubtless, because more inclusive
in application to the most general type of motion of which a rigid
body is capable; but the second has weight, too, in attacking the
conditions of pure rotation that are made prominent, for in-
stance, in common forms of the gyroscope.
The local acceleration of a pure rotation given by differentiating
equation (44) is
v = (d> x r) + (to x v). (72)
Let us make this form our text and starting-point, remembering
that in the other circumstances it is to be recast into
66 Fundamental Equations of Dynamics
u = (d> x r') + (<o x u), (73)
with continuations where (r') replaces (r) everywhere and (u)
replaces (v), while (u) is read the local acceleration of the rotation
and is the excess of (v) over (v). The vector () gives the
velocity of the extremity of (<>), of course; and its base-point
will be taken conventionally at the origin with which our idea of
rotation is associated. Then the process modifying (<o) by (u)
is one of continuous parallelogram composition for intersecting
vectors, though equivalent indeed to addition in a triangle.
The vector sum in equation (72) deserves close attention,
because though the two types of its terms are on one count an
incident of the algebra, it happens that they conform remark-
ably, first to the kinematical elements, and later to a certain
plane of cleavage in the dynamics. The form of the second term
connects it conclusively with change of direction only for its
velocity; and the first term enters and vanishes with angular
acceleration. If (a>) retains direction (<b) must be colinear with
it; and then first inspection can identify the terms with the
tangential and the normal acceleration respectively of the local
(dm) in its circle perpendicular to (<>). But the complete separa-
tion of changes in magnitude and in direction for (v) that then
exists should not be assumed more generally; it is always true,
however, that the first term in the acceleration bears the same
relation to the axis of angular acceleration (w) that the corre-
sponding velocity (v) does to the axis of rotation (to) .
55. Multiplying equation (72) by (dm) yields the effective
force-element, which, because it is exhibited locally, must have
a moment to be found by taking that force in vector product
with its (r). The total moment then demanded by the localized
forces must, as we have seen, be furnished by the impressed
forces; and this amount is expressed by the integral
M - / m [r x (( x r) + ( w x v))dm]. (74)
The Fundamental Equations 67
Denote the two main constituents of this moment by (M') and
(M"); and let us take up the second part for examination.
Expand the triple vector product, omit the scalar product of
perpendicular factors, and finally write for (v) its known value.
This shows
M" = / m v(<o-r)dm = / m ( x r)(o>-r)dm. (75)
Next form for comparison the product
<o x H = / m (c,> x [to(r-r) r(<o-r)]dm)
= - /m( x r)(o>-r)dm, (76)
and we see that the extreme members are identical. Hence we
conclude that the office of thus much of the force-moment is to
produce a change of direction in the vector of total moment of
momentum so regulated that the latter would move with the
body or retain its position in the body. This is a simple corollary
of the interpretation of (CD) according to section (47). If (to)
and (H) were in every case colinear, their vector product at the
value zero would become formal and meaningless. But it appears
plainly in equation (66), first that (H) may be thrown out of
line with (co) by the term
- / m r(<o-r)dm,
which does not in fact generally vanish nor become colinear
with (to); and secondly, that (H) and (<>) cannot become per-
pendicular by compensations within the first term, because every
product (r r) is essentially positive. That they never are perpen-
dicular we shall conclude presently (see section 58) ; the general
obliquity of the rotation-vector and the moment of momentum
vector is one characteristic in rotation, and is operative to cause
effects to which there is no parallel where a kinematical vector
and its dynamical associate are colinear, like momentum and its
velocity. If angular acceleration is absent, every element in (M')
68 Fundamental Equations of Dynamics
is zero, but (M") is not affected, since it depends upon the (o>)
of the epoch, and not upon the past or future history of (to) . If a
rigid body is spinning steadily about a fixed axis even, (M") is
called for, as a directive moment, whenever (G>) and (H) diverge.
For the case of rotation about the center of mass, (M R ") will be
furnished by a couple. These moments are recognizable as the
centrifugal couple of the older fashion in speech. Like forces
normal to a path, they disappear from the power equation by a
condition of perpendicularity, as is visible from equation (68),
when we have noticed through equations (75, 76) that (M") is
perpendicular to (o>).
56. What has been determined about (M") presents it in such
relation to the (G>) of the epoch that an impressed total force-
moment of that value is adjusted exactly to continuance of
constancy in the rotation-vector ((>); the zero value of power
and the consequent constancy of (E) being an evident con-
comitant of that as primary condition. It is further acceptable
on commonsense grounds that (H) whose divergence from (w)
is fixed by the mass-distribution when (w) is constant, as the
form of equation (66) proves, must then accompany that mass-
distribution through its changes in azimuth round the rotation-
axis, so as to describe a right circular cone and keep up with any
originally coincident radius-vector of the body. And the shrink-
ing of such a cone into its axis provides for the singular case of
non-divergence, with no (M") required for adjustment.
With the above details in hand, the part (M') of the force-
moment appears in the light of a disturber of adjustment, and
that opens for it an indefinite range of possibilities or puts away
the expectation of particular conclusions, except two: that it
must supply, first, all power and all changes in magnitude of (H) ,
and secondly, any change of direction that displaces (H) rela-
tively to the body.
57. At this point the chance offers for a pertinent remark
The Fundamental Equations 69
about all equations like (74) in their type. They exhibit an
impressed physical agency (here of (M)) in terms that compare
it for excess or defect with an adjustment that is not compensa-
tion as equilibrium is, but calls for positive action (such as (M")
here exerts). It is an ambiguity inseparable from the algebra,
especially where the total available is numerically less than the
critical value, that an adjustment disturbed is indistinguishable
from one not secured. In other words we can be sure only that
(M') and (M") are mathematically represented in (M), when the
latter has been assigned arbitrarily; using again the present
instance, we know nothing of (M') and (M") separately as active
agencies. Neither of the forms
M = M"; M - M" = 0; (77)
indicates equilibrium, but both express a fulfilled adjustment,
much as equation (36) was read. Both of the forms
M = 0; M' + M" = 0; (78)
apply the condition of equilibrium to (H) in the sense of making
it a constant vector. In these circumstances an angular acceler-
ation that underlies (M') will appear in the equations unless (M')
and (M") are zero separately, which can be true only specially;
and there is some trace of mathematical suggestion that this
angular acceleration arises by give-and-take between (M') and
(M") that diverts the latter from its original office of keeping
(to) constant.
Doubtless that instinctive view, if it exists, receives some
support from knowledge of other conditions in which an active
assignable force-moment is indispensable to the appearance of
angular acceleration; and that is the root of the inclination to
see paradox in the phenomena that realize the conditions of
equation (78). But in consequence of the divergence already
spoken of, if the (H) vector preserves its direction in the reference-
70 Fundamental Equations of Dynamics
frame while the body is in rotation, the vector (to), oblique to it,
will not be constant also, and accordingly there will be angular
acceleration. This occurs spontaneously we might say, (M) be-
ing zero, in the absence of control that would be effective to
keep () constant and shift the burden of change upon (H). It
makes the reasons for the apparently abnormal results more
obscure, that the kinematical aspects depending upon (<o) and (o)
are often patently visible, whereas the dynamical elements that
really dominate are hidden from view. 1
58. While we are laying emphasis upon the general separation
of directions for (o>) and (H), it is proper to be aware how this
works out only for the body as a whole through the mass-summa-
tion of (dH) and the introduction of the common rotation-vector,
and does not appear in the local elements, that it is the object of
that plan and its advantage to handle in one group and not
individually. It was observed already in equation (2) which
had not yet been narrowed to rotation, that for each (dm) its
(dH) and its (Y) are coincident vectors, the latter lying in the
normal to the plane (r, ds) and being attributed to the local (r) as
its particular angular velocity. This lesson can now be repeated
from equation (51) or (66), if we denote by (toi, TI, fi) the unit-
vectors of (to) and of (r), and of the perpendicular to (r) in the
plane (&>, r), noticing that for instance equation (66) can be
written, if (a) is the angle (to, r),
dH = (i(cor 2 ) ri(o>r 2 cos ))dm
= Yi(oor 2 sin a)dm = "f(r 2 dm). (79)
It is instructive to see, next, how the body as a whole retains
for its total moment of momentum in relation to its rotation-
vector the same type as equation (79) shows; and this can be
done by assembling the projections of every (dH) upon the
direction of (G>). The result to be recorded for use is
1 See Note 18.
The Fundamental Equations 71
H( u ) = / m o>i(cor 2 - (<oi-r)(o-r))dm = wl ((o) , (80)
expressed as we find, also as the product of an angular velocity
and a moment of inertia about its axis, but both these factors
now refer to the whole body, and this form excludes perpendicu-
larity of (o>) and (H).
Because (H) is a sum into which the differently weighted ele-
ments (Y) enter, and the weighting depends upon what happens
to be the mass-distribution, the final result cannot be forced
completely into any one mould, beyond the point here estab-
lished; only we know that the rest of (H) must be in the plane
perpendicular to (G>). Therefore according to equation (67) we
learn that
E = i(-H) =f Iw, (81)
which may also be inferred directly from equation (52), by a
slightly varied reduction of the last member but one. Let us
use the occasion to renew the reminder that the rotation relative
to (C') involves only a transfer to its notation of the details here
attached to the other case.
59. A similar trend can be marked in the other partners (o>)
and (M') which bring kinematics and dynamics into connection :
an elementary type of expression which appears differentially
then persists in application to the body as a whole, but with a
supplement governed by the particular mass-distribution that
produces obliquity of (M') and (). For the local element
(dM') equation (74) leads by expansion to
dM' = (d>(r-r) - r(<b-r))dm, (82)
which it will be noted reproduces equation (66), except that (w)
has replaced (o>) throughout. Consequently equation (79) can
be paralleled in the form
dM' = (o>i(cor 2 ) - r^cor 2 cos j3))dm
= pi(cor 2 sin 0)dm = (w srin /3)pi(r 2 dm). (83)
72 Fundamental Equations of Dynamics
But (i, TI, pi) are now unit-vectors for (o>), (r) and the per-
pendicular to (r) in the plane (<!>, r), and (/3) denotes the angle
(, r). It is plain that (co sin /3)p x is for each (r) the effective
part of (<o), as ( sin a) fi is the locally effective projection of (co),
and that (r 2 dm) is a moment of inertia for the axis (pi). Thus
the type is set for the corresponding expression in terms devised
to apply to the body; and in fact we find
M' (i) = / m <b(r 2 - (<bi-r) 2 )dm = <Z>I (cS) , (84)
whose form excludes perpendicularity likewise for (<b) and (M')-
60. It can be conceded as one legitimate purpose of equations
(80), (81) and (84) to extract from the more general treatment
of rotation what residue of correspondence remains with those
simpler forms that are met in uniplanar dynamics. Looking in
that direction, the main difference can be localized in the addi-
tion of an independent axis of (o>) to stand alongside the previous
axis of (o>). But the greater enlightenment in the discussion
comes from the insistence upon putting foremost the powerfully
direct analysis, by means of the dynamical vectors (H) and (M)
and their connections. This tends to make the kinematical
vectors, and especially (6), rather subsidiary until restrictions
upon the problem restore to them more nearly equal weight.
61. If we start again from equation (66) and enter upon the
semi-cartesian expansion for the vector (H) the first results found
are
H (x) = (D( x) / m (r-r)dm - / m x(o>-r)dm; 1
H ( y) = fa> ( y)/ m (r-r)dm - / m y(o>-r)dm; > (85)
H( Z ) = o)( Z) / m (r-r)dm - / m z(wr)dm. J
These continue to assume pure rotation round the origin, the
body being in a general orientation relative to the reference-
frame (XYZ). Retaining one value of (w) given in relation to
(XYZ), the last terms in the second members are seen to depend
upon the body's orientation, but the first terms are invariant for
The Fundamental Equations 73
all such orientations. By a definite choice of orientation the last
terms can always be remarkably simplified, and what are known
as the principal axes of inertia for the origin will then coincide
with the axes (XYZ). We presuppose the proof that there are
never fewer than three orthogonal principal axes at every point
that is in rigid configuration with a rigid body, and ordinary
acquaintance with properties of the ellipsoid of inertia or mo-
mental ellipsoid; this material is standard and accessible.
In all three equations expand (o>-r) and reduce to the forms
H (x) = i{o>( X )I (x ) aj (y) / m xydm aj (z) / m zxdm} ; 1
H(y> = j{w(y)I(,) - w (z) / m yzdm - w (x )/ m xydm}; \ (86)
H (z) = k{co (z )I( z) oo (x) / m zxdm &> (y) / m yzdm}. J
The property of principal axes determines the disappearance of
six integrals at the orientation where those lines of the body
coincide with (XYZ). Supposing that coincidence, therefore, it
becomes true that
H = o>( X )I( X ) + ( y )I(y) + (z)I( Z ). [Principal axes.] (87)
But (H) can be represented invariantly by an indefinite number
of groups of orthogonal projections, and for one group, which
can be chosen at every epoch and for every (&>), the coincidences
that simplify equation (87) will occur instantaneously. How
and on what terms the advantage of the simplification can be
made permanently available is a question to be taken up here-
after (see section 118); but some useful decisions follow immedi-
ately here.
62. And first, the possible extent is made evident of the can-
cellation ensuing through the difference between the two con-
tributions to the second member of equation (66) . It is indicated
by the present remainder, in which all the terms are essentially
positive, if we take the vector factors absolutely. Secondly, if
we turn to kinetic energy, the aid given by adopting principal
74 Fundamental Equations of Dynamics
axes, there too, is apparent in reducing the number of terms in
the expression. For whereas the expansion of equation (67) on
the basis of equation (86) will yield nine terms that do not
coalesce into fewer than six, the reduction of these to three is a
consequence of equation (87), from which follows
[Principal axes.] (88)
This again by deleting subtractive terms has regained parallelism
with the case of translation and three orthogonal components of
velocity except for the difference, irreducible in the general
expression, between the uniform mass-factor (m) and the indi-
vidual inertia-coefficients like (I( X >).
63. In the third place, that similarity in type between equa-
tion (66) and equation (82) which has been relied upon before to
abbreviate details can be employed again. Like equations (86)
for (H) we can write for (M')
M' (x) = i{w (x )I( X ) - w(y)/ m xydm - w (z) / m zxdm} ; 1
M'( y ) = j{w ( y)I(y) W( z )/ m yzdm - w (x )/ m xydm}; j- (89)
M' (z) = k{co (z) I( Z ) - co (x )/ m zxdm - co (y) / m yzdm} ; J
in which the same six integrals occur that the choice of principal
axes eliminates. Consequently if we use at the epoch the pro-
jections upon the principal axes, we obtain
M' = <b( X )I (x ) + o>(j)I (y ) + <)!(). [Principal axes.] (90)
This adds one feature to the previous conclusion in equation
(84). and makes evident that (M') cannot vanish while (<b) differs
from zero, as a limitation upon the subtractive element of equation
(82) . And it throws stronger light upon a possible constancy of (E)
while both (M') and (M") are active, for which the condition is
that (M') as well as (M") should be perpendicular to (<>). This
is compatible with the presence of (<o) since the latter may have
any direction relatively to (to). Where the time-derivatives of
The Fundamental Equations 75
equations like (80) play a part in such considerations as the fore-
going, of course it may be necessary to take account of variable
moment of inertia as being important in reconciling the presence
of (w) and the absence of (M).
Should the rotation that is under investigation be about the
center of mass of the body, the force to be brought in for the
accompanying translation or to accelerate the particle of the
combination is calculable as (mv), where any value may have
been assigned by other elements to the second factor. But if the
case is one of pure rotation round any origin or fixed point, it is
plain that the acceleration and velocity of the center of mass are
prescribed at the values
v = (<b x f) -f- (<o x v); v = (o> x f), (91)
requisite locally under the rule of equations (44, 72). Then the
total force brought to bear must be accurately adjusted to produce
this acceleration, and a constraint at the origin may have to be
made active in order to give exactly the requisite force. For
reasons of that nature, the constraint may need to be calculated
or expressed, although it can contribute nothing to the moment
(M) about the origin, and can in that respect be ignored. It
rests upon the general understanding about sections 45 and 51,
that all the leading equations like (86, 88, 89) are adaptable to
center of mass as origin without formal change, and by mere
substitution of the values then effective.
CHAPTER III
REFERENCE-FRAMES: TRANSFER AND INVARIANT SHIFT
64. Let us recall now the fact that the exercise of choice of
reference-frame must be an assumed preliminary to determining
any definite working values for the fundamental quantities, and
consequently for all quantities calculable in terms of them.
This is not interfered with as a truth by our predominant habit
of making the earth's surface locally the tacitly adopted basis of
reference. The circumstances then bring with them quite natu-
rally a recognizable need of deliberately guided inquiry into the
extent to which such values are affected by an allotted range in
selection and specification for our reference-frame. This will
afford the necessary machinery for correct transfer from one
reference-frame to another as standard when that is dictated by
an effort at greater precision or by reasons founded in an ad-
vantage of convenience.
The line of thought to be taken up next will trace out those
matters of material consequence connected with the chief kine-
matical and dynamical expressions which require for their settle-
ment a collation of values resulting when particular frames are
chosen among a group that are in assigned conditions of relative
configuration and motion. The fullest survey belonging to that
discussion embraces much that would be scarcely relevant on
the scale laid down for our present undertaking. But by allowing
the more practical interests in these directions to set the limits,
we shall confine our scope to methods that are in most frequent
use for translating the important expressions into convertible
terms of familiar type and ascertaining their mutual dependence.
In so far as vectors can be made the vehicle of expression, they
76
Reference Frames 77
are likely to deal directly with resultants and totals, and then
we are concerned with the amounts by which these change at a
transfer from one frame to another. Yet because we must at
times prepare more completely for computation, this alone would
constrain us to sacrifice to those ends the compactness of vectorial
statements. Other reasons also compel us to find place for the
partials or components that are characteristic of various coordi-
nate systems whose peculiar advantages make them useful
auxiliaries to the reference-frame; and this will raise a second
group of questions. Some close intrinsic connections will be
found, however, to make interdependent the two branches of the
inquiry, relating one to the uses of coordinate systems and the
other to comparisons among reference-frames, which occupy this
chapter and the next.
65. First as to transfers and comparisons among reference-
frames. Since scalar mass that is unaffected by position and
motion becomes by that supposition neutral to the main issues
here, something can be done toward clearing the ground by
noticing at once how many important decisions must then turn
upon the kinematical factors; solely upon these in the differ-
ential elements, though as we have found at certain points in
the preceding chapter, the mass-distribution continues to play
some part through the integrals that are related to the center of
mass and to the moments of inertia. Accordingly we are enabled
to restrict ourselves in the first steps to kinematics, essentially
to radius-vectors and velocities and accelerations, the properly
dynamical phase being covered finally by introducing the neces-
sary mass factors.
As one aid to brevity, we shall outline a notation by way of
preface, to be used consistently throughout the combinations and
comparisons that we must make. Let one reference-frame estab-
lished by its origin (O) and its axes (XYZ) be constituted the
standard, the axes being orthogonal and in the cycle of a right-
78 Fundamental Equations of Dynamics
handed screw. By affording to our thought one term common
to a series of comparisons, this frame will furnish a means of
coordinating their individual results. Let any one of the other
reference-frames with which we may happen to be concerned
alternatively, either under suggestion from special conditions or
for the purpose of more general discussion, be determined
through its origin (O') and its axes (X'Y'Z') and be distinguished
as a comparison-frame. All the frames are supposed congruent.
We shall preserve a helpful symmetry of notation by assigning
regularly primed quantities to comparison-frames and unaccented
symbols to the standard. But we must not fail to remember
either that the distinction which sets off one frame as standard
is for convenience of correlation only, in the first instance, and
it retains its arbitrary element until physical reasoning can be
seen to converge noticeably or convincingly upon one frame, or a
set of frames meeting formulated conditions, as the basis better
accommodated to the ultimate statement of any physical laws
or regular sequences among phenomena. We have touched on
this point in sections 6 and 7. In the preliminary view every
frame is qualified for selection to be standard, in relation to
which all the others fall into their status of comparison-frames.
66. The configuration of any (0', X'Y'Z') relative to the stand-
ard can be specified as though it had arisen in virtue of a dis-
placement from original coincidence with (O, XYZ), without
needing to imply, however, that the coincidence once existed in
reality and that the final configuration has developed pro-
gressively by a time process, but also without excluding the
latter possibility. In order to dispose of certain aspects of the
matter, let us at first conceive definitely all these individual
configurations to be permanent, each comparison-frame being
taken in a configuration that it retains. Then any continuous
transitions within an arrangement of such frames will associate
themselves rather with grouping it into a space locus, and no
Reference Frames 79
idea will be imported into it of those other features belonging
distinctively to motion and a path. But we must expect to find
here as elsewhere, that the two points of view run easily one into
the other, with those groups of virtual displacements, indicated
as possible without violating the conditions for the locus, becom-
ing an actual series in time when the paths are described. One
moving frame can mark the positions of all members of a group
that are in permanent configurations, as it coincides with them
in succession. In point of fact, several similar modulations of the
thought here hinge alike upon that dual conception of the
elements that enter.
67. The assignment of its relative configuration will involve in
general for any frame both a difference of position between (0')
and (O) and a difference of orientation between (X'Y'Z') and
(XYZ). Moreover these two data are assignable independently,
and it is intuitively true that the actual localization of (O',
X'Y'Z') is reproducible from coincidence with (0, XYZ) by com-
bining them in either order. Let the parallel displacement or
translation of the axes with the origin (0') be specified by the
vector (OO') which we shall denote by (r ). And the changed
orientation is equivalent to a subsequent displacement by rota-
tion of (X'Y'Z') as a rigid cross, because they are congruent with
(XYZ) and remain orthogonal. Using the notation of section 45,
we can indicate the result by the vector sum
T = /d Y , (92)
with the possibility attaching to resultants in general, of repre-
senting equivalently many sets of components.
If the idea of succession enters the last equation, the present
connection confines it to a timeless series of elements (dy), in
each of which the constituents (^.yv) or substitutes for them are
coexistent. Where it will not cause confusion, the term rotation-
vector can be applied to (dy), as well as to (y) of the earlier
80 Fundamental Equations of Dynamics
section. For any comparison-frame accordingly its configuration
is given with the requisite definiteness by the two total displace-
ments taken in either order,
r = /dr ; r = /dy- (93)
68. Let us introduce next any point (Q) having at a given
epoch radius-vector (r) in the' standard, and (r') in a comparison-
frame. The difference of orientation alone while (0') coincided
with (O) would leave the radius-vector invariant for all per-
missible sets of axes, the expression of which condition can be
put into terms of the two sets of unit-vectors,
r = ix + jy + kz = iV + j'y' + kV; (94)
where the invariance is noticeably obscured until the vector
algebra brings it into full relief. The alternative relation
accompanying separation of (O') and (0) is
r = r + r', (95)
whose form obviously excludes equality of (r) and (r') so long
as (r ) differs from zero. It should be observed about the last
equation that it is based rather upon a triangle as graph than
upon a parallelogram, because the conception of (r') makes it a
localized vector with (0') for base-point.
Regarding now (Q) as typical in any continuous or discon-
tinuous assemblage of points, and (Q') as any other such point
whose radius-vectors in the two frames appear in the allowable
forms (r + Ar), (r' + AT'), we have for the vector (QQ')
Ar = Ar', (96)
throughout the group of points, independently of the points
chosen and of the particular comparison-frame employed. This
records the patent truth that the arrangement of members in
any point-group, or their configuration relative to each other,- is
expressible invariantly by means of the standard frame and of
Reference Frames 81
every (O', X'Y'Z'). With that meaning the remark is to be
accepted that " Position coordinates appear in our equations by
a convenient fiction only, they being parasitic and auxiliary
variables that can be eliminated." 1
69. If for sufficient reason we maintain the discrimination
between (Q) and (Q') as two individual points and locate each
permanently in its configuration with (O, XYZ) ; or let each be
fixed in the space attached to the standard reference-frame in the
words of one current phrase; no questions about time-derivatives
of (r), (r'), (Ar) or (Ar') can arise, so long as the configuration of
(O', X'Y'Z') is also by supposition permanent. The source of
those reasons and their cogency will depend upon the case in
hand; they may be physical in their nature and extracted by
interpretation and analysis from observation, or their origin,
may be frankly due to a feature in the mathematical treatment.
By associating other such individual points with (Q) and (Q')
we may build up a continuous group as a limit, for which the
general radius-vector becomes in length a function of its orienta-
tion but the essentials of the description remain timeless.
However in any unforced survey of other particular circum-
stances and their plain suggestions a competitive view must find
recognition, that will regard both (Q) and (O', X'Y'Z') as indi-
viduals somehow identifiable through a series of changing con-
figurations in (0, XYZ), and consequently any account that aims
at practical completeness cannot neglect coordinating the two
alternatives. There is the elementary fact, for example, that the
same dependence of radius- vector upon its orienting angle as before
can be presented with both variables made functions of time. But
the fruits of that idea are not exhausted in one announcement at
the threshold of the matter. For when in our view (Q') becomes
a subsequent position of the point (Q), or whenever, more inclu-
sively, the varying position of a moving point is matched at each
1 Quoted from Poincare.
82 Fundamental Equations of Dynamics
epoch with the permanent position of a coincident point, the
twofold relation of the same symbols to which this leads with
such a double point will reappear perpetually. This can make
either aspect of the coincidence a continuous indicator or marker
for the other, by means of some connecting rule that formulates
from either side the relation of consecutive values here of the
radius-vector. Neither phase of the combination can be ignored
or subordinated, without losing hold upon ideas that are central
in evaluating any variable quantity by legitimate transition to a
substituted uniform condition. 1
70. These considerations confront us with the necessity of
preparing here for that kind of transition, and conceiving (Q)
and (O', X'Y'Z') to be individual and moving. This can be
executed conveniently by subdividing into steps, and taking first
the one that affects (Q) alone, while we retain for the time being
that permanent configuration of (0', X'Y'Z') in the standard
frame which is afterwards to be abandoned. If we accept for
(Q) and (Q') a fusion of identity in the sense that they are now
adopted as two positions of the same moving point, terminal for
any time-interval (At), the mean velocities for that interval will
be equal in our two reference-frames, and also the instantaneous
velocities at the epoch beginning the interval. This conclusion
finds expression in sequence with the requisite new reading of
equation (96) as
/Ar\ /Ar'\
v - Lim At=0 ( - ) = Lim At=0 (^ J - V; (97)
or in semi-cartesian dress,
since both sets of unit-vectors are by supposition constant here,
as well as (TO). And further,, because these simultaneous veloci-
1 See Note 19.
Reference Frames 83
ties of (Q) are thus continually equal vectors, it is an evident
corollary that the accelerations of (Q) in the two frames are
always equal at the same epoch; or
(Av\ / Av' \
I = Lim At =o 1 ) = v', (99)
ii V / \ /it /
whose expanded equivalent again is
.d 2 x , .d 2 y , d 2 z
71. Taken together, these statements make clear for every
epoch the in variance of velocity and acceleration that holds
good throughout any group of reference-frames that are in
permanent relative configuration. Also the consequences in
application to the same system of bodies at the same epoch are
apparent. Each local velocity and acceleration being un-
affected, the six fundamental quantities show in the standard and
in any comparison-frame of the group thus correlated:
Q = Q'; R = R'; E = E'; P = P'; 1
H = H' + (r x Q'); M = M' + (r x R'), I
which it may be well to compare for likeness and difference, say
when (r = f), with the corresponding relations exhibited in
section 51, the contrast between (C') there and (0') here lying in
the freedom of the former point to move with velocity and ac-
celeration. The less narrowly limited connection of center of
mass with force-moment and moment of momentum should be
realized.
72. The foregoing results are sufficiently practical in their
bearing to incite us to appropriate, without delaying, the possi-
bilities that they illustrate. These lie in the direction of a certain
liberty to employ what amounts to a whole series of different
84 Fundamental Equations of Dynamics
reference-frames at successive epochs of the same problem, or
inside the range covered by one discussion, and yet avoid pro-
hibitive complications that might be due to such repeated trans-
fers to new standards. Provided only that we observe those
restrictions which underlie the invariance of any particular
quantities with which we are dealing, the frames become inter-
changeable in respect to them; and freedom prevails to depart,
at later epochs and as often as may prove desirable, from the
initial choice of reference-frame. At least it is evident how
there will be no danger, on relinquishing one frame and adopting
another subject to the proper conditions, of dislocating ruinously
by breaking into it the expression of a continuous series of values
for any quantity that the change leaves invariant. Dislocations
of minor scope can be reckoned with otherwise, or often dis-
regarded, where they enter.
Such procedure remains clearly valid, always within its limita-
tions, whether its revisions of choice involve configurations
separated by steps that are finite or that are made with finite
pauses between them, or whether the group of frames used melts
at the limit into a continuously consecutive arrangement. It is
equally permissible, besides, to regulate the employment of
members in a group of frames according to a time-schedule, or
to effect timeless transitions among frames and to concern our-
selves comparatively with simultaneous values of different quan-
tities, or finally of the same quantity when we break the barrier
of invariance. The actual working out of the main thought
rings the changes on all these offered chances, so that several of
the combinations will come before us prominently for specific
examination.
73. We proceed next to remove the limitation that has held
us to permanent configuration for (O', X'Y'Z')- We relax this
permanence relative to (O, XYZ) by admitting, first changes in
(r ) alone while (y) is unchanging in equation (93), and after-
Reference Frames 85
wards the full freedom with changes in (y) also. It seems ad-
vantageous to attack this phase of the matter, too, through what
we have spoken of as fusion of identity; but now for comparison-
frames that like the points (Q) and (Q') can from another
approach also be distinguished as separate individuals. Return
then to that original view of those points, include some second
comparison-frame (O", X"Y"Z") and carry on the notation by
adding
O'O" = Ar ; O"Q' = r". (102)
The relations associating (Q) with (O') and (O), and (Q') with
(0") and (O) are
r = r + r'; r + Ar = (r + Ar ) + r", (103)
showing by their difference
Ar = Ar + (r" - r'), (104)
whose verbal equivalent can be read from the broken line
(QO'O"QO that is equal as a vector sum to (QQ') and closes a
quadrilateral that may be of course either gauche or plane.
We may now retrace the previous track further, whenever we
can attribute to the frames (O', X'Y'Z') and (O' 7 , X"Y"Z")
some adequate basis of continuous identity similar to that which
was made to unite (Q) and (Q'), so that the entire group of
discrete frames of permanent but differing configurations is
replaced by the conception of one representative frame (O',
X'Y'Z') in continuously variable relation to the standard. First,
confine attention to the origin (0'), deferring a little the intro-
duction of changing orientation, suppose (r ) to vary with time
and read equation (104) to correspond. The originally un-
related vectors (r') and (r") coalesce under one symbol (r')
when that is used to signify a vector drawn always from the
position of (O') at any epoch to the simultaneous position of (Q).
It is therefore a vector to be rated in the standard frame as
86 Fundamental Equations of Dynamics
localized, but variable in all three particulars of length, orienta-
tion and base-point. In pursuance of that thought write
r" - r' = AT', (105)
divide equation (104) by the elapsed time (At) and proceed to
record the limiting ratio in the form
=(if) = r-*o = v-v , (106)
if (v) and (v ) denote the velocities of (Q) and of (0') in the
standard frame. The formal repetition in this first member of
(v') as specified in the terms of equation (97) is significant of its
unconstrained meaning here too as the velocity of (Q) reckoned
in the frame (O', X'Y'Z'), but under an extension that allows a
supposed motion of (0'). Duly observing the imposed condition
of unchanging orientation for (i'j'k') that is still maintained,
confirm this feature of the development by writing the time-
derivative of the permanent relation in equation (95) in the form
and compare with equation (98). It is plain that (v') and (v)
are equal at any epoch when (f ) is zero.
74. These thoughts harmonize in another respect with equa-
tion (106) if we see registered there a consequence of a double
process of incrementation for the vector (r'), now completely vari-
able in the standard frame, with rate (v) at its forward end and
with rate (VQ) at its base-point. In every such combination, so
long as these rates are equal, the vector retains its length and
orientation in the reference-frame; as a free vector it remains
equal at all epochs, though as a localized vector it experiences
change of position determined by the common value of the two
rates. In the less particularly chosen suppositions where the two
rates are unequal, only their difference such as (v v ) is avail-
able to give change of tensor and of orientation.
Reference Frames 87
But to take account of these latter elements for (r') and to
ignore or drop out the change in position for (O') substitutes
effectively (O', X'Y'Z') as reference-frame, the orientation of
(i'j'k') having first and last the requisite permanence, so that the
transfer is uncomplicated in that respect. And since the part
(v ) applies simultaneously or in common to all points (Q), the
readjustment of velocity values made necessary by this type of
transfer to a new reference-frame (0', X'Y'Z') can be summarized
as the subtraction of a translation with the velocity of the new origin
in the first standard frame. In connection with this the thought
frequently finds expression that each frame carries its space in
rigid attachment to it, and these interpenetrating spaces will
have in the present case at each coincident pair of points the
relative velocity ( Vo) at any epoch.
The effects upon acceleration of a similar transfer while
(i'j'k') remain constant show plainly on forming the time-deriva-
tive of equation (107). This gives
dV dV dV
^ si/ dt^ +j '^ + k 'dt^ = *~ to; /-v' = v o; (108)
and the proper allowance shows again in terms of a translation
with the new origin (0'), whose acceleration, however, is now
essential and not its velocity. In the light of equations (107,
108) the combinations become self-evident by which velocities
or accelerations or both may be left invariant under a change of
reference-frame. The bearing upon the segregation in sections
21, 31, 48 and 49 will not escape attention.
75. In order to embrace finally the transition to axes (X'Y'Z')
whose orientation is changing in the standard frame, while they
are accompanying their origin (O'), we can use our knowledge
that the rotation-vector of sections 45 and 67 specifies such
changes adequately, and thus complete under the wider play of
these conditions the time-derivative of the relation that remains
valid,
88 Fundamental Equations of Dynamics
i = r + r' = r + (iV + j'y' + k'z'). (109)
Upon the supposition that the group (i'j'k') are at the epoch
varying in direction relative to (XYZ) as determined by the
rotation-vector (y), we are led by the differentiation directly to
the equation
+/+*, (no
from which it follows that
v-v' = v + (txr'); v = v' + [v + (t x r')]. (HI)
Typical special cases under this equation can be decided by
inspection. Note the form now taken by the idea of inter-
penetrating spaces in section 74, connecting it with the general
motion of a rigid solid in section 48. The last group of terms in
equation (110) must still be recognized as the velocity (v') of
(Q) in (O', X'Y'Z'), because the transfer to the latter as the
standard cancels perforce from admission into (v') every change
in orientation attributable otherwise to (i'j'k'), in addition to
ignoring changes in the position of (O').
76. Various equivalent verbal formulations beside those al-
ready suggested can be devised for equations (107, 108, 111),
that all amount in principle to a superposition of relative veloci-
ties or accelerations. And it will be seen how the same idea can
be applied repeatedly and can carry us through a chain of trans-
fers to a final result that accumulates in itself all the contributions
at its several steps. Remembering that forces are bound to
superposition also, as they enter successively with the acceptance
of their accelerations into physical status, trace there a line of
advance in precision that would parallel our discarding one
reference-frame in favor of another. 1 The same possibility of
superposition lies open as we go forward from equation (111) to
i See Note 20.
Reference Frames 89
consider the similar transfer for accelerations, though the com-
plications soon cut down any advantage of a verbal expression
for it.
Formal routine yields for the time-derivative of the general
relation in equation (110) or (111) the result
v = v + (r x r') + 2(t x v') + (r x (r x r')) + v', (112)
in which (r'), (v')> (v') specify the position, velocity and acceler-
ation of any point (Q) by means of (O', X'Y'Z'); that is, to
recapitulate,
r' = i'x' 4- iV 4- kV- v' = i' 4- i' 4- k'
dt ^ 3 dt dt
(113)
> - i' 4. V 4. k ' -
d~ + J W ~ ' k d tr >
(y) is the angular acceleration belonging at the epoch to the
rotation-vector (y), and (v ) denotes the acceleration of (O')
in (0, XYZ). Interest will center here upon the terms affected
by the rotation, into which the elements (r') and (v') individual
to the point (Q) enter; and for the latter, the connections shown
in equation (111) must be duly heeded. It will cultivate control
of details in the method to carry through its application to such
combinations as (Y = 0), (Y = 0), separately or conjointly, in
preparation for the summary that follows. And then to work
out lists, comparable with that in section 71, for the general
conditions of equations (107, 111, 112), showing how the different
quantities are affected by the transfers from one reference-frame
to another that have been brought under review. It is always
a reciprocal interdependence that is in question, and a procedure
for transfer in either direction.
77. To round out this stage of the inquiry, we can now formu-
late for velocity and acceleration the suppositions necessary to
their invariance, that will put the frames for which these are
90 Fundamental Equations of Dynamics
satisfied to that extent on an equal or indifferent footing. We
begin with acceleration, whose invariance necessitates con-
formably to equation (112),
to + (r x r') + 2(f x v') + (r x ( r x r')) = 0. (114)
But (v ), (Y) and (y) are to be assumed independently of each
other; and further, the search is for a general relation covering
all points (Q) in all phases of their motion, which puts aside as
insufficient every particular adjustment or singular value like
r' = 0; v' = 0;
or colinear factors in some individual vector products. Hence
the proposed invariance of acceleration demands all three con-
ditions,
v = 0; t = 0; Y = 0. (115)
These permit the comparison-frame to have unaccelerated trans-
lation with (O'), but forbid changes in orientation (y) as indicated
by its time-derivatives of the first and second order.
The invariance of velocity imposes different limitations deriv-
able by inspection from equation (111) as being
v = 0; r = 0. (116)
The second of these conditions, therefore, is common to the
invariance of velocity and of acceleration. But as regards the
translation with (0') equation (116) excludes any velocity (v<>)
though allowing an acceleration (v ), while equation (115) inverts
these relations. The double condition for invariance of velocity
bars at the epoch motion of (O', X'Y'Z') in (O, XYZ), but gives
freedom as to subsequent states. The triple condition for
invariance of acceleration maintains the exclusion of -changing
orientation and sharpens it by (y = 0), but allows any constant
value of the vector (v ).
The above conclusions coupled with the discussion that led
Reference Frames 91
to equations (98) and (100) bring out how (O', X'Y'Z') if treated
as moving in the standard frame must always sacrifice in some
degree the invariant properties in regard to velocity, acceleration
and the dynamical quantities dependent upon them; though
these are, nevertheless, preserved intact by a succession of frames,
each in coincidence with the moving frame at one epoch. The
permanent values of (r ) and (y) for the stationary frames are
marked off, one by one, in the series of instantaneous values for
those elements belonging to the moving frame. In this sense
and to this extent, the presence or absence of an invariance that
happens to be in question can be made to turn upon the point
of view, which because it affects values also raises issues that need
to be decided in the light of clear statement of the position our
thought has occupied. Consequently it is likely to repay us,
if we enforce this main idea by approaching it in reliance upon
the frames of permanent configuration, the mathematics being
modified to match.
INVARIANT SHIFT.
78. Whereas the radius-vectors (r) have been handled in the
preceding equations as functions of time alone, directly in (0,
XYZ) and in (O', X'Y'Z') through the relation
r = r + r', (117)
this second mode of making a beginning will disguise the same
radius-vectors (r) into functions of three independent variables
(t, r , Y). And this will evidently lead toward fixing attention
upon a whole group of comparison-frames inclusively, to be
constructed by assigning continuous, but otherwise arbitrary,
values to (r ) and (y), perhaps in connection with equation (93),
while (t) remaining unchanged gives simultaneous currency to
those values.
The exact differential of (r) indicated according to the new
terms is
92 Fundamental Equations of Dynamics
(118)
This form might indeed be denominated rather sterile of meaning
in respect to (r) itself, for it is apparent enough from many of
the expressions that we have been laying down that (r) is not
intrinsically dependent on either (r ) or (y). Similarly if we
use equation (117), and after omitting the terms that are neces-
sarily zero, on our assumption about independent variables, write
"-^*** (119)
appeal to equation (94) seems to tell that (r') at any epoch does
not change with (Y)- But after admitting that
dr dr dr' dr'
= + = 0; = 0; (120)
dr dr dr dy
equation (119) is found, notwithstanding, really helpful for the
end sought, as a starting-point for collating different sets of
components within our group of frames, though it might be
superfluous did we restrict ourselves to resultants. In order to
develop this idea more fully introduce the semi-cartesian equiva-
lent
r' = i'x' + j'y' + kV, (121)
whose second member is intended for a comprehensive notation
applying both tensors (x'yV) and unit-vectors (i'j'k') generically
to the whole group. They are then variables as affected by
passage from one frame to its neighbors, and in addition the
tensors are variable with time in the same frame.
This temporary identity of the variables in the one frame, which
may pick that one out or enable us to recognize it, and yet be
evanescent for the group of frames as a whole, lies close to the
heart of the thought in equation (119), as contrasted with a
completer convection of identity with one moving frame, whose
Reference Frames 93
tensors and unit-vectors are consequently functions of time only.
For the present purpose, on the other hand, and in its adapted
mathematics, the tensors (x'yV) must be considered functions
of (r ), (Y), (t); but the unit-vectors (i'j'k') and (r ) do not at
this stage vary by mere lapse of time ; nor the former by reloca-
tion of the origin (O') they must be functions of (y) alone.
Under the suppositions and the reasons for them thus made
explicit, we execute the differentiation of equation (121) in
combination with equation (119) and obtain
ar / , ai' ,aj' , dk'\
dr = dr fl + I x' + y' + z' T ) d?
dr \ dy dy dy )
,/az' , az' , az' \
'( dr + d7 + -rrdt ).
\ar dy dt )
79. This expansion supplies material to interpret profitably,
when it is observed that the imposed condition for the partial
time-derivatives with the set of variables now adopted is the
same in effect as that for invariant velocity to which equation
(97) is subject. Consequently the three terms on the left are
properly equated to the velocity of any (Q) in the standard frame,
when we write
i' ^ + r * , k , M. - v a23)
at + J at at
The double use of this equality is apparent, either in obtaining
projections of known (v) upon the (X'Y'Z') of the configuration,
or in determining (v) by means of its projections upon whatever
particular comparison-frame is designated by the stationary
values at which (r ) and (y) are arrested while the partial change
with (t) is recorded.
94 Fundamental Equations of Dynamics
Thus no essential in regard to consistent expression of velocities
would be sacrificed if we depended upon any such comparison-
frame momentarily to replace (O, XYZ) in its service as standard,
and did likewise for new stationary values of (r ) and (y) with
velocities at other epochs. This comment will infuse its due
quota of meaning into the .equality
f = v = |' (124)
and parallel expressions, whenever similar opposed total deriva-
tives and partials are made to play their roles as the basis of a
regular procedure, in which a resultant vector is to be con-
structed or evaluated by means of components parallel to axes
that differ systematically, or in which the projections of a given
vector upon such axes appear naturally.
It is readily apprehended, at this point, how such plans are
effectively equivalent to a continuous process of transfer to new
standard frames that is kept simple by its preservation of invari-
ance, while it may secure a permanence of form or other ad-
vantage in addition. The indispensable resolution of accelera-
tion along tangent and normal of the epoch in treating curved
paths is one case in point; and the compact forms obtained by
introducing principal axes will suggest strongly some similar
scheme in continuation of sections 61 and 63 with expectation of
profit from it. It seems convenient to have a brief name for
contrived plans of this character, so we shall refer to them here-
after as shift of reference-frame, implying always invariant shift
in so far as some quantities are not thereby modified from the
simultaneous value indicated in the standard frame. 1
80. The three terms put down in equation (123) are then
seen to reproduce accurately in the combinations of equation
(122) the actual displacement (dr) for the time (dt) of the
moving point (Q) in the standard frame; and therefore, the
i See Note 21.
Reference Frames 95
remaining entries in the coefficients of (i'j'k') must be illusory
if taken by themselves, as regards describing what is thus
happening at (Q). In fact, as their form involving constancy
of (t) indicates clearly, they are attendant upon comparisons of
corresponding and simultaneous pairs in two sets of projections
determining or determined by the same (r'), but connected with
two sets of axes differing in orientation by (dy) and having
origins separated by (dr ). The complete coefficients of (i'j'k')
being evidently the exact differentials for the present inde-
pendent variables of the tensors (x'yV), equation (122) can be
rewritten
dr = dr + (d Y x r') + (i'dx' + j'dy' + k'dz'), (125)
if we bring in the consequences of the rotation- vector (dy) in
the form
(126)
Accordingly equation (125) in its second member is so arranged
that it includes within its last group deviations from the true
value of (dr) through apparent or spurious changes in the
tensors, and finally offsets these by the corrective first and
second terms.
That exactly the compensating adjustment shown must exist,
can be argued summarily, in line with our remark upon equations
(119, 120), from the independence of actual -changes in (r) of
mere subheadings in our accounts of them, but some few details
are worth inserting for emphasis. The first of equations (120)
is self-evident, for (r') must lose whatever (r ) gains, while (r)
is held at its value by unchanging (t). Let us therefore analyze
only the second of those equations in regard to the dependence
of the tensors upon (Y). We must have
96 Fundamental Equations of Dynamics
x' = i'-(x - x ) + i'-(y - y u ) + i'-(z - z n ). (127)
Then because neither (xyz) nor (x yoZo) in the standard frame
are dependent upon (Y),
- x ) + (y - y ) + (z - z 3 ))
(128)
= I -r-dv I -r'.
Consequently
^d 7 = (d Y xi')-r' = -- (d Y xr')-i'; (129)
and similarly
^d 7 = - (dYxr')-j'; ^-d 7 = -- (d r xr')-k'; (130)
which together prove consistently with anticipation,
81. Let us next return to equation (110), with the reminder
that it occurs in a general procedure of substituting a new
reference-frame to be standard, by making necessary allowance
for the relative motion of the two frames. Multiply both
members by (dt) and verify that its form then becomes identical
with equation (125), although the latter was deduced under
more special limitations that we propose to distinguish as shift.
and that keep the velocities invariant. In other words, the
sum of the last three terms in this equation will differ by the
same amount from an actual displacement (dr) in the standard
frame, whether (dr ) and (dy) designate differentially changes
of configuration observable in the one moving comparison-frame,
or whether the same elements express the shift in passage to a
consecutive member of the invariant group of frames.
Reference Frames 97
These two relations distinct in their conceived source are
joined into a formal identity, primarily because together they
embrace a series of coincidences, as displayed in sections 69 and
77, for each aspect of which the same symbols can be given
coherent meaning. But that fact though patent is no good
ground for obliterating either one of the serviceable conceptions
out of which the equation that we are now discussing has arisen
for us. We should rather grasp firmly the thought that two
successions are here instructively coordinated: one ensuing by
movement of an identified frame into new positions, and the
other by timeless shift to new stationary frames. These con-
clusions refer in this first instance, of course, only to the velocities
for which they have been established ; but they are conveniently
capable of extensions. In the measure that these are unfolded,
they will lend finally to the otherwise trivial identity
A = (A - B) + B (132)
that equation (125) may suggest, a value for working needs
through practically advantageous selections of (B). Note, for
example, that equation (74) is scarcely different in type.
82. As the last remark might imply somewhat plainly, the
exploitation of the dominating idea in shift will look to govern
its course and its extent by special phases of adaptation con-
trived to meet combinations that do occur. Analysis that we
shall undertake of several coordinate systems may be expected
to illustrate and repeat that lesson. What the instances quoted
in section 79 show is more generally true: That the plans for
shift require various adjustments to be renewed continuously,
and keep modulated pace with conditions that develop velocity,
acceleration and the closely related dynamical quantities. Thus
the progress of the shift must accommodate itself to a regulative
time-series of other values, and this in turn imposes upon the
shift process itself a necessary rate in time. That situation the
98 Fundamental Equations of Dynamics
mathematics handles by recognizing (r ) and (y) to be functions
of time, instead of treating them as independent variables subject
only to timeless change; so linking them with each other and
with the salient phenomena that are to be followed up that some
line of advantage sought is most nearly secured.
Nevertheless since the previously independent increments still
form a background, these additional functions of time will differ
in certain respects from those that yield, for instance, the veloci-
ties and accelerations of the moving points (Q). One formulation
of the critical difference declares that the latter class of time
functions is dictated altogether by an objective element; they
must conform to the phenomena studied and express them, their
own nature and form being to that important extent not under
control. Those of the former class are open to free choice,
although we may grant, indeed, that this control is exercised
normally in bringing to pass some mode of subordination to what
is occurring in other sequences, to the end of attaining simpler
models in equations, or the like removal of complications. This
employment of time functions in dynamics that are distinguish-
able in their nature, has long been commented upon and provided
for, though the discrimination is stated variously and not always
in clearest terms. 1
On a foundation of the foregoing explanation or some equiva-
lent, we are brought to accept a two-fold dependence upon time
in equation (122) and in any statements that disclose to examina-
tion the grounds for a similar distinction. Thus we gain the
liberty to regard the partial processes as simultaneous, to divide
equation (125) by (dt) and so to establish an exact formal identity
with equation (110) by allowing for shift rates that are inde-
pendently assignable. Yet the alternative readings diverge still
in the direct meanings associated with (f ) and (Y); these are
alike, however, in standing equally among the controllable time
i See Note 22.
Reference Frames , 99
rates, because the one definite frame to which transfer shall be
executed may move at will, save as outlook toward convenience
guides or special circumstances demand. Perhaps it is not
over-subtle either to insist upon a second residual difference:
The plan of equation (110) aims primarily to connect properly
two sets of values for velocity, each correct and complete for its
own conditions; but equation (125), on the contrary, entertains
only one set of values as correct, that are made to reappear
finally from being obscured under a transient distortion.
83. We should not have elaborated these ideas with equal
fullness had the results borne solely upon the narrower issues
gathered about the radius-vector, and had not Hamilton's hodo-
graph given a clew toward making the radius-vector repre-
sentative of other vectors, and the velocity of its extremity a
key to the general vector's time rate. The vector algebra having
fallen heir to these methods and enlarged them, it is natural to
look upon the previous section as a preface and proceed to trace
again its characteristic connections when any vector (V) has
replaced (r), and its time-derivatives are offered in parallel with
(v) and (v). In the course of such extension, we may expect
correspondences and fruitful grafting of larger ideas upon the
parent special case, all along the line of development whose
details are now fairly before us.
But when we come to examine and sort the material that pre-
sents itself under such headings, we find the two chief operations
that we have been comparing very unequally represented in
practice. The circumstances of unrestricted change from one
reference-frame to another do reappear in connection with all
physical vectors and other types of quantity; and as we have
seen exemplified repeatedly already, those changes when they
are made necessitate a deliberate reconsideration of all these
quantitative values. Yet besides, the occasions that compel
such revisions are, at once, comparatively rare and apt to be
100 Fundamental Equations of Dynamics
made for conditions that have become more strongly specialized;
although the process is important as regards flawless execution,
it shows few features that give it the weight of a procedure that
holds its place among the routine methods of frequent use.
The alternative conception that we call shift, however, has
been introduced and given preliminary analysis here to a degree
that may seem not quite called for, because in the first place it
is implicitly or explicitly involved when a number of the standard
coordinate systems in dynamics are employed, which is a routine
procedure; and because secondly, there has been some failure
in clear apprehension and announcement of just those conse-
quences of the restrictions upon the process of shift that bring it
into close alliance with the prevailing purpose of coordinate
systems. For these are, in the main, adapted to the one central
idea of expressing equivalently or invariantly, through some
convenient dissection into parts, a resultant or total quantity
that relations in a standard frame have first actually or potentially
settled upon. When therefore we dismiss in a few sentences the
subject of changing reference-frame for the general vector (V),
and yet expand the idea of shift on its broader lines, the explana-
tion is to be sought in the reasons that have just been given.
84. If we look again at equation (95) with a view to generalizing
upon it, we must describe (r ) as the difference between the
values in the two frames of the vector that is under considera-
tion. Similarly if we write the equation
V = V + V (133)
in beginning an attempt to extend the validity of previous con-
clusions, it is clear how (V ) is to be read. It is also apparent,
or verified by easiest trial, that one obstacle to indicating here a
more general rule for change of reference-frame enters because
the value of (V ) depends upon the quantity represented by (V),
as instanced by the conditions for invariance in section 77. But
Reference Frames 101
it was also forced upon our attention, from equation (94) onward,
that (r') in the standard frame is invariantly given by all frames
whose origin is at (O') in its position for the epoch. And while
this too draws the lines closer for (V) and limits narrowly the
usefulness of results attached to derivatives of (r), (FO) and (r'),
in doing that it points convincingly toward the process of shift,
if we are to generalize, in which this very invariance has been
made a prominent characteristic. When we look at the matter
from another side, and observe how near an assigned behavior
of (i'j'k') comes to furnishing completely the compensating or
corrective elements in an equation like (125), once more the
conformity of a coordinate system to some rule of displacement
can be seen. Thus polar coordinates are essentially a shifting
orthogonal set, and a scrutiny of the standard expressions for the
components there shows that they meet (r') on an equal footing
of reproducing a resultant invariantly.
85. We shall begin the definite inquiry about shift in its larger
relation to coordinate systems by supposing that we have to do ,
with any free vector determined in the standard frame as (V),
postponing the mention of localized vectors. Then (V) may be
associated legitimately with the origin (O) as base-point, and
any element that might correspond to (r ) will be suppressed.
With the usual unit-vectors, here taken at a common origin for
convenience, we must have at the epoch, whatever range in
orientation may be permitted for (i'j'k'),
V - iV (x) + jV (y) + kV (l) = i'V (x ' } + j'V (y ') + k'V (l '>. (134)
This relation, to repeat with emphasis an incidental remark of
section 79, may face in either of two directions, according as the
data make (V) itself or its three constituents directly known.
The next equation derives much of its importance from the
fact that the algebra so seldom furnishes a resultant vector im-
mediately, unless the superficial geometry happens to fit.
102 Fundamental Equations of Dynamics
Express now the time-derivative of (V) ; it will be consistently
specified for the same standard frame as (V) itself, and it ap-
pears as
k'V (z '))
k' -~ (V ( .'>). (135)
It is to be observed about tensors like (V( x ')) that they are
differentiated on that comprehensive understanding about them,
spoken of in section 78, which is favored by an algebra that
attends to magnitudes alone and can neglect orientation. In
the first group of the third member in this equation, it is the
vector algebra with its equal attention to directions which is
repairing that deficiency in the other algebra. In order to
follow up and express this idea, we adopt the notation for all
such cases,
(X '>) + j' (V (y ')) + k' (V ( .',), (136)
(m)
intended to suggest that only the tensor magnitudes of (i'j'k')
have been differentiated. Omitting the second member of equa-
tion (135), and in reliance upon section 80 for a reduction of
the first group, the third member can be rewritten in the more
nearly standard form,
V = (Y x V) + V (ra) . (137)
But equation (134) would not be modified if the origin for
(i'j'k') were at any distance (r ) from (0) and were moving in
any way. Our last result would still hold, provided the same
(Y) were retained, because it is a sheer relation for projections
upon which it stands. Further, whenever (y) is zero, both (V)
and (V) are represented indifferently by their respective compon-
Reference Frames 103
~ents in (XYZ) or in (X'Y'Z') ; and this harmonizes with the invari-
ance found by using the permanent configurations of the coinci-
dences and the idea of shift. Otherwise even when (i'j'k') fall
in (ijk) and make the two sets of components for (V) the same,
the total time-derivatives of any algebraic expressions for the
tensors of (i'j'k') would not agree with the projections of (V)
on (X'Y'Z'). But note that the proper partial derivatives of
those tensors would give correct values for (V), as we discovered
from equation (123) in the case of (v).
There is one condition of special arrangement that cancels the
difference between (V) and (V( m) ) though (y) is not zero; namely,
colinear or parallel factors in the corrective vector product. And
since (Y) as applying to (i'j'k') rests on a supposition subject to
a certain control, there is a strong hint in the above possibility
of cancellation, which several coordinate systems have found
their own ways to adopt. We can give a first illustration from
our original discussion of the rotation-vector. For if we multiply
equation (137) by (dt) and identify (V) with (dy) the two
members show equality to the second order, in confirmation of
section 47.
86. Let the vector (V) be represented graphically from (0) as a
base-point, in the manner of the velocity vector for the hodo-
graph, then the derivative (V) will be given as the velocity of its
extremity in (O, XYZ); and on comparing equations (111, 137),
the former in application to a common origin, the other derivative
(V(m>) is seen to give similarly the velocity with which the
extremity of (V) moves in the frame (X'Y'Z'). Consequently
we find forms like (V( m )) described sometimes as derivatives
relatively to the moving axes (X'Y'Z'), and, to be sure, they are.
But we must not neglect the other fact that this uncompleted
derivative is applied to a quantity that like (V) has been speci-
fied for the standard frame, and that itself does not stand in any
one particular relation to the frame (X'Y'Z'). These schemes,
104 Fundamental Equations of Dynamics
if thus viewed, are composite; or they straddle between the
standard frame for (V) and a comparison-frame for (V( m ));
but they are less disjointed if interpreted as shift. The above
denial, of course, runs only against a general truth, and does
not exclude special conditions under which the same term covers
both a shift and the other form of transfer. It is plain for
example, in giving velocity by means of polar coordinates in
uniplanar motion as
v = T I ^ + (w x r), (138)
that the first term in the sum can be read either as (V( m )), or as
(v') for the frame consisting of (r) and a perpendicular, with the
second term equally adapted to either sense.
It contributes much to the serviceable simplicity of equation
(137) that it observes always the limits of a one-step transition
from a vector to its first derivative, while a radical change of
reference-frame must rebuild from the beginning by as many
steps as are necessary. Let us exemplify how contrasts appear,
by taking (v) as the vector of equation (137) and placing the
result alongside equation (112), from which (v ) has been removed
by the supposition of a common origin, and in which, for closer
parallelism, we have substituted for (v') in terms of (v). On
one hand we find
v= ( Y xv) +v (m) ; (139)
and on the other
v = (Y x r') + 2(t x v) - (Y x (Y x r')) + v'. (140)
It is evident how the latter equation has accumulated compli-
cations in its two steps that we followed earlier, and that the
last terms in the two equations are not reduced to equality even
by making (Y) constant.
87. With this exposition accomplished, of the consequences
for free vectors and their first derivatives of their inclusion in
Reference Frames 105
plans of shift, we can proceed to add for localized vectors those
supplementary particulars which the localizing factor makes
necessary in relations like
(rxV) = (r xV) + (r'xV), (141)
when account is taken of the change in (r ) due to shift of the
comparison-frame into some new but permanent configuration.
This allowance is obviously required in order to complete the
details for the effective momentary replacement of (O, XYZ)
by successive members in the group (0', X'Y'Z'). And it is
most easily disentangled from other elements, by using that
superposition applying to similar cases which was indicated as
far back as section 67.
Using the temporary notation
M^(rxV); M' = (r'xV); (142)
the special question that concerns us here is the relation between
(M) in the standard frame and (M')> the latter quantity being
expressed under the guidance of ideas that it will be well to
make quite explicit. First, the vector (V) enters both products
invariantly; and secondly, its total time-derivative appears
without distinction in both, because changes in (i'j'k') being
now put aside in order to consider changes in (r<>) alone, the
corrective term of equation (137) disappears. But thirdly, with
(Y) dropped from the list of section 78 for the reason named,
(r') becomes a function of the two variables (r , t). Then its
exact differential is for the present shift
dr'=^dr + ^dt; (143)
CT() C/tr
and if this is timed to march with the actual changes during (dt)
we get
^L'-^^f^ ^2 = ^
dt ~ <9r dt + dt ' dt = f ' dt ~ *'
106 Fundamental Equations of Dynamics
the last equality having the same validity as in equation (124).
Hence
M' = x y +( r 'xV) = (( r_ fo ) xV ) + (r'xV); (145)
M = (f x V) + (r x V) = M' + (f x V) + (r x V). (146)
Consequently, though (O') coincides with (0), if there is dis-
placement of the former with shift rate (to) the values of (M)
and (M') as defined will still differ by the term (t x V).
We may restate the last equation by arriving at it through
M - M' = (r x V); M - M' = (f x V) + (r x V), (147)
if that is deemed a sufficient analysis of the conditions for the
differentiation; and there is precedent for calling (M') the
moment of (V) for a moving base-point. It is only iteration
here, however, to make the comment that the directer thought
holds in view the stationary points (0'), for which the coincident
moving point serves as marker at beginning and end of the
interval (dt).
Let us make application of this development to moment of
momentum and its derivative, as being the localized vectors
among our fundamental quantities. We are still confining
attention to shift of origin alone; and we shall not go beyond
the expressions for the representative particle at the center of
mass. Write then
H = (f x Q) = (r + ?') x Q = (r x Q) + H (0 ' ); (148)
M' = ((v - to) x Q) + (f x Q) = H (0 ',;
and reduce by omitting the product of colinear factors. But
for the moment about (O') of the force measured in the standard
frame we have
M(o') = f' x Q = H( ') + (f x Q), (149)
Reference Frames 107
which thus replaces with these conditions of shift the relation of
equation (VI).
88. For establishing the theorem of equation (137) and pre-
senting its bearings and a few of its consequences, reliance has
been placed almost exclusively upon the vector algebra; yet
those ideas were manageable to the other algebra also, though
it cannot fail to be apparent how much the absence there of
direct indication for orientation renders the operations in
matters like these more cumbrous, and the expressed results less
perspicuous. If, therefore, it seems profitable to go over part of
that ground in terms of the older method, that is not at all
with wasted effort upon verification, nor in order to gain reward
in fuller insight, except as seeing the cross connections is likely
to prove instructive. But coordinate algebra is indispensable
for calculation; transition to more succinct treatment, where
it can finally displace the older method, is still in progress, which
is keeping some comparisons temporarily that will fall away
later; and moreover, the next chapter is concerned with coordi-
nate systems as its chief topic. Consequently in preparation
for that material and for these other reasons, it seems well to
put in a link of connection; we shall, therefore, proceed to parallel
section 85 with the algebraic equations that offer the same
meaning under other forms.
It is unnecessary to carry a separation of origins into this
development, because as we have noticed repeatedly its effects
are in themselves easy to record, and are cared for, completely
by uncomplicated superposition. Thinking of (X'Y'Z') and
(XYZ) as having common origin (O), (x'yV) and (xyz) are, in
the first instance, the coordinates of any point (Q). But we
can draw advantage in two ways from previous experience;
first, (Q) can locate a representative particle of finite mass as
well as one mass-element of a body, and secondly, (x'y' z/ ) and
(xyz) can be made to denote the projections of any vector (V)
108
Fundamental Equations of Dynamics
with base-point at (O), by extension of their relation to the
particular vector (r) that is now identical with (r'). Unless the
contrary is said explicitly, (V) is to be regarded as determined
in the standard frame (XYZ), and introduced invariantly into
any connections with (X'Y'Z'). This vector can be regarded
as localized at (O) either by its property as a recognized free
vector like (Q) and (R), or by a convention agreeing with its
nature in cases like the rotation-vector (<o) and its companions
(cb), (H), and (M) when pure rotation about (0) is supposed.
The symbols are to be endowed with the wider valid meanings in
the equations constructed according to the adjoining table that
shows the direction cosines of the relative configuration.
X'
Y'
Z'
with
ll
m,
ni
X
1 2
ni2
n 2
Y
I
m 3
n 3
Z
89. The usual transformation equations when made explicit
for (xyz) are
x = lix' + miy' + niz', "
y = I 2 x' + m 2 y' + n 2 z', V (150)
z = I 3 x' + m 3 y' + n s z'. .
And the companion forms derivable by an elementary process
are
x' = hx + Uy + Uz,
y' = mix + m 2 y + m 3 z,
(151)
z' = nix + n 2 y + n 3 z.
Together these are known to depend upon or to express the
mutual relations of projection between two sets of components
of the same resultant vector. When the direction cosines are
Reference Frames 109
invariable, the correspondence with constancy of (i'j'k') is evi-
dent, and the same mutual relation runs on into all the deriva-
tives, giving invariance whose obvious details need not detain us.
A change of configuration, however, makes in general all the
direction cosines vary, and there the same alternatives recur
that were brought out in sections 78 and 82. One of these will
make (x', y', z') each a function of three independent variables
that are time and two direction cosines, the third of the latter
being removed by a standard connection like
li 2 + 1 2 2 + 1 3 2 = 1. (152)
The second point of view will set time in its place as the one
independent variable of which all other quantities are functions;
but here it will be just as desirable as before to put into properly
conspicuous relief the modified relation of time to variables
like (x, y, z) and to others like (li, la, la).
90. Equations of the same type as
dx' , dx dy dz
ar" k S H 1 S' + li 5
can be read in the light of equation (123) ; and what remain to ex-
amine are the complete time-derivatives of the quantities (x'yV),
principally in order to detect the rotation-vector (y) of (X'Y'Z')
by penetrating its disguise of direction angles and their deriva-
tives. Adopting the fluxion notation, for ease in writing total
time-derivatives, we have first
x' = (l lX + I 2 y + I 3 z) + (I lX + I 2 y + I 3 z). (154a)
Note in passing, as consequences of equations (151, 154) that
may prove suggestive later, 1
_,
dx> l ~dtdx ~dx'
which are typical of similar relations running all through the
1 See Note 23.
110 Fundamental Equations of Dynamics
sets of equations, when we add to the value of (x') its com-
panions
y' = (mix + m 2 y + m 3 z) + (ri^x + m 2 y + m 3 z),
z' = (n x x + n 2 y + n s z) + (r^x + n 2 y + n 3 z). J
Concentrating attention upon the last groups in these equations,
because the effects of changing configuration appear exclusively
in them, and introducing the necessary direction angles in order
to prepare for the connection with (y), expand into the forms
[xdi sin ai + yd 2 sin 2 + zd 3 sin a s ]
- [x/3i sin /3i + yi8 2 sin /3 2 + z0 8 sin j8 8 ];
[xei sin ei + ye 2 sin e 2 + ze 3 sin e 3 j.
But the normal to the plane (X', X) must be the axis for (<ii);
and with the direction cosines of those intersecting lines given as
1, 0, 0, (X); l lf 1 2 , 1,, (XO; (157)
the direction cosines (X, AC, v) of the normal to their plane worked
out by the standard method gives
(156)
X = M = ~ -- , v = -: -- . (158)
sin ai sin ai
But as explained in section 46 the rate at which (X 7 ) is turning
about that normal must be the projection of (y) upon that line,
or equivalently,
di = XT (X ) + /*T(y) + viw, (159)
from which follows
di sin i = 7 (y ) cos a s 7 (16) cos a 2 . (160)
Proceeding similarly with the eight other terms which complete
the group of that type in equations (154), it is seen after simple
reduction that they make up in the first, second and third equa-
tion respectively
~ (T(y')Z' - 7(,')yO; - (7( Z ')X' - 7(x')Z'
Reference Frames 111
Since the first members of those equations correspond to the
total derivatives of the tensors obtainable from equation (125),
we find after orientation and forming the vector sum that equa-
tions (154) yield consistently with equation (137)
Von, = V - (Y x V), (162)
on our understanding about the broader meaning of (x'yV)
and (xyz).
It is left as an exercise, modeled on the above plan but con-
tinued into the formation of second derivatives, to reach by the
algebraic routine the coordinate equations which together repre-
sent the result recorded in equation (112), if we suppress there
all terms depending on a separation of origins. Where the
quantity (Y) occurs in executing this, it is of interest to realize
what has been alluded to elsewhere; that (Y) and (Y) may be
connected with either (XYZ) or (X'Y'Z'), since the difference
term in equation (162) is zero when (Y) is (V).
CHAPTER IV
THE MAIN COORDINATE SYSTEMS
91. The standard frame itself has an additional office of
providing a coordinate system that is basic in certain ways, and
that is in fact tacitly utilized for the semi-cartesian expansions
in terms of (ijk), both in immediate relation to vector quantities,
and for the expression of constituents in work, kinetic energy
and power, where vector factors occur in scalar products. To do
these things has become so much habitual or even instinctive
that we learn with some surprise how Maclaurin is given credit
for invention here, as Euler is for inventing the concept of fluid
pressure, which at this date might also seem part of external
nature.
The standard frame, too, has one lead in advantage over other
resolutions through the unqualified permanence of its origin
and of its unit-vectors, which enables us to submit its tensors
unhesitatingly to algebraic operations, and pass over to vector
algebra by merely supplying the ellipsis of the unaffected ori-
enting factors. The disturbing influences in other combina-
tions, where (r ) and (i'j'k') make more caution advisable, have
been forcing themselves upon us repeatedly. But as we have
seen illustrated for mean values, and as is not unusual, the
presence of such desirable elements as we find in the standard
frame may be also a drawback. Within the complete projection
on a standard axis, distinctions of source in changes of magnitude
or of direction may be lost, that are vital in the vectors that play
a part. The net force parallel to (X) and its work, if written for
a particle
X = m^; W = /Xdx; (163)
112
The Main Coordinate Systems 113
hide, in the first, the fact that normal force (N) and tangential
force (T) are coalescing in the one sum, and in the second, that
part of this work is illusory in so far as the projection of (N)
enters the sum (X), and does work in the algebra though not in
the mechanics. At one other point we have been enabled to
compare the principal axes of inertia with (XYZ) and ascertain
that all advantage does not lie with the latter, for expressing
compactly either the scalar energy or the vector force-moment.
And these considerations, in sum, may justify us in leaving the
resolution into constituents according to the standard axes to
one side, except where we touch upon it for some special con-
nection. Then we are free to devote detailed attention to other
coordinate systems that are chiefly current, and make due
analysis of their intention and of the scope of their success.
It seems quite enough therefore if we collect here the indicated
partitions for (XYZ) that are reasonably self-evident rewritings
of the totals to which the preceding text has given most weight:
Q = iZ/ m xdm + jS/ m ydm + kS/ m zdm; (164)
H = iS/ m (yz - zy)dm -f- jZ/ m (zx xz)dm
+ k2/ m (xy - yx)dm; (165)
E = |2/ m x 2 dm + S/ m y 2 dm + 2/ m z 2 dm; (166)
R = i2/ m xdm + j2/ m ydm + kS/ m zdm; (167)
M = iS/ m (yz - zy)dm + jS/ m (zx xz)dm
+ kS/ m (xy - yx)dm; (168)
P = 2/ m xdX + 2/ m ydY + 2/ m zdZ. (169)
It will be found profitable to compare equations (165) and (86) ;
also equations (166) and (81, 88), including the comment preced-
ing the latter. Since the first three equations in the above group
are mere expansions of the forms in section 15, they have the
same scope as those. Similarly the validity of the last three is
114 Fundamental Equations of Dynamics
coextensive with that for equations (16, 17, 18) of which they are
the expansions.
EULER'S CONFIGURATION ANGLES.
92. Because it deals directly and exclusively with the recurrent
element that is found at the root of so many particular results,
we shall take up next those orientation angles for specifying
configuration which were devised by Euler and by custom bear
his name. They have not yet been displaced from a conceded
position of value in use for their purpose. There is an added
reason for giving these angles proper discussion in that the
expression of them as vectors has scarcely been attempted; we
find their connections with other specifying elements almost
exclusively in the form of purely algebraic equations. It is a
curious fact that angle in prevailing practice has not arrived at
legal recognition as a vector, though the vector quality of its
first and second time-derivatives, angular velocity and angular
acceleration, was announced and employed a number of years
ago. So we need to do something consciously toward incor-
porating angle-vectors into our scheme of treatment on a parity
with other vector quantities, in order that real symmetries of
relation may not be seen distorted.
Supposing that one end of a line (r) is fixed and that it moves
into a new position, its second configuration in relation to its
first can be given by a vector-angle normal to the plane of the
two positions. This vector is axial, and related to an area with
duly assigned circulation; and the area is in the plane located
by the extreme positions of (r), its magnitude being twice that
of the sector of the unit circle limited by those positions. But
such a direct representation of this total would be no more
convenient for use in all cases than other resultants are, so its
projections according to Euler's plan are substituted, which
amounts to giving the latitude and the longitude on unit sphere
The Main Coordinate Systems 115
centered at the fixed point or origin (O), in which (r) cuts that
surface. Assuming next that (r) is a definite line of a rigid solid
that is limited to pure rotation about (O), a third angle added
will enable us to complete the description of a new configuration
for the solid, and this last angle will denote a rotational dis-
placement about (r). We shall follow usage in assigning the
symbols (ft) to the latitude angle, and ($) to the longitude angle,
while () is added for the rotation about (r) ; it remains only to
agree upon zero values of the three angular coordinates. It
suits our purpose in its general course better, to think in terms
of a displaced rigid cross (X'Y'Z'), which may here be made
equivalent to the rigid solid named above, and then coincidence
of (X'Y'Z') with (XYZ) yields the natural zero. We identify
(Z) with the earth's polar axis in its relation to latitude and
longitude.
93. Beginning with resultant angular displacement (y) at zero,
and (X'Y'Z') coincident with (XYZ), let the plane (Y'Z')
separate from (YZ) by angular displacement (t|r) about (Z), in
which that vector angle must then fall. Next let angular dis-
placement (O) occur about the displaced position of (X'), in whose
line therefore it must lie as a vector angle ; and finally let (X', Y')
turn with angular displacement ($) about the final position
of (Z'), with whose line this third vector angle must then
coincide. To make the conditions standard, (tjr, d, $) are
all to be taken positive by the rule of the right-handed cycle.
The order of the three displacements has been chosen so that
each is made about one of the three axes (X'Y'Z') as found at
the beginning of that stage. It is verified without difficulty
that the summed projections on (XYZ) are
Y(x> = i(# cos ^ + <f> sin # sin ^
Y(y; = K# sin 4> v sin & cos ^); (170)
<p cos #.
116 Fundamental Equations of Dynamics
And if we resolve on the final orientations of (X'Y'Z'), those
projections are
Y(x') = i'(# cos tp + ^ sin & sin <p);
Y(y') = j'(-& sin <p + $ sin tf cos <?); L (171)
Y (I ') = k'(p + ^ cos tf).
These two sets of projections are orthogonal; but if we state the
supposed displacements directly, and let (tjri, #1, represent
unit-vectors agreeing with those suppositions, the set is oblique
to the extent that the angle (ijri, $0 is (ft) and not in general a
right angle. We add accordingly,
and have secured three equivalent forms of expression for the
resultant angle-vector (y). Observe also the differences among
the three in regard to the unit-vectors; (ijk) are permanently
oriented, (i'j'k') are capable of displacement by rotation, for they
remain orthogonal, but (ifci, fti, $1) must be considered indi-
vidually. It is seen, if we hold definitely to the terms of the
description, that (tjri) is of permanent orientation in (Z), that
(fti) depends for orientation upon (i|r), being always normal to
the displaced position of the (Y'Z') plane, and that ($1) depends
similarly upon both (ijr) and (ft), because the () displacement
begins where the second stage leaves off. All three quantities
(i|r, ft, $) are rotation-vectors applying to the axis-set (X'Y'Z')
as representative of a rigid body, and standing to the changes of
direction of individual lines in the relation established by sec-
tion 46. This needs to be borne in mind if any question should
be opened about changing the sequence of the three steps, so
that (ft) and ($) though equal to their first magnitudes are con-
nected as vectors with different axes.
The above forms of statement are mathematically on the same
footing as a means of determining (y), but there can be no real
The Main Coordinate Systems 117
doubt where the preference would fall on the score of ease in
application or execution, when the three plans are compared.
The second is especially intricate because its projections are
associated with that very terminal configuration of (X'Y'Z')
which it may be the object to locate, but which must somehow
become known before the scheme can assume full definiteness.
It should be inserted however for the sake of its subsequent
uses.
94. The emplojTnent of the standard angles (ijr, #, $) is not
confined to expressing configurations, and is therefore not
exhausted in equations (170, 171, 172). Indeed the primary
service of Euler's so-called geometrical equations has begun at
their developed connections with the rotation-vector or angular
velocity, and found a natural continuation in dealing with
angular acceleration written (Y) or (<b). As we now undertake
to make those connections clear, combinations will occur at
first or in later application, that make it advisable to retain (Y)
and (Y) for use with comparison-frames like (X'Y'Z'), and let
the meaning of the parallel quantities (<>) and (<b) refer ex-
clusively, as in sections 45, 55, 62 and 63, to a rigid body's rota-
tion, either about its center of mass or about some fixed point.
To maintain this consistent distinction will avoid confusion where
both pairs of elements are presented in the same inquiry.
The expressions for (Y) that we have just obtained are con-
trived to show its value at the advancing front of a progressive
angular displacement to which (i{r, #, $) can be considered to
belong. Consequently it is adapted to differentiation, with a
view to exhibit either a systematic succession of partial differ-
entials or simultaneous time rates in a total derivative; and
previous discussions have laid a foundation for interpretations
leading in both directions. In the first instance we are most
nearly concerned with the derivation of (Y) from the three several
equations (170, 171, 172) and the collation of results with sec-
118 Fundamental Equations of Dynamics
tion 85 as bearing upon the current algebraic forms. And because
this has some little flavor of revising the latter, the fuller infusion
of vector peculiarities into these matters having not yet worn off
its novelty, there seems to exist a stronger reason for detail,
than the mere arrival at conclusions for handy use might
demand. 1
95. As in similar comparisons elsewhere, the (ijk) projections
furnish reliably through pure and total tensor differentiation an
unquestioned standard to which alternatives must conform if
correctly formulated. So the first straightforward step is to
employ equation (170) in this test; and we prepare the way
with the expansion
P/d# d<p . \
Y = 1 I TT COS ^ -f- TT Sin & Sin \f/ I
L \ dt dt /
(di/' d$
# sm \l/ + <p cos & sm \1/ -r-
dt at
+ <p sin & cos y/ -rr
dt
)]
(173)
(ft COS ^ TT <P COS # COS ^ T
dt dt
d^\1
+ <p sin & sm ^ -7- I
But we have been remarking from section 79 onward that the
partial time-derivatives in equations like (171, 172), when the
unit-vectors are made variables, must reproduce the standard
frame values obtained through (ijk). Let us accordingly write
See Note 24.
The Main Coordinate Systems 119
out those two sets of partials and proceed toward comparing
them with equation (173). Observing that the conditions of
the differentiation exclude trigonometric functions of the angles
from varying, though they permit the angles as magnitudes to
change, we find
It (i/ ' j/ ' k/) = *' (f cos * + It sin * sin *
- (174)
jj. \ri> "*j vu
The value directly apparent in the last equation can be noticed
by inspection to agree with that of the equation preceding, if we
assemble mentally from the latter the items falling respectively
along (ii, Oi, 1). And this coincidence is next to be recognized
similarly in the first groups marked off under (i, j, k) in equation
(173), with the single variation that the latter appear as total
derivatives of the angle magnitudes. The patent conclusion is
that proper allowance for the difference between these total and
these partial time-derivatives must exactly offset the remaining
groups in equation (173); and that outcome might be accepted
on the fair ground that it harmonizes with equations (126, 131),
without going further. Yet the completed analysis of how
that compensation is in fact brought about here, has an im-
mediate bearing and interest that justify setting down its several
steps.
96. The last groups of terms in equation (173) can be brought
together and rearranged so that they are identified as the vector
products to which they are equated below:
9
120 Fundamental Equations of Dynamics
TT sn \f/ + cos ^
J n
+ <p -T- (i cos & sin ^ j cos # cos ^ k sin
Qt
TT
Qt
(176)
sin # cos i + j sin # sin ^)
The verification as regards magnitudes, directions and order of
factors in the vector products is ordinary routine devoid of arti-
fice, due regard being paid to the specifications of direction in
the sections immediately preceding. The character* of the
second member is plain: it consists of allowances for changing
directions of the two unit-vectors (#0 and (1), the former being
affected by the turning about (i|ri), and the latter by the two
turnings about (1^1) and (#1). It is instructive to notice that
these individual consequences of the changes in the unit-vectors
preserve their type and enter singly in parallel with the develop-
ments of sections 47 and 80, although there is here no common
factor, the rotation-vector, related equally to all three unit-
vectors (I{TI, #1, $1). This line of attack has been adopted partly
in order to extend in that direction our earlier proof.
In preparing to demonstrate that the differences between
(d#/dt) and (dtf/dt), (d<p/df) and (dp/dt), exactly nullify the
second member of equation (176), it is most direct to start from
explicit values of (\f/, &, <f>). By a process of elementary elimina-
tion applied to equations (170) it follows that
cost?
=: T(z) ~ ~ ^ 7(x) Sm ^ ~" 7(y) C S '
7(x) COS ^ + 7(y) Sin
1
Sill
sn " T(y cos
(177)
The Main Coordinate Systems
121
It is to be remarked as regards these equations that in order to
arrive at their partial time-derivatives, we must include as
variables only (7(x>), (7( y >), (T())> and for the total derivatives
we must include also all the other factors as functions of time.
It is therefore possible to write these indications of the differences :
dt at d\f/
dt? aj? _ at?
dt at d\f/
d<p dtp dtp
dt at? dt '
(178)
Evaluating the second members from equations (177) and finally
adding the orienting unit-vectors we derive these expressions:
d- cost?
- *fei - (7(x) cos ^
ty
sin
dt
(cos t?
~sin~^
dt
at?
dt
dt
sm ^ " T(y) cos
7(y) C S
COS
dt?
dt
dt?
dt
sm
dt
/
= * x V ~
COS t?
dt?
T(y) C S * dt
cost? dt?
(179)
122
Fundamental Equations of Dynamics
After forming them into three groups as shown below, they can
be recognized as constituting the vector products to which they
are severally equated ;
cost?
- &i<p- sin &= - --
(180)
The first quantity of these three is known by the first parenthesis
to be perpendicular to (tjri) in the plane of (ijri, $1) ; so the second
quantity is perpendicular to ($1) in the same plane; and (<h)
is by supposition normal to that plane. The directions match
the order of factors and the signs.
97. When the established conclusions of equations (176, 180)
are united with what was found to be true on casting up into a
vector sum the three first groups in the coefficients of (i, j, k),
equation (173), the registration of all these connections yields
the continued equality
. ( d$ dp \
y = i I cos ^ + sm # sm ^ I
d& dtp
sin ~ ~ sin * cos
,( w , w \
J I - ^ sm * + ^T sm * cos ^ )
dt + a! cos
(181)
The Main Coordinate Systems 123
tfCh x tO .*.
The last member is a specially plain demand of the vector algebra,
in order to reconcile the value of (y) obtained by means of
(XYZ) with the terms of equation (172) and its vector angles,
and uphold the condition for invariant representation of (f)
as the angular displacement proceeds. With this invariance put
beyond critical doubt such vectors as (y) take their place under
the procedure of equation (137), and we have detected here the
earmarks of an invariant shift. A closer superficial agreement
with that equation results from the coordination of derivatives
calculated from equations (170, 171), because the axes (X'Y'Z 7 )
remain orthogonal and rotate. With some watchful avoidance of
confusion in the notation, the reasoning of section 80 can be
duplicated, and the result confirmed without difficulty,
* = dt ( ' i ' T(x ' ) + ^ T(y/) + k/7(z/) )
+ (r x r ), (182)
where (Y) in the vector product must denote the shift rate for
(X'Y'Z'), and the rest of that member shows the type of (V( m) ).
We do not need now to transcribe the details of that develop-
ment, with a less particular value for the shift rate.
98. Having made the beginning in section 93 with angular
coordinate which may be placed in parallel with coordinate
lengths, the above relation that introduces an angular velocity
124 Fundamental Equations of Dynamics
is liable to the same sort of double reading that was insisted
upon in section 81, so that the change of reference-frame for
angular velocity would also come to the front. Then using the
third member of the last equation for illustration of a more
general case, its first group can be said to present angular velocity
relative to (X'Y'Z'), while the vector product added transfers
correctly to (XYZ) as a standard. If this second branch of the
idea is before us, a continuation of it in close likeness to the
working out of consequences into equation (112) suggests itself
naturally, in order to make a transfer between reference-frames
that covers angular acceleration, as the previous equation pro-
vided for such a change in respect to linear accelerations. But
that general provision will be omitted, with the intention of
considering any special instance under its plan in the light of
its own circumstances; and what attention is now to be given to
angular acceleration will enter with the repetition of the one-
step shift process, in which the original vector (V) is an angular
velocity, and the derivative that appears in particular to replace
the general derivative (V) of equation (137) is an angular acceler-
ation, with the one standard frame retained, and no departures
from invariant values finally tolerated.
That policy meets the requirements most frequently made in
this field, and indeed the material that has grown to be classic
and devoted to the relations of rotation-vectors and their deriva-
tives to dynamical quantities, expressed especially by means of
Euler's angles, marks its initial stage at the point that we have
now reached. One feature of it, that we have once alluded to,
is letting angle figure as an algebraic magnitude, but constructing
a sequel where its two derivatives become vectors, effectively or
with full recognition. It cannot be surprising, therefore, that
those distinctions in respect to angular quantity, between its
partial and its total time-derivative, nowhere need to appear
in the classic equations; though we have been compelled to
The Main Coordinate Systems 125
give them weight in the interest of correct work. Because both
compensating elements in equations like (173) have their source
in orientation, a view that excludes orientation needs neither;
and the one magnitude derivative with respect to time that is
retained may within certain limits raise no issue whether it is
partial or total. There is however one place where comment
has been the habit upon something of defect in the algebraic
linkage, and where it is interesting to discover that the concept
of vector angle does a little to make a better joint. We shall
attempt to dispose of that minor matter in this pause between
two steps of the more important progress. 1
The comment in question hinges upon equations that the
algebraic methods have always written equivalently to
/ t * * t\ *
i -f = -r- cos <p + -77- sm $ sm <p;
d? . d^
j'.^. = - sm tp + TT sm t? cos ^;
d<p d\(/
k -t = JT + zrcos??;
(183)
and where our sequences of thought have caused the substitution
of time partials everywhere in the second members. If we pick
out one equation for a sample, multiply by (dt) and write
(i'-y)dt = d# cos <p + d\}/ sin & sin <p, (184)
the usual and perfectly true remark about it and its companions
is to this effect: The second members not being exact differen-
tials under the ordinary test, because the equalities are not
satisfied that would give for instance
*\ f\
(cos <p) = (sin & sin <p), (185)
i See Note 25.
126 Fundamental Equations of Dynamics
there is some drawback upon using the first members. But if the
vector plan retains the total derivatives in equations (183) and
completes them, equation (184) becomes, as we have seen,
(i'-y)dt = d$(cos <p + ^ cos $ sin <p)
sin <p) + d<p( & sin <p + ^ sin # cos <p), (186)
in which the coefficients of (d$, d^, d<p) do make the first member
an exact differential by conforming to the standard rule, as direct
test verifies. That particular drawback was removed by using
vector angle in deriving the rotation-vector, and by aiming in
our calculus deliberately to preserve the exact differentials that
occurred naturally.
99. For the kind of inquiry that comes next in order, rotation-
vectors in the standard frame are an assumed basis in the state-
ment, being either given outright or brought within reach by
such data related to Euler's angles as the foregoing sections have
set forth. The undertaking looks toward expressing angular
acceleration-vectors for the standard frame in terms of the same
angles (t|r, #, $) and consequently in connection with some
auxiliary frame like (X'Y'Z'). In its main outline this must
stand as a parallel illustration of the method introduced before;
but in order to vary from mere repetition, let there be one
rotation-vector (<o) applying to a rigid body that is in pure rota-
tion about the origin (O), and a second (y) for the axes (X'Y'Z'),
with whose aid (<b) is to be determined through its projections
upon them. We shall choose special assumptions, that will be
found profitable because they anticipate one set of data met in
real requirements of investigation. Let that definite line of
the body, which is to have the angular coordinates (ijr, #) and
thus specify those elements of the body's configuration, always
coincide with (Z'); and to complete the assignment of relative
configuration for body and axes, let ($) be permanently zero for
the latter. Therefore (Y') is contained permanently in the
The Main Coordinate Systems 127
plane (Z', Z), and (X') in the normal to that plane. Dis-
tinguishing the angles applying to the axes as (tf, #', $') the
conditions are
%' = fy &' = O; *' = 0; 6 (any value). (187)
100. The rotation-vectors (w) and (y) are now to be expressed,
but that cannot be done by borrowing the forms from sections
95 and 97. For it is essential to the present circumstances that
the sets of projections of each rotation-vector must give that
quantity invariant^, as before it was exacted that the angle
(Y) should be so expressed by equations (170, 171, 172). For
every range in this use. equation (134) is to be made funda-
mental and characteristic. Going to one root of the matter in
equations (111, 116), and holding to the leading thought of
section 86, it becomes formally clear that no term like (y x V)
of equation (137) can appear in forms adapted to the new inde-
pendent start. And in reason it is convincing that projection
at the moment is indifferent to past and future, and its results
must be mathematically independent of a continuing process to
which it is indifferent. All this fits perfectly our conception of
each set (X'Y'Z') as fixed, and (y) as a shift rate among the
fixed sets. Bringing to equation (173) the modifying idea that
(Y) equal to zero must accompany the projection upon the
individual set of axes for the epoch, we find first that the second
groups in the coefficients of (ijk) drop away because they repre-
sent projections of a term like (Y x V), and secondly that the
difference between total and partial time-derivatives disappears
in view of equations (178, 180). To be sure this detail is only a
roundabout consequence of discarding at the one projection that
which belongs only to a unified series of such projections as a
whole; but it has bearing in dispelling lingering obscurities on
the formal side of these matters. The point would not need to
be labored so, were not misapprehension fostered by the mis-
nomer reference to moving axes in speaking of them.
128 Fundamental Equations of Dynamics
This is preface to writing the values
dt
dt"
t-*S + *5f! < 188 >
in order to proceed from them to the value of (<o) that is con-
nected with the projections of (w) on (X'Y'Z'). It seems worth
noting that these may be corroborated by considering the par-
ticular configuration when (X'Y'Z') fall in (XYZ), for which of
course equality of projections must ensue. From equation (173)
we see for that case and for the projections of (to),
..,
dt ^dt
and =
(189)
It is true that the cancellations of terms arising from the type
(Y x V) now follow from (y) being zero, but they show con-
sistency in the final outcome. The sum in (Y( Z )) is contributed,
part by turning of the plane (Y'Z) about (Z), and part by turning
relatively to that plane about (Z') coincident with (Z). Finally
we can summarize in a brief rule the office of the two derivatives
in connections like the present one: The partial time-derivative
of the tensors enters where projection has preceded differentia-
tion, and the total derivative where differentiation has preceded.
101. By projecting the rotation-vector (o>) upon (X'Y'Z') we
find
dt
dt
(190)
the tensors being comprehensive or general values as explained
The Main Coordinate Systems
129
in section 78, and therefore open to differentiation, whose execu-
tion yields .
dt 2
sm
dt dt cos
(191)
d
dt (a)(5/)
-
dt
^ '(<'>) = dl + d* cos * " dT dt sin
The differentiation of equations (190) needs for its completion
the terms introduced by changes of orientation in (i'j'k'), which
are
. d?? d</<
d??
\ 2
j si
JO sin ^ + (*! x j') - sin 0;
dt +dt cos
dt
+ (*]
d?? d^ dt? d'i'
TT -VT + TT TT COS
dt dt dt dt
(192)
Next resolve the vector products into (X', Y', Z') and assemble
the terms for each one of the axes, which shows for the results
when reduced by some cancellations
130
Fundamental Equations of Dynamics
.,dV
= J
dt dt"
_
dt dt COS ?
(193)
In making -the resolution the components of the vector products
to be used are shown by
= $1 ( " 9 + ^ T cos # j r sin t> I .
\ dt dt w at at /
= cos &
= *i; ii x
sn
= *i sn
x = i cos
*i x ^i = -
(194)
Having obtained by these operations the projections of (<b) for
the standard frame upon (X'Y'Z'), as corrected for the assumed
shift of the axes, the total (<b) given by the vector sum of the
second members is easily seen to be
dt? d<p
dt dt
di?
(195)
And this last form of the value for the angular acceleration of
the body is finally to be compared, on the one hand with the
result of differentiating directly
dr?
(196)
and on the other hand, with the standard relation in equation
(137). The first of these comparisons is no more than a matter
of inspection, because the derivatives of the tensors appear
immediately, and the known changes of orientation for (ijri, 0-j, ^)
are exactly accounted for in the vector products of equation (195).
In order to carry through the other comparison we need for
The Main Coordinate Systems 131
(V( m )) the derivatives of the tensors that are already recorded
in equation (191), and whose vector sum can be thrown into the
form, when the parts are duly oriented,
*-*>5? + *'5F + *>3?+tt><'>|j;.
To this must be added
whose expansion reduces to
and confirms through the sum of equations (197, 199) the former
value of (o>). Notice the difference in the segregation for the
two groupings, by which the same term can be attributed at will
to change of direction or of magnitude.
The components of (<o) in (XYZ) are obtainable in the forms
. / dt? d<f> . . \
tooo = i I TT cos \I/ T ~rr sin $ sin \f/ ] :
\ dt dt /
. /d$ . d<p . \
= j I 3 sin \f/ 3 sin r? cos ^ I ; r (200)
\ at at /
d^ d<p
through which another plain road is opened to determine (<b);
but we shall not go further here than to indicate it.
POLAR COORDINATES.
102. The system known as polar coordinates is a fitting sequel
to what has just been done, because Euler's angles that we have
denoted by (t, #) are universally employed to orient the radius-
vector (r) whose pole is then taken at our origin (0) . The angle
132 Fundamental Equations of Dynamics
($) is obviously superfluous when we are concerned with one
line only and not with a body, even when (r) moves in three
dimensions; and when a limitation to the uniplanar conditions
is imposed the pole is most often located in the plane of motion,
and then of the three angles (t!/) alone needs to be used. We
shall guide the development toward the relations for three
dimensions, and afterwards call attention to some briefer state-
ments for the uniplanar case. .
If we write the radius-vector (r) as the product of its unit-
vector (FI) and its tensor (r), according to one normal scheme of
the vector algebra, the time-derivative (f) takes on the form
f-rxjj+te, (201)
with unforced separation of the entire directional change from
that which refers to the algebraic magnitude. By means of the
results now at our disposal, the vector (y) in application to the
single line (r) would lead straight to the expression for the
velocity of (Q) at the extremity of (r),
v = r 1 ^+(txr) =r 1 ^+(thxr 1 )r^+(# 1 xr 1 )r^. (202)
From the second member, we infer at sight the truth of one usual
statement about (v) : That it includes simultaneous motion on a
sphere centered at the pole of (r), and growth of (r) in length.
So long as we think strictly in the terms indicated, there is no
rotation according to our use of that word; we deal with (y)
merely as the angular velocity of the one line. But when the
third member of the last equation is drawn in, the set of axes
(X'Y'Z') as laid down in section 93 reappears, since the three
parts of the velocity constitute always an orthogonal set, of
which (r) itself would be (Z') in our adopted convention, coin-
ciding with (Z) for zero values of (i{r, ft). The completed con-
The Main Coordinate Systems 133
sistent identification of axes and their true rotation-vector gives
,,dr .,
V(z/)=k 5 V(y/)=;l " rSm
Vfx'1 =
cty dtf
TT + #iTr ; = permanently.
(203)
It is self-evident that these three projections are an invariant
equivalent for (v), because they are in their source only the
three parts of (f) in the standard frame. But we can also repeat
the remark attached to equation (138), and enlarge it in the
direction of presenting these polar coordinate relations for velocity
in the light of a narrowly specialized instance within more elastic
conditions.
Instead of binding (X'Y'Z') to coincidence of (Z') and (r),
let the axes rather move about the origin (O) as allowed by any
general value of the rotation-vector (y). The configuration of
(r) in the frame (X'Y'Z') will be shown generally by
r = i'x' + jy + k'z'; (204)
and for those suppositions the general values of (V) and (V( m ))
in equation (137) will assume the form
The effect of that particular choice for the rotation-vector in
equation (203) is then put clearly in evidence: the velocity of
(Q) at the extremity of (r), but reckoned relatively to the frame
(X'Y'Z'), is thrown exclusively upon the axis (Z'), while (x', y')
remain permanently at zero, and the term (Y x r) is left to bring
134
Fundamental Equations of Dynamics
in all of both components that (v) shows parallel to (X 7 ) and
to (Y'). Or in the alternative reading, the correction for shift of
orientation being perpendicular to (r), it is segregated com-
pletely from the only change in tensor magnitude that is allowed
to become realized in (X'Y'Z').
103. The natural order proceeds next to take up, with polar
variables as instruments, the task of expressing the polar com-
ponents of the acceleration with which (Q) moves relatively to
the standard frame, and which can be determined otherwise,
as we know, by projecting the resultant (v) upon tfre directions
of (X'Y'Z') at the epoch. However these projections may be
written originally, the translation into functions of (r, i|r, #) is a
matter of algebra only. Leaving that method aside, the details
will be worked out in two ways, both moving with reasonable
directness toward the end in view, and each having its own
interest through the vector algebra of it. Let us carry out
first the application of equation (137). It gives
dr d^ di/ 1 d$ - \
Trsin # + -j- -7- r cos # I
dt dt dt dt /
(206)
As a help in expanding the second equation these relations enter:
(*h x TI) = &i sin #;
ifci x (ifci x TI) = i' sin & cos & TI , .
ti x (^i x TI) = ^i cos ft] (ft i x TI N *'' '
&i x (t|ri x r : ) =0; di x (<h x TI) =
The Main Coordinate Systems
135
Summing the items in their proper orientation, the polar com-
ponents of (v) are found to be
t(*') = i'l r + 2
Vfv'l = l'
dtdt \dt
d^dr .
V \
I sin # cos # 1 ;
(208)
The second development picks up its thread at equation (201),
and differentiates that again as it stands; so the first stage shows
immediately
r^t^T^+Zi^+r*; (209)
and carrying out some of the indicated operations yields
f i = (t x ii) ;
r'i = (Y x ri) + (t x rO = ( T x r x ) + (y x (r x rO);
Y =
dt
'dt
with 1^1 constant, &i = -rr (^i x #0;
, dV / , d^ dt? \
TI x rO -^ + ^ (th x *0 ^ j- 1 x
(ir
Y x (Y x rO = [tti x (iti x rO] I -
(210)
10
136 Fundamental Equations of Dynamics
Substituting these values in equation (209), it is recognizable
readily with the aid of equations (207) that the results of the two
methods are in perfect agreement.
104. The adjustment of the foregoing analysis to the simplified
conditions of uniplanar motion, where the pole for (r) is taken
in the plane of the motion, will make (#) constantly a right angle,
so that (r) revolves in the equatorial plane of the sphere whose
polar axis is (Z). In adaptation to that case the velocity com-
ponents are
dr / d^\
v (z ') = = v (r) = ri^; V(y') = J'^r^ J ; v u ' } - 0; (211)
and the acceleration components become
/d 2 r
= V(r) = Tl
')<
\dt 2
(212)
v (x ') = 0.
Even on this simpler level, and after removing those complica-
tions which belong to the freedom in three dimensions, the
same feature remains prominent through all the results; in one
sense the idea of superposition fails. For though the resultant
velocity contains neither more nor less than the parts due to the
radial motion by itself and the revolution by itself, we cannot
build up in that fashion the acceleration (v) of equations
(208), nor yet of equations (212). In the latter, the second term
in the coefficient of (j') does not belong to the radial motion, nor
to the circular motion, but it appears only when these two types
coexist. And under the broader conditions, the coexistence in
pairs of the three component velocities asserts itself through
the terms in the acceleration :
dtdt' dtdl'. dtdt
The Main Coordinate Systems 137
In view of their obtrusive symmetry, it is somewhat surprising
that the force depending on the third of the group should have
invited and fixed nearly exclusive attention: it is the famous
compound centrifugal force with which the name of Coriolis has
been associated. 1
Approaching along the line now laid down to follow, these
terms can be traced intelligently to a common origin in the
nature of the coordinate system that is being employed; their
appearance is connected essentially with the changes of direction
peculiar to the descriptive vectors that are used. On that side,
the parts of the force that match such accelerations may be
declared mathematical, though it must be granted that they can
become sound physics too, whenever those descriptive vectors
are closely fitted to the physical action. In a centrifugal pump,
a force that goes with the coefficient of (j') above does work
and strains the structural parts. But the same term shows in
the algebra, when constant velocity is referred to a pole lying
outside the straight line path, although no net force at all can
then be active. It is also a significant fact that the factor (2)
in each case makes its appearance because two terms coalesce,
whose function is different in respect to the vector quantities
that they affect. It is half-and-half change of magnitude in
one vector and change of direction in a second distinct vector,
as our process of derivation demonstrates. So the force of
Coriolis cannot give a definitive account of gyroscopic phenomena
on the basis of an incident in the algebra; first, it must be
exhibited to correspond with traceable dynamical action. The
same lesson is enforced here as by the matters broached in
sections 35 and 57, of which the latter is peculiarly pertinent in
that it brings forward the idea that angular acceleration, and
therefore the coexistence of rotations about (ijji) and (<h) that is
characteristic of the compound centrifugal force, may come about
i See Note 26.
138 Fundamental Equations of Dynamics
in the absence of all force-moment, as a symptom that control
is absent, not that it is present and is producing these effects.
105. The general values of equation (208) cover as a special
case, it is plain, the condition that (r) shall be constant in length
which goes with a pure rotation about (0). Consequently if
we make that assumption here, the special value of (v) that is
obtained must be reconcilable with the determination made in
sections 54 and 101. Only the latter, in its turn, must be special-
ized for a point situated in its axis of ($1), which is now also that
of (ri). The notation in the two sections is consistent with the
same supposition about the rotation-vector (y) of (X'Y'Z');
and the axis (Z') is common to both inquiries. But it will be
observed that (<h) of section (101) is identified with (i'), and (#0
of equations (203) is paired with (j')j and hence a comparison
of results must adopt in correspondence
(i'); (JO; OO ; [Equations (193)]
(j'); (-!'); (k'); [Equations (208)]
in order to preserve the right-handed cycle.
If (r) is constant in length the terms remaining in equations
(208) are
(214)
And the vector sum of these must agree with equation (72) after
the latter has been adapted to the point
z ' = r; x ' = y' = 0. (215)
We have for use with equations (72, 188, 193)
The Main Coordinate Systems 139
<a(-r) - r(co 2 )
(216)
d>A \ I dt? \ 2 / d^>\ 2 d^ d^> \
dtJ+VdtJ+Uv + 2 dF cos '>)-.
When the multiplications are carried out and the items duly
oriented by the plan explicitly recognized for equations (214),
the values are found in agreement at all points.
The special circumstances to which equations (214) conform,
make them express the acceleration of a point in the symmetry-
axis of a top or gyroscope when it is spinning about that axis
while the latter is executing any motions that change (ft) and
(tjr) . Beside the utility of this value in application to the problem
of the top, and the consolidation that the conclusions attain
through the comparison, it is particularly instructive to follow
carefully and in detail the appearance of terms in the acceleration,
and their various disappearances by cancellation. Then one
learns to cross-examine the mathematics and to discount sensibly
its evidence or suggestion as to just what dynamical processes
are in operation.
106. The fact that the resolution into polar component shapes
itself in accommodation to each individual radius-vector prevents
the introduction of any usefully general integrations to include
extended masses. As a substitute recourse is had, where the
radius-vector enters naturally, to plans like that worked out
for the rotation of a rigid body, which has contrived to extract
the common elements (<o) and (<b) for use with all radius-vectors,
and the moments of inertia as factors that cover the whole mass.
The polar components that have been deduced are then limited
practically to one mass-element or to the particle at the center of
140
Fundamental Equations of Dynamics
mass of the body. For the latter case, there is no difficulty in
writing down for the six fundamental quantities the parts of
their standard frame values that match the orthogonal polar
projections. These are:
Q = m
E = im[vV> +
H = m- i'
R = m[i'v (x ') + j'v (y ') +
P = R (X ')
M = - i'
x ri)r ~
drl
dtj
[H (z ') = 0];
R (y')
+ j'(rR (x ')) = M (x ') + M (y ',;
[M ( .') = 0].
(217)
As an addendum to the separation of power or activity (P) into
its parts it is worth noting that the total force corresponding to
the heading (y x v) of equation (206) can finally contribute
nothing to the work done. It must of necessity be perpendicular
to (v) and therefore ineffective in the product (R-v). Amounts
of work per second may be yielded in the parts of (P) by the
inclusion of these directional forces, but they must be self-com-
pensating and give zero of work in the aggregate. Their behavior
in both respects toward power is similar to that of normal force
that is confined to changing direction in resultant momentum.
Under (V( m )), other elements of force may be entered that also
give change of direction to (mv) ; this function it may share with
(y x v). But (V( m )) has monopoly, as was pointed out earlier,
The Main Coordinate Systems 141
of bringing about all changes of magnitude in (v), and hence in
(mv). It is plain common sense to confirm these conclusions by
the observation that what happens to coordinates merely to
the descriptive vectors as we have called them cannot affect
the physical data that they are devised to describe.
HANSEN'S IDEAL COORDINATES.
107. By the trend of the standard illustrations, it cannot fail
to have grown conspicuous already, how varied the available
combinations must be and how many kinds of adjustment to
special purposes are rendered possible, when once such resources
and expedients have been brought under fair control, and a
definite formulation of the ends sought has been arrived at.
The next instance in order, the ideal coordinates so named by
Hansen who proposed them, is adapted to strengthen that per-
ception. 1 The invention of the plan seems to have been con-
sciously directed by a purpose, and it finds a place here because
it has made its standing good for certain fields of application.
As would be natural to surmise, the proposals that have won
acceptance have been gleaned by the sifting of actual and con-
tinued trial among the larger number submitted for general
approval. Ideal coordinates are made to follow upon the polar
system here because the radius-vector still remains a prominent
element in their specifications; and on this account, they too
have no immediate range beyond tracing the motion of one
particle or mass-element. It will be recognized that they pursue,
like the other coordinate systems that have been discussed, the
object of stating standard frame values, but in more elastic
partition of the totals than (XYZ) itself can furnish.
The chief concern of ideal coordinates is with velocity, and its
main course may be called a response to the question, in what
direction can the restrictions upon the frame (X'Y'Z'), that the
1 See Note 27.
142 Fundamental Equations of Dynamics
polar system has been seen to impose, be loosened without im-
pairing the in variance of (v) that the polar components retain.
That point being secured, the other consequences entailed are
left in whatever form they may happen to appear. In this way
it 'becomes part of the inquiry to ascertain how the expression of
acceleration is affected by the assumed conditions. The frames
(X'Y'Z') and (XYZ) continue with a common origin (O).
108. If we add to the suppositions of section 102 a rotation of
(X'Y'Z') about (Z') that can be of any assigned magnitude,
equation (202) will be written, when as before we identify ($1,
k', and rO,
dr d<l/ d# d?
v = ii - + (th x rOr ^ + (*i x rOr -^ + (fc x r^r ~ ; (218)
but the difference introduced is only formal since ($1, TJ) are
identical unit-vectors, and in this frame (X'Y'Z') it is still the
coordinate (r) or (z') alone that can differ from zero, while the
same corrections make the previous invariant representation of
(v) persist. This puts before us the nucleus of Hansen's idea, as
vector algebra allows us to condense it. Now it will not be
overlooked that (V(,'), V( y '), V( z '>) as determined by equation
(203) are the components of (v) in that frame of permanent
configuration in (XYZ) for which, with ($) equal to zero, the frame
(X'Y'Z') is the indicator at the epoch. But it follows from the
form of equation (218) that a whole group of fixed frames which
at the epoch have (Z') in common and are distributed through
all azimuths round that axis for the range (0, 2?r) in (<p), satisfy
first the relation for the vector sum
d^ d#
V(x') + V( y ') = ;i x rOr ^ + (tf i x r x )r ^- , (219)
and accordingly for the invariance of
v = V(x') + V( y ') + v ( ,'). (220)
The Main Coordinate Systems 143
Whatever the direction therefore, in which the extremity of (r)
is instantaneously moving parallel to the (X'Y') plane, it is
possible to select at that epoch among the group mentioned
above one frame for which (V( X ')) is zero, and another for which
(V(y')) is zero; and whichever alternative is chosen of these two
it is further open to attempt determining the rate of the rotation
about (Z') so that this one component remains permanently zero.
We shall return presently and develop consequences of those
possibilities, after pausing to insist a little upon equation (220)
which has not yet been particularized in that sense.
109. In order to come nearer to the form of statement that
Hansen was compelled to employ, go back to section 89, where
equations (150, 151) express the invariance of (r) in frames
having a common origin. Let us pass on to consider equations
(154), noticing how the added invariance of (v) necessitates the
vanishing of the last group of terms in each of them, for which
one condition extracted from equation (162) is seen to be that (y)
though differing from zero is colinear with (r). For our benefit
just now, this signifies that if two frames give equivalent sets of
components for the same resultant velocity, the equivalence will
not be disturbed by allowing one of them to be subject to a shift,
provided that the axis of it lies in the radius-vector at the epoch.
Then, as Hansen puts it, equations (151, 154) will exhibit the
same type in their forms, with velocities replacing everywhere
the corresponding coordinates, and the ideal for (x'yV) has been
reached. As we have approached it there are two stages: the
shift of (X'Y'Z') in the angular coordinates (t|r, #) is not without
influence upon the relations, but it has been compensated in
equations (203), and adding then a supplementary shift about
(ri) that is also ($1) leaves this compensation untouched.
The zero value of ($) having been standardized for equation
(203) with (X') in the plane (Z', Z), for the more general value
of ($) that is now contemplated we should write
144 Fundamental Equations of Dynamics
,, ( d# <ty . \
v (x ') ==i I r^ cos <p + r ^- sin & sin <p 1 ;
(221)
., f d# . d<A . \
V(y') =] I r-r- sin <p + r -r- sin # cos <p 1 .
And if we settle upon making (V( y ')) zero, the proper value of ()
at the epoch is determined by
r -r- sm
r cTt
(222)
Let us retain (y) for the rotation-vector of (X'Y'Z'), and dis-
tinguish by (o>) the angular velocity of (r), so that in the subse-
quent details
i ^ J v dt > _u A d(f> I d * a * d *
r ==^dt +^dt +^dt ; 0) = ^ 1 dt +dl dt- (223)
Then under the condition of adjustment shown by equation (222)
we have
(224)
dr
v (y') = ; V(,') = r i^-
110. The execution of this manoeuvre reduces the statement,
so far as velocities are concerned, to one of motion in an instan-
taneously oriented plane (Z'X'), with a resolution of (v) for the
standard frame along the radius-vector and the perpendicular to
it in that plane. The values of the components conform per-
fectly in type to those of the similar projections in the permanent
plane of uniplanar conditions; and the prospect is opened for
success in determining such a rate of rotation about (r) as will
perpetuate the instantaneous relations in exactly this form
when they have been established at some one epoch; this involves
The Main Coordinate Systems
145
keeping the values of (V( y '>) continuously at zero, though it is
always reckoned in the normal to the shifting plane (Z'X').
The examination of the arrangement requisite to that end is
connected with the question about components of the accelera-
tion (v), and we shall make our beginning there.
Recorded in equations (208) are the projections of (v) for
(XYZ) upon the (X'Y'Z') axes as located by (4 = 0) ; and from
them can be calculated the equivalent set of projections upon
the axes (X'Y'Z') located by the general value of ($), precisely
as equation (221) does this for velocity. Those projections can
finally be particularized for the angle ($') assigned by equation
(222) to satisfy its announced condition. Distinguish the last
named components of (v) temporarily as (t( X "), ^< y "), ^("))l
they are given by
v (x ") = i"(v (x ') cos <p' + V(y') sin <p');
V(y") = j"(- V(x') sin <p' + V(y') cos <?');
V) = k"(v (z ')); with k" = k' = n,
the new unit-vectors being (i"j"k").
(225)
In the text of section 103, the components of (v) happen to find
expression through polar variables, but that is plainly only an
incident of the sequence in which they were developed; they
might just as well have been derived from
(226)
') = k" ( k"-i
or in some other equivalent fashion, the choice depending upon
how the data are presented. It is another consequence of this
146 Fundamental Equations of Dynamics
idea, that the original shift of (X'Y'Z') in (i{r, #) belonging to
polar components is unessential; in effect it drops out of con-
sideration through the allowances for its presence when equations
(208) were made correct, as we saw also in speaking about
equations (203). The vital element in these ideal coordinates is
the accompanying rotation about (r) which has been relied on
at critical points to secure at once invariance and simplification
in the relations for the velocities, and whose consistent intro-
duction into those for acceleration we are now prepared to finish.
In order to accentuate the real dissociation from the polar
scheme, let us think definitely in the terms suggested for equations
(226), of two coincident frames in the configuration with (XYZ)
designated by (t|r, #, $'), of which one is fixed, while the other is
departing from coincidence by rotation about the (r) of the epoch.
We will temporarily call the rate of this departure (u) in substi-
tution for the time rate ($1 d<p/dt).
111. Then the specialization of equation (137) to these cir-
cumstances gives, if we particularize the velocities also as (v (x "),
v (y "), v (z ")), and remember
u (z "> = u; u (x ") = u (y ") = 0; v (y ") = 0; (227)
.. d ,
(uwr) ;
v (z ") = k" -: (v ( ,")).
(228)
Hence, in order that these values may be reconciled with a
permanent zero value for (v (y ")), the magnitude of (u) must be
adjusted to the acceleration parallel to (j") of the epoch, which
for the present purpose we may suppose to be one among the
data, as well as the velocity component (i"(r)). At the same
The Main Coordinate Systems 147
time, as the forms of the last equation show, the accelerations
parallel to (k", i") are reckoned as though those were constant
unit-vectors. But it is plain that the existence of shift cannot
disappear completely from acceleration and from velocity too,
because the necessary conditions
(u x r) = 0; (u x v) = 0; with (u) not zero; (229)
are incompatible, so long as (v) and (r) are not parallel.
There is a strong natural suggestion, through the connection
and the form in which these ideal coordinates have come to our
attention, that they bear by their intention upon the astronomical
problems that occupy themselves with orbits whose differential
sectors are drawn out of one containing plane by disturbing forces.
To this conception of a continuous succession of osculating orbits
the method is ingeniously accommodated, with a separation
that is of practical advantage between the forces (mv (x ")),
(mv (z ")), whihc, as it were, control the orbit-element of the
epoch, and the force (mv (y ")) into which the distorting influence
is collected. Yet interest in the method should not be confined
to astronomers, because its device is repeated with only the
modifications that the new conditions impose under the next
heading, when the osculating circle of curvature is brought into
relation with any curved path of a moving point; and the
parallelism is an instructive feature for our discussion.
RESOLUTION ON TANGENT AND NORMAL.
112. The local tangent and normal to the path of a moving
point afford a coordinate system that has been in general use
since the days of Euler, but its employment for velocities could
not be carried beyond the rudimentary stage of indicating the
set of values (0, 0, v) in every such application. It is clear that
this remark includes with equal force momentum and kinetic
energy that contain no other kinematical factor than velocity.
148 Fundamental Equations of Dynamics
The resolution tangentially and normally has that ground for
concerning itself solely with acceleration and with dependent
dynamical quantities like force, power and work. In this it
differs from the coordinate systems that have been occupying
us hitherto: by not being serviceable in more than one stage of
differentiation, whereas the terms of the other systems have
linked with two derivatives at least. How the tangent-normal
plan branches off from the radius-vector series appears when we
write
dr
r = iir; f = v = f : r + r x -^ ;
(230)
dv
r = v = v lV + Vl ^ ;
and compare the last equality, that realizes the separation along
tangent and normal to the path, with equation (209) that con-
tinues the polar component scheme. Because one stage does
isolate itself thus, it becomes feasible for it to remain bound by
the invariance test for a quantity with which it connects, and
yet take on the quality of a mixed plan in other respects. A
plan mixed or composite in regard to the standard frame, by
dealing with comparison-frames (O', X'Y'Z') whose (r') and (r')
are not invariant with (r) and (f), though (r') and (f) are thus
related. Section 77 furnishes all needed reminder about like
combinations.
Such realities as the exclusion of normal force from effect
upon power have thrown tangential force into stronger relief;
and the more impressive function of the latter in changing mag-
nitudes. Some plan or other of resolution for acceleration is
favored, because the resultant quantity finds in general no
visible geometrical element falling in its line as the tangent to
the path does with velocity. The projections on tangent and
normal form the simplest set that contains any segregation, for
The Main Coordinate Systems 149
as we have once noticed, the (XYZ) set does not discriminate
but speaks always of its own tensors. The separation on the
basis that tangential acceleration changes velocity through its
tensor alone, and the normal part changes the unit-vector alone,
is the most important early and familiar instance that general
ideas of vector algebra had to pattern after. The last of equa-
tions (230) has, as we are aware, grown into a general handling
of any vector derivative.
113. The polar components of acceleration have been found
to involve in comparative complexity the distinctive traits of the
velocity vector as exhibited by its derivative, because their
formulation is guided by elements foreign to (v) and borrowed
from the behavior of the other vector (r). And as we see illus-
trated repeatedly, the changes in any vector indicate themselves
most directly by analysis of its derivative according to some
leading idea inherent in the vector itself. It did not escape us
that the vector (H), for example, is but indirectly described by
use of (to) and () in sections 56 and 57, and that there is likely
to be a gain when the more direct connection of (H) and (M) is
utilized.
Before entering upon any new considerations, let us once more
pick up the thread at section 89, and renew the thought that
(xyz) and (x'y'z') can be read as projections of any free vector
such as (v). Then equations (154) or their alternatives made
explicit for (x, y, z) are the algebraic statement of shift for
acceleration, (v) for the standard frame being given indifferently
by
v = ix + jy + kz; v = i'x' + j'y' + kV. (231)
Also the details worked out for (r), beginning with section 78,
are translatable for (v), and justify for instance, as we can use
now the Euler angles, and are paralleling (r = 0),
dV dV dV dV
150 Fundamental Equations of Dynamics
i
whose meaning reproduced more briefly in
v = v (m) + ( r x v) (233)
gives foundation for our next useful conclusion.
114. In a plane curve that is the path of a point (Q), the
successive orientations of the tangent can be said to arise by a
continuous turning, whose axis is the normal at (Q) to the
plane of the path. And this turning to which we assign the
angular velocity (w), and which accompanies the progress with
velocity (v) of (Q) along the curve, is registered in its effect upon
(v) through the normal acceleration that is written
v 2
v (n) = (w x v) = - pi - . (234)
P
The order of factors in the second member is seen to direct this
acceleration toward the local center of curvature of the path,
and the known geometry introduces the radius of curvature,
whose standard unit-vector points away from that center.
Complementary to this is the tensor change in (v) provided for
by the tangential acceleration whose natural form is
dv
v ( t) = V! ^ . (235)
In order to recast these statements in the language of shift,
let comparison-frames be conceived distributed along the path
and with origins 'in it, each in a permanent configuration with
the standard frame, its (X') axis pointing forward along the
local tangent and its (Y') axis inward along the normal, (t>) being
standard as positive. All such frames will give both velocity
and acceleration invariantly with the standard frame, and for
each one as (Q) passes its origin the same conditions prevail at
the epoch:
V( X ') = v; v (y ') = v (z ') = 0. (236)
But the shift of origin alone, as we have noticed elsewhere, being
The Main Coordinate Systems 151
without effect upon the projections as vectors, the application of
equation (233) will yield
., d dv
'
V) = j'(v); V> = 0;
consistently with equations (234, 235).
115. But a space curve differs from a plane curve very much
as the instantaneous orbit spoken of in section 111 differs from a
plane orbit, in that its differential sectors, bounded now by radii
of curvature and not by radius-vectors, are not coordinated into
one plane. Each is treated typically like the uniplanar case,
however, but in the plane of its epoch. A gradual change of
this plane can always be accomplished by an added turning
about some axis contained in each plane element, the displace-
ments due to which being normal to that element are merely
superposed on whatever process is being completed within the
plane of the element itself. The direction of each such axis in
its individual plane will be chosen according to the particular
condition that it is desired to fulfil.
In the account of Hansen's coordinates it was proved that the
designated axis left both component velocities (<> x r) and
(ri(dr/dt)) unaffected by a rotation about it; and also two of
the three component accelerations. In the example before us
now, it becomes desirable to leave unchanged the one velocity (v)
that enters unresolved, and the entire acceleration. It soon
appears how this is attained by letting each differential sector
turn about an axis that is the line of (v) at the epoch. This will
add no new velocity at any point like (Q) in that axis, and it
leaves the acceleration components unaltered because the supple-
mentary term (Y' x v) would in any event be normal to the
plane element, if (Y') denotes a rate of rotation about any axis
in that plane, and this term vanishes for every magnitude of (Y')
11
1 52 Fundamental Equations of Dynamics
when the latter is colinear with (v). Consequently if we apply
equation (233) again, writing
t - ( + t'), (238)
equations (236, 237) are continued in validity for any space
curve, though derived originally from uniplanar motion. It is
plain in what way the shift process is to be modified when it
must include a varying plane (X'Y') for the osculating circle;
and also that the tensor of ($') must be fitted to the tortuosity
of the curve, while (w) is determined by the circle of curvature.
The vector magnitude (y') is, to the extent shown, external to
the acceleration problem stated; and in this it goes beyond the
corresponding vector (u) of Hansen's system, as reference to
equation (228) confirms. The geometry of space curves, in
which our axis (Z') figures as the binormal, is seen to build with
similar ideas to those just developed.
116. If a comparison-frame (O', X'Y'Z') is moving as a whole
relative to the standard with unaccelerated translation whose
velocity is (v ), and the velocity of (Q) relative to (O', X'Y'Z')
is (v'), the last of equations (230) gives for
v = Vo + v', v = t/v' + v/ ~ . (239)
And since by supposition (i'j'k') are here constant vectors, there
is no distinction between (v/) relative to (X'Y'Z') and (XYZ).
Hence comparing the paths of (Q) relative to the two frames, it
is clear that the sum is invariant, if we add together each tan-
gential acceleration and its partner of normal acceleration,
although the velocities in the paths are different, as is the appor-
tionment of the acceleration between the two components. Such
indifference as exists to the inclusion or the exclusion of constant
velocities is often a helpful fact in treating of accelerations.
But its other limitations must be observed beside the one just
indicated, as applying for example to power (R-v). If in this
The Main Coordinate Systems 153
product (R) is retained, and (v) is changed to (v'), the product
is altered unless (v ) and (R) happen to be perpendicular.
As the summation
/'
t/ n
ds = (s - so) (240)
constitutes a rectification of the path, so the other legitimate
summation
jT (m^dt) = m(v- v ) (241)
might be termed a rectification of momentum. In each opera-
tion we may see, by one way of viewing it, the accumulation of
tensor elements upon one shifted line that becomes parallel in
succession to the vector elements whose tensors are thus summed.
But it does not explain fully why the second summation is
mathematically as valid as the first, just to remark that each
element of momentum is colinear with an element (ds). The
tensor factors may be in any ratio that varies from one element
to another and distorts the graph. In addition to whatever
else can be said, we may return to the idea of comprehensive
tensor running through a process of shift and observe what
condition makes an element of actual displacement and the
exact differential of such a tensor equal, by obviating that fore-
shortening of each element and the telescoping of their series
that shift in general causes. If we take for instance equation
(122) in connection with its context, the condition is seen to be
that the vector product denoted generally by (y x V) should be
perpendicular to the line on which the tensor in question is laid
off. This becomes a specially simplified relation when the plan
of shift is such that only one tensor occurs. The polar scheme
contains only the length (r) of the radius-vector; the tangent
and normal resolution only the tensor of (v), which may indeed
be identified with (r) by the thought of section 88. In forming
154
Fundamental Equations of Dynamics
the derivative of (r) or of (v) under the form of equation (137),
(V( m >) comprises nothing but the total derivative of the tensor,
and the mathematical test for integrability is met. If it were
practically easier to devise plans of the type instanced, without
sacrificing other advantages, there would be less hindrance to
forming integrated values of tensors in working out results of
shift.
117. We shall close this summary of our last system of point
coordinates by gathering for record its most serviceable relations
to the fundamental quantities, and here again with a representa-
tive particle at the center of mass of a body definitely in mind.
They show in terms of projections parallel to the (X'Y'Z')
specified for equations (236, 237), with (XQ', y</, z</) added for
the coordinates in the standard frame of the particle caught in
passage through the (0') of the epoch.
Q (x ') = Q = mv; Q (J ' } = Q (l ' } = 0;
E (x ' } = E = mv 2 ; E (y ' } = E (z ' } = 0;
H = (x ' + y ' + z ') x Q = + JWQ) - k'(yo'Q);
R( y ') = j'(mvco); R (z ') - 0;
(E
dt dt '
y,/
-*(
dv ,
- m dt" yo
-f j' f m ^- z ' } - i'(mcovzo').
(242)
The expression written for (M) should be compared with the
direct vector derivative of (H) as given above in terms of the
shifting (X'Y'Z').
The Main Coordinate Systems 155
EULER'S DYNAMICAL EQUATIONS.
118. The configuration angles (t{r, #, $) have been associated
with Euler's name already; and once more we follow the estab-
lished custom in speaking of the next plan to be examined as
Euler's, describing the statements of it as his dynamical equations,
and so contrasting them with the purely geometrical or kinemati-
cal ideas brought forward under the other title. 1 This second
group of Euler's equations constitutes a system of resolution for the
dynamical quantities that departs in one important respect from
all the others that have preceded it in the order that we are
following. It has been constructed with specific reference to a
rigid body as a whole, instead of being shaped for one element
of mass, or at most for a particle at the center of mass. The
summation covering the entire mass has been incorporated into
the expressions, as an integral part of their standard form; the
field of use for them is particularly among those parts of the total
quantities that must fall outside all plans that are limited, in
conception or in effective and convenient adaptation, to a par-
ticle's translation. Therefore it will be anticipated that we shall
deal in these equations with that element of rotation in the most
general type of motion for a rigid body, which is the obligatory
remainder after deducting a translation with its center of mass.
The explanations on this point in sections 48 to 63 may be re-
ferred to; also those in regard to the dynamical independence
of the rotation and the translation, and the connection of a pure
rotation about an origin with one about a moving center of mass
(see sections 52 and 53) . Let it be remarked, in order to cover
this aspect of the situation, that Euler's dynamical equations once
developed for the conditions of rotation, are applicable equally
to either occurrence of it.
119. A junction with previous results can be made by bringing
together the equations for the values of (H) and of (M), since
1 See Note 28.
1 56 Fundamental Equations of Dynamics
it has been proved that moment of momentum and force-moment
furnish central clews to guide inquiry among the phenomena of
rotation. Let the understanding be that our analysis attaches
primarily to rotation about a center of mass (C'), and that any
necessary transitions to pure rotation are to be adequately
indicated.
On returning to equations (86) the signs of mass-summation
are in evidence, and also of the general interrelation between
each component of (H) and all three components of the rotation-
vector (G>), when an unguided choice of (XYZ) has been made,
to which axes those located at (C') will be assumed parallel for a
beginning. The concept of (CD) as properly applicable to the
complex of radius-vectors lying within the body has been adopted
profitably, but it is not to be overlooked that a changing con-
figuration of body and (XYZ) makes the inertia factors variable.
Neither does parallelism of the axis of (to) with one of (XYZ),
permanent or transient, introduce the lacking symmetry into
these equations. Note, however, the form of equation (80),
regard (o>) as parallel to (Z), and complete the set of component
equations thus particularized. They are for (X'Y'Z') at (C'),
H (z ' } = k' ((,')!(,')); H(y') = j'(- (l ')/ m y'z'dm) ;
(243)
H (x ') = i'( w( z ')/ m z'x'dm).
Observe the form of the last two components, and the fact that
the orienting factors in them are coordinates.
120. The commentary of the last paragraph can be duplicated
essentially in respect to equations (89), replacing (H) by (M')
and (to) by (to). Thus if we next suppose the axis of (<b) parallel
to (Z), all three components of (M') persist, and a similar differ-
ence in type reappears, between the first component and the two
others. Again for (X'Y'Z') at (C'),
M' (z ') = k'(w (z ')I (z ')); MV) = j'(- tt ( ,')/ m y'z'dm) ;
M'( X ') = i'( w (z ')/ ra z'x'dm).
The Main Coordinate Systems 157
Bringing in the other part (M") of the total force-moment does
not better the symmetry, neither of the last equations nor of
their parent equations, since in reliance upon equations (75, 76)
we find
M" = (oxH). (245)
These observations multiply reasons for appropriating the
principal axes at (C') in a selective choice of (X'Y'Z') for any
one epoch, and then perpetuating whatever advantages are
reaped, by introducing a shift that is so regulated that the same
three lines of the body which are its principal axes for (C') shall
always be taken to mark or indicate the configuration of the
fixed frame, in terms of whose projections or components of the
quantities in question the equations are to be written. The
case for these principal axes is strengthened when equation (88)
adds kinetic energy to the expressions in this way simplified;
and when we reflect that within the scheme now proposed, the
inertia factors are reduced from six in number to three that are
the principal moments of inertia, and that the triplet retains the
same values as the axes under this scheme shift. The general
case is to be supposed, where there are no more than three
principal axes at (C')> and the momental ellipsoid is not one of
rotation.
In view of the role about to be assigned to them, a specialized
notation referring to principal axes is called for, and we shall
meet that need first by using (A, B, C) to denote both the
magnitudes of the principal moments of inertia and the axes
with which they are associated. As magnitudes, (A, B, C) are
scalar factors in equations. They are associated with lines and
not with either one direction in those lines, so they are not
vector tensors. As axes for specifying configuration, (ABC)
designate by convention one direction in each line. The cycle
order is as they stand written, so that in the zero of configuration,
(A) is parallel to (X), (B) to (Y), and (C) to (Z). The axis of
158 Fundamental Equations of Dynamics
(C) is then (Z') of our preceding notation, and it has at any
epoch the angular coordinates (ijr, #). The third angular dis-
placement ($) is about the (C) axis itself. (See section 93.)
Secondly, projections of any vector upon the principal axes will
be denoted as illustrated for (o>) and (to) thus:
0) = <0(a) + W(b) + 0)( C >; G> = fa)(a) + <i(b) + "(c),' (246)
and the corresponding unit- vectors by (ai, th, Ci).
Utilizing this notation, the equations brought under review
above are reduced to the forms
H = o>(a)A + to (b )B + w( C )C;
(247)
M' = t>(a)A + w(b)B -f <>(<!) C;
M" =
+ bi(cO( c )CO( a )A 0)( a )OJ(c)C)
+ Ci(()(b)B co(b)aj (a )A); (248)
E = |[Ao; 2 (a ) + B w 2 (b ) + Cco 2 (c) ]. (249)
And this yields for the similar components of the total moment
(M)
M- a f ' A I fC* "DM . "
(a) ai[O)(a)A -f~ W(b)W( c ) \\J &)]>
M (b ) = bi[w (b )B + co (C )co (a) (A - C)]; (250)
M( C ) = Ci[o)( c) C + W( a )W(b)(B A)]. .
The sequence of ideas by which these specialized equations have
been reached should be attentively scrutinized, also the inter-
pretation of the combinations at this stage. Equations (250)
are evidently valid at any one epoch, and can be evaluated if
these elements are known at that epoch:
(1) The orientation of the axes (ABC) in (XYZ), and the
magnitudes (A, B, C);
(2) The vector (<i) in tensor and orientation;
(3) The vector (w) in tensor and orientation.
The Main Coordinate Systems > 159
121. In order to supply some other profitable details, and to
put another link in the connections of these equations with
general forms, we shall recur to equations (86) and differentiate
with regard to time, the first of them for a sample. It is funda-
mental that the result must represent the projection of (M)
upon (X), the latter being taken arbitrarily; and that with base-
point at (C') all moments must be reducible to couples, all net
force being absorbed into the translation. (See section 51.) The
conspicuous complication in this derivative is a lesson about
what principal axes avoid, for we find
H (x) = M (x) = i
- -^ (co ( y))/ m xydm - co (z) / m ^ zdm
co (z )/ m x -7-dm j- (<y( z) )/ m zxdm f . (251)
J
In the third member, the third, fourth, sixth and seventh terms
are to be further expanded by use of the velocity relations for
rotation,
dx dy
~j~7 = CO( y )Z w (z)yj "TT = W( Z )X CO( X )ZJ
dz (252)
When the axes (XYZ) are particularly chosen to be the set (ABC)
in its position at the epoch, all terms can be struck out that
contain as factors the integrals known as products of inertia.
And this choice cancels the second term in the third member
also. Because for all sets of orthogonal axes at the same origin
we have
160 Fundamental Equations of Dynamics
I(x> + I( y ) + I( Z ) = 2/ m r 2 dm (an invariant magnitude) ; (253)
and hence during relative displacement of body and (XYZ),
d d d
But for the longest and for the shortest axis of the momental
ellipsoid, corresponding to the least and the greatest principal
moment of inertia, the condition of maximum or minimum re-
moves two terms separately from the above equation of condi-
tion, which then proves that a stationary value of moment of
inertia enters for the third principal axis also.
After removing all the terms of indicated zero value, there
remains
,V j T d
H (x) = ai \ I (x) -7- (co (x) ) + co (y) o;( z )/ m (y 2 + x 2 )dm
co( z) co (y )/ ra (z 2 + x 2 )dm \ , (255)
for comparison with the first of equations (250) . The two state-
ments harmonize completely, if we insist upon the identity of
meaning for the expressions
<b(a), i : (co'( x) ) ; [(A) and (X) parallel.]
Lat j
they are both representative of the projection of the vector (<b)
upon (A) or (X). The comparison for the two other pairs of
equations is to be made similarly.
122. The next step in progress releases equations (250) from
this one reading of their symbolism, and lays a foundation for
the equivalences
d . d ,
d
6>(c) = Ci T7 ((c)),
where the second members are to be recognized as components
The Main Coordinate Systems 161
of (V (m )) in equation (137), for application to the derivative (<b)
as expressed under a process whose shift rate is marked by the
axes (ABC). Since these are definite lines of the body, they
must conform to its rotation-vector (&>), and we have in this
shift another example of cancelled correction, for
" = 0>(m) + ((> X to) = El T7 (C0(a)) + t>i 77 (o)( b ))
+ C 4 ^ ((.)), (257)
where the tensors in the third member have taken on a new
shade of interpretation. They have become the generalized
values for the shifting axes, instead of being particularized single
values.
But there is one more consequence in this direction that still
remains to be formulated, and that can be drawn from the
expression in equations (247) for moment of momentum which
can now be conceived as continuously valid and differentiated,
due allowance being included for the changing orientation of the
projections that make up the total. We can write
H = ^aiA ^ (co (a) ) + biB ^ (o> (b) ) + CiC ^ (co (c) ) I
+ ( x H), (258)
whose separation into components restates equations (250), after
incorporating into the latter the transitions of equations (256).
The forms derived by either line of procedure are Euler's dynami-
cal equations, whose establishment with the means at their
inventor's disposal must always be rated as a remarkable achieve-
ment. It is in addition moreover remarkable that the segre-
gation according to the terms of equation (258), which is more
nearly mathematical in its origin, is also a separation that splits
the force-moment into parts with a plain and important difference
162 Fundamental Equations of Dynamics
of physical effect; and the beginning made in section 55 was
with design selected in order to dwell upon that fortunate chance.
A conclusive proof of the equations in very few lines can evi-
dently be extracted from the material that has been discussed
here with greater expansion; but a demonstration may become
too brief to be effective for insight, in a matter that has wide
general bearings, so the detail is probably not superfluous.
123. Among the uses of Euler's equations, the predominant
type of rigid body whose rotation is to be investigated is likely
to show a certain symmetry, whose representation in the mo-
mental ellipsoid gives equality to two axes of the latter. This
must convert the general ellipsoid into one of rotation with a
symmetry axis; the known consequence being that all per-
pendiculars to that symmetry axis at the center of the rotational
ellipsoid become principal axes with equal moments of inertia.
This combination arises if the rotating body itself, being homo-
geneous in material, has an axis of symmetry; and bodies
designed for rapid revolution are usually turned in a lathe. But
it is clear that a prism of square cross-section, as well as a circular
cylinder, would manifest its symmetry in a momental ellipsoid
of rotation. And Euler's dynamical equations, being concerned
with distribution of mass only as recorded in principal moments
of inertia, would not discriminate between the two cases, granted
the magnitudes (A, B, C) are severally equal in them.
It is proposed next to reconsider equations (250) in the light
of this possibility, designating (C) as the axis of symmetry of
the momental ellipsoid for (C'), with the corollary that the
magnitudes (A) and (B) are equal; their common value we can
call (A). If now the axes of (A) and (B) are still definitely
located as lines of the body, whose rotation-vector (y) is identical
therefore with (<o) for the body, no essential change appears in the
equations except dropping out the last term of the third. Espe-
cially equations (256) that are determined by the equality of
The Main Coordinate Systems 163
(Y) and (to) are available as before. However all lines of exposi-
tion in reaching Euler's equations must set the adoption of
principal axes in the central place, and not the equality of the
rotation-vectors. So by multiplying the number of principal
axes the condition of symmetry enables choice to be variouslj r
exercised and yet range among them, though the auxiliary equal-
ity be abandoned and a relative motion through the material
of the rotating body be permitted to the principal axes that have
been selected. It is clear that the assumed relations limit the
difference between (y) and (<o) to a turning about (ci) that is
also ($1); but to this element it remains free to assign any
magnitude. The expression of that freedom is
di/' d$ d<p
' ^ l dt * dt ^ dt '
(259)
where (k) may have any positive or negative value. Euler's
equations proper given for (k = 1) have been put before us
already; and we shall add for consideration, among the gener-
alized Euler forms suggested by the last equation, only that
modification which becomes necessary when the value of (k) is
taken at zero. This supposition happens to offer some special
advantage in handling combinations like a gyroscope under
control by weight moment, and the earth as affected by a gravita-
tion couple due to its spheroidal figure.
124. Let us mark the change of plan by using (A', B') to
denote those principal axes that are now substituted for (A, B),
recollecting first that as moments of inertia all four magnitudes
are equal, and secondly, that (C) is common to both sets of
axes. Then as a reminder of the needed revision in equations
(256) we can write
164
Fundamental Equations of Dynamics
d ,
_d_
dt
Cl
dt
+*' xu - (260)
In equations (193) are recorded values for those components
of (o>) which accord in directions with the present specifications
for (A'B'C); and in equations (191) of section 101 the line of
development caused us to put down in terms of (i|r, #, $) the
first three entries on the right hand of equation (260). It seems
advisable to clinch the comparison in respect to equations (256,
257) by developing here for that resolution the general com-
ponents of (<b), and lastly confirming the harmony of the two
sets at their coincidence that occurs for ($ = 0). These are the
first details:
= ai TT (&>()) =
,
sm <p + -T- -rr cos t? sin v?
dt at
.
+ -rr -;-- Sin t? COS
dt dt
COS <p + TT -v COS t? COS
d$ dp
rr T
dt dt
1
sin ^ sin v?
W(o) =
d Td 2 <
-TT (W(c)) = C
dtd
sm
(261)
The Main Coordinate Systems 165
the values to be differentiated in the second members being duly
identified in the survey that equation (181) has put together.
What remains of the results last written when they are particu-
larized for the condition ( = 0); with (td( a ')), (w(b')) obtained
by a corresponding resolution of (o>), fills out the more general
form of Euler's equations,
M(a') = ai'[co (a ')A' + co (b ')co(c)(C - B')];
M ( b') = b/U^B' + eo (o) eo (a ')(A' - C)]; (262)
M( C ) = Ci[o>( c) C];
the necessitated companion being the equalities of magnitude
A = A' = B = B'. (263)
Finally the components of (to) that match the above statement
being added:
/ ( d# \ , / <ty . \
w(a') = ai ' I -j^- j ; <a ( b') = th' I ^ sin & J ;
(264)
dpdt
" + co
such advantage as this alternative formulation possesses on the
kinematical side is made to appear. Dynamically something is
contributed to a preference for it when the resultant force-
moment is a vector that lies continually in the line of the axis
(A'). A preliminary examination of the instances quoted above
shows that they lend themselves unconstrainedly to this analysis
which will be found applied in section 127.
125. On the surface the constant reference to (CD) and (6),
either in their totals or through differently designated sets of
their components, is apt to leave a misleading impression that
they are pivotal quantities in any investigation where Euler's
equations are employed. It seems worth while, therefore, to
put in stronger light the primary emphasis of equation (258)
166 Fundamental Equations of Dynamics
upon changes that are going on in the moment of momentum
vector (H). The separation in the second member there fits a
line of demarcation between changes in magnitude and in direc-
tion, since the first group of terms is by the connections that have
been established for it a magnitude derivative of
H = ai(Aw (a) ) + bi(Aw (b )) + Ci(Cw (c) ), (265)
though distorted from its value as reckoned in the standard
frame by shift of the axes (ABC). But just that shift is in-
dispensable, as we have insisted, in order that the properties of
principal axes may prune the cumbrous algebraic expansions
into maximum brevity. Where a corrected segregation for (H)
into changes of magnitude and of direction entails a sacrifice of
the gain by using (ABC), the balance of choice leans always one
way; that much of dynamical indirectness in Euler's equations
is condoned. But there is an increasing tendency and a whole-
some one, to put their dynamical sense to the front, letting (o>)
and (o) fall into a subordinate importance, derived in large
degree from the clews they furnish to (M) and to the course of
events for (H). It was less easy to do this under the older forms
of Euler's day, but it is facilitated, as has perhaps been made
convincingly apparent, by a vector algebra that follows so
intimately the history of vector quantities.
126. Naturally the thought has suggested itself to inquire after
a scheme modeled upon the resolution of force into a tangential
and a normal component, for application to moment of momen-
tum. One main obstacle is not difficult to detect, for after indi-
cating the start in parallel to the other procedure,
H = h!(H); H = fct(H) + fct- I ;
it is noticeable first that (H) cannot be assumed to fall in a
principal axis, and secondly that no data for (hi) are available
The Main Coordinate Systems 167
from geometrical sources. Therefore the longer forms, for (H)
in equation (86) and for (dH/dt) in equation (251) must be used,
and the expressions must be encumbered with an added angular
velocity for (hi). Introduction of (XYZ) gives no help, nor of
the partial time-derivatives that rely upon holding (ABC) sta-
tionary. Either leaves commingled the parts that are sought
distinct.
But one resolution of force-moment can be carried through
that is different from Euler's and yet has aspects that recom-
mend it. This is contrived so that one component is taken in
the axis of (o>) at each epoch, and arranged otherwise as will be
explained presently; approaching in plan the tangential resolu-
tion of force in so far as (<>) and (v) can be said to bear similar
relations to the two aims. It has the merit besides of piecing
out the usual discussion of rotation about a fixed axis, by giving
recognition to those supplementary terms which disappear on
fixing the axis about which the body is rotating.
Return to the value of (M") in equation (75) and of (M')
formed by mass-summation of equation (82), and assemble their
respective contributions. Let (u) denote the rate of change in
direction of (<>), so that with unit-vector (i) we have
-(uxo>); (267)
where (u) must be perpendicular to (<>); and subdivide (r) as
shown by
r = r u) + r'. (268)
Then
fdco 1
M u) = wi -7-/ m (r-r)dm - / m r (a ,)(6-r)dm , (269)
C.x.)).fo., + .0 :
= ( d~f r(w) ) + ( U X W ) * r '- ( 27 )
12
'x'dm
168 Fundamental Equations of Dynamics
Identify (Z') with (o>), and (u) with (Y') in direction, giving
M (u) = M (z ') = on jr/m(r 2 z /2 )dm uco/ m z'
Tdco 1
= wi T-I(z') - uco/ m z'x'dm . (271a)
L
In the plane (X'Y') we have to consider
- /m(w x r)(o)-r)dm / m r'(tvr)dm + (u x co)/ m (r-r)dm, (272)
from which are gathered without difficulty
(x' } = i' [
M
M (y ' } =
co 2 /my'z'dm -
dco
dt /E
^'x'dm + ucoI( X ') ,
dco
~dt
/ m yVdm
- uco/ m x A y'dm .
(271b)
Noteworthy is the extent to which equations (271) are reduced
by symmetries, though (u) is not zero, as well as the reappearance
of the elementary form when (u) vanishes. Dissection of these
moments shows almost immediately the force elements at (dm)
in components parallel to our (X'Y'Z') to be
(273)
dR ( ,-, = i'[ -^3
dR (y ,=j'r^x'-
LUI
dR( z ') = o)i(ucoy')dm;
which should be connected also with equation (72) by direct
projection upon (X'Y'Z') ; and by applying the proper shift
process to (H), determined by the elements (<o, u) on the same
line as sections 111 and 115 develop.
The Main Coordinate Systems 169
REGULAR PRECESSION AND ROTATIONAL STABILITY.
127. The aim and scope of these discussions could not attempt
to include many particular requirements of individual problems
without transgressing the boundary set by their intention, which
is guided rather toward preparation for more generic or recurrent
needs. It is, therefore, only because the dynamical features of
gyroscopic action are generally acknowledged to be typical
within a comparatively broad range, that some space is con-
ceded to examination of them. But though this carries us
beyond the stage of laying out a plan and somewhat into exe-
cution of it, it is proposed not to go far in that direction, nor to
speak of more than two topics that are critical points in the
general perspective. The first of these takes the form of a
deliberate inquiry into the circumstances of that adjustment to
steadied motion which is described with a phrase of wide ac-
ceptance as regular precession, and about which as a center so
much else can be made to figure as a disturbance of it or a de-
parture from it. And the second is devoted to laying bare the
play of dynamical factors that operates to produce rotational
stability. 1
The arrangement of the gyroscope is assumed to give it a pure
rotation about a fixed point (O), that is now taken as origin for
axes like (A'B'C), the last named being an axis of symmetry,
the shift rate for the set being as agreed in section 124, and the
zero of configuration being marked by coincidence of (A'B'C)
with (XYZ), where the (Z) axis is chosen vertical and down-
wards. The total controlling force-moment is supposed to be
furnished by weight, the standard frame being fixed relatively
to the earth, and the gyroscope has universal joint freedom at (O).
For its rotation-vector (o>) then, the two equivalents have been
supplied,
i See Note 29.
170
Fundamental Equations of Dynamics
For regular precession the conditions that obtain are
d\l/ dtp
dt dt ' *' constant ;
or
. (274)
(275)
w(a') =0; W(b'), co( ), constant.
And in order to standardize values, attach the further conditions
A'>C.
(276)
Then the weight moment is negatively directed in the axis (A'),
and with understandable notation the application of equations
(262) to this adjustment shows the following scheme of specialized
values :
d<A dp
= a/ A^ ^sin ??.
i'( Wr sin??) =
/dt \ f d<o dt M
(277)
/ \ ~" ~~ / _i
= bi' [zero];
= Ci [zero].
It is a clear matter of algebra that the first equation is satisfied for
sin # = 0;
or for
co (c )C dr
4WrA cos
dt 2A cos
or in another expression of it for
4Wr(A - C) cos
dt
2(A - C) cos
> (278)
The Main Coordinate Systems 171
Putting aside for the moment the first root, ou? questioning
begins with ascertaining the dynamical double process that
finds expression in the two signs of the second root and that
shows to inspection in either form under the assumed relations
of value, a quicker rotation about (Z) and a slower rotation of
opposite sign as possible adjustments.
128. It lies on the surface that while regular precession con-
tinues the vector (H) can be changing its orientation only and
not its tensor, and that since (H) must always be contained in
the plane (B', C), the applied force-moment must in the adjust-
ment meet the condition
= ( ^
ai '( - W? sin 0) = ^ x H (279)
equally at the quicker rate and at the slower rate of rotation
about the vertical axis. For the explanation how this can
occur, we shall look upon the moment of momentum as built up
by superposition, following the second member of equation (274)
in its elements which are now the first and third only. The
contribution from the principal axis (C) and its horizontal part
effective here in (M) let us write
N' = ni(c^sin0). (280)
Then having excluded (Z) from being a principal axis by the
suppositions laid down in the inequalities (276), the second instal-
ment of (H) must allow for both a vertical and a horizontal part,
the latter being contained in the plane (Z, C) ; and it alone is ef-
fective in (M); call it (N"). The total effective component of
(H) for the vector product of equation (279) is accordingly an
algebraic sum
N' + N" = ni | C^ sin (A - C) ~ sin & cos 1 , (281)
the part (N") being readily evaluated to confirm this.
172 Fundamental Equations of Dynamics
129. It is next apparent from the cycle order that the rotation
about (Z) must be negative in order that both terms within the
parenthesis may first point the same way relatively to (ni) for
our fixed assumptions, and secondly, give by the vector product
that negative orientation in (A 7 ) which the operative and nega-
tive weight-moment demands. So the standardized form in the
circumstances becomes
M (a ') =
- (A - C) (~ Y sin cos #1 . (282)
It is patent how elastic the constancy of this algebraic sum can
be made, or of its equivalent vector product; large (N' + N")
and slow rotation, or smaller (N' + N") and quicker rotation.
With equation (281) besides to show reversal of the rotation about
(Z) converting a numerical sum into an algebraic one, all other
elements being held unchanged. But leaving those details as
covered sufficiently, it behooves us to note in equations (278)
that each double value has its own common quantities that are
not entirely reconcilable. Since
the first member, together with both (#) and (d<p/dt), cannot all
remain unchanged while the rotation about (Z) is made fast or
slow. Equation (281) has tacitly taken one choice; but ((<.))
is a standard-frame quantity, whose constancy in magnitude
moreover is assured under the third of equations (262) whenever
(M( C )) is zero. We might then attach our thought preferably to
the first form in equations (278), and recast the result thus:
M ( .', = (ih^ x H) = ^ x (N' + N")
; (284)
The Main Coordinate Systems 173
in which the possibilities of varying factors in a constant product
reappear, with ($) and ((<.)) barred from change. It will be
noticed finally that either more direct derivation of result corre-
sponds exactly with the terms to which the first of equations
(277) reduces, so our analysis reversed could be applied im-
mediately to the latter. It ought to be said about the realization
of conditions, that the spin round the (C) axis is usually pre-
ponderant heavily in magnitude, and for this reason the observed
rotation about the vertical with a negative weight moment is
normally retrograde, the necessary high rate for the contrary
rotation being practically unattained.
130. Let equations (262) next be released from their restriction
to that adjustment whose relations are now ascertained. Then
with repetition of the idea put forward in the connection of
section 56 there can be a rearrangement in this instance, too,
that will describe the general action in terms of a deviation from
adjustment as a convenient basis for exhibiting the consequences
in the light of a disturbance. Re-establishing their unspecialized
character, equations (277) will be written
a/(- Wrsintf)
..,[
dtA 4 <ty . 1
rr cos #A -j sin $ :
dt dt J '
=
= (
|_ cii
+ jT A TT sin & TT sin #A -j .
dt dt dt dt J
But all the items there put down only elaborate still the one
(285)
174 Fundamental Equations of Dynamics
dynamical fact that no vector change in moment of momentum
is ever being produced except the increment along the instan-
taneous position of the (A') axis, which is that of (#1). Denote
the projection of (H) upon the plane (Z, C) by (H'), and the
first of the three expressions can be put in these equivalent forms :
The first statement is read that the weight moment devotes to
changing magnitude for the component of (H) in its own line
whatever margin remains after providing for continuance of
change in direction for the rest of (H). And the second, that
the deviation of the actual moment (M) from the adjustment
moment (M( >) required for prevailing values is registered in a
process of change for (ft). The indicated preemption claim of
the changes in direction has a certain figurative shading, we may
allow, but a certain truth also; because those affect quantities
at their existent values for the epoch, whereas the quantities
that are changes in magnitude are called into being and not
present already. And so with the second form of statement:
the section referred to concedes that the subtracted force-moment
in the first member may be declared nominal or mathematical;
but both points of view above are dynamically suggestive and
to be entertained as a mental habit.
The other equations of the group (285) set forth the kinematical
complications that ensue because nothing dynamical is effective
in those lines. They give foundation for important and inter-
esting studies that are, however, only to be alluded to here;
we shall content ourselves with insisting once more upon the
thought of sections 56 and 57. At the regular precession adjust-
ment every term in the second members of these equations
The Main Coordinate Systems 175
vanishes separately and they become a blank recording nothing.
Now they sum up algebraically to zero, though the individual
terms need not vanish; but they are, in a sense to be understood
with due limitations, as empty of physical content as ever; they
chronicle only formal and internal readjustments of expression.
131. The topic of rotational stability is also at its core dynami-
cal, and it is approachable most directly through the considera-
tions that we have been attaching to regular precession, when
the possibilities are examined of securing that type of adjust-
ment with the (C) axis directed nearly in the upward vertical.
We shall confine inquiry, on this side as well, to outlining the
connections; their essentials being grasped, the exhaustive
treatment of details offers no other obstacles than the inevitable
mathematical difficulties.
The first pertinent thought is derivable from equations (278)
when a range into the second quadrant is permitted to (&),
and a discrimination needs to be regarded between real and
imaginary values of the rotation about (Z), or between adjust-
ments that can and that cannot be accomplished. Selecting the
first alternative form for the solution, this dividing line is to be
drawn where the values denoted here as special yield the relation
= (Co/( C )) 2 + 4WrA cos #'; cos &' < 0. (287)
And the critical magnitude which (w( C )) must at least reach if
imaginary values are to be excluded completely is given by
(288)
so that if the spin about (C) equals or exceeds this rate, the
attainment of regular precession at every position in (#) is only a
matter of providing the companion value of the spin about (Z).
With this simple mathematics clear the next step is, as in the
previous combination, to detect and assign the dynamical
176 Fundamental Equations of Dynamics
reason that must underlie it. The first stage in meeting that
requirement starts with the merely reshaped equation
^ = sintff -- W? - ^Co> (c) + A^YcostfY
(289)
This can be made to tell us that if the axis (C), having been
directed vertically upwards, moves away from that position, and
changes ($) by a small amount from the value (TT), it will be true
that
/ d#\ (290)
Aft = fti( -r- Jdt; cos & = - 1.
In words, the rotation rate (d$/dt) will always be subject to
reduction in magnitude when the above parenthesis is itself a
negative quantity; and we have discovered a cause for this
reduction by seeing how the weight moment meets a first claim
for guiding directional changes in (H); a special case under
equation (286) is before us now. The stronger such absorption
of force-moment, the more rapid becomes that check upon the
initial motion in (ft), which will begin straightway as (C) leaves
the upward vertical whenever the parenthesis is in the aggregate
negative. Therefore we are led by these considerations to look
at equation (284) in a somewhat new light after rewriting it
( 2 S4a)
Then a zero value of the parenthesis when its factor is not zero
marks the transition between favorable and unfavorable con-
ditions for checking an existing motion in (ft). In application
to the second quadrant, the third term must be a positive mag-
nitude always, but it decreases as (C) approaches a horizontal
position. It is clear that cases may occur where the first member
The Main Coordinate Systems 177
has unfavorable sign as (#) leaves the value (TT), and becomes
favorable only after a finite drop of the axis (C). Also it has
been seen that the unfavorable interval can then be narrowed
by quickening the spin about (C), and it disappears at the
critical value indicated by equation (288). Because (sin # = 0)
is always one solution, there is a discontinuity possible here
between the two types of solution, similar to that for the conical
pendulum obtainable by assuming (d<p/dt) zero in the second
form of equation (278). The classification sometimes made of
gyroscope tops as weak and strong follows the line of thought
just traced.
132. The factors in the second term of the parenthesis that is
under examination are never quite independent so long as (d^/dt)
occurs in (co(^); but their dependence assumes a special phase
when the (C) axis and the vertical can become coincident, for then
there will be only two different expressions for the same (vertical)
component of (H). In order to develop the latter relation and
to reduce the parenthesis accordingly we shall begin with the
more general statement and afterwards particularize it. By
projecting from (B') and from (C) on the vertical and adding
we obtain
-j- + -T- cos
dt dt
) C cos # . (291)
Consequently
H Ul) - ih(H (c) -*i) ~ fc A sin 2 ; [B' = A] ; (292)
with the general value for the tensor ratio
d\f/ H^j) -- H (n) cos
dt- A(l-cosM) -
which gives under the equality attendant upon coincidence in
178 Fundamental Equations of Dynamics
the upward vertical, the conventions for signs being duly recon-
ciled,
Substitution in equation (290) shows as a condition that the
right-hand member should be negative when (C) leaves the
upward vertical with positive (At?)
(Cc^e)) 2 > 4AW?. (294)
The greater this inequality the stronger the retardation, the
sooner the departure is brought to a halt. The mathematics of
equation (288) has found thus a foundation in the dynamical
process initiated when (C) leaves its vertical position.
133. In what precedes, the emphasis falls upon moment of
momentum in relation to force-moment. The thought is not
complete however until the work of the weight moment has
been connected with changes in kinetic energy. For the case in
hand we find by using the principal axes,
E-iAlYgY + fgrinfYl + iCV,.,; (295)
and the last term being constant, the variations or interchanges
consequent upon work done are confined to the two other terms.
Now referring to equations (285) examination soon convinces us
that the initiative, so to speak, centers in the quantity that is in
the line of the resultant force-moment. So long as (d#/dt) is
zero, no change can occur in (co(b')); but the vanishing of
(w(b'), co(e)) separately or simultaneously might not prevent
changes in (d#/dt). It is characteristic of the stability here in
question that the action depends vitally upon the actual oc-
currence of a displacement; and this accounts for the known
feature of gyroscopic mechanisms, that their efficiency is nullified
by removing the degree of freedom upon which their functioning
depends.
The Main Coordinate Systems 179
For the power as the derivative of the kinetic energy, we can write
p - A [ (a? ) 5F + "> SE <">> ] - M <-'> 5? ' (296)
Let the conditions be such that positive work is done, negative
moment being accompanied by negative displacement. Then
the first term in the second member will be negative for opposite
signs of its factors. And we see diverted from their appearance
in the coordinate (#) the magnitude changes in both (H) and (E)
that (M) would make visible there, were there no gyroscopic
interactions.
The general agreement of the equation (288) and the inequality
(294) in their formulation of a critical value is obvious; and it
ought not to be longer obscure why the same truth is at the
foundation of each criterion. The essence of the adjustment to
regular precession is the insufficiency of the available weight
moment at a certain value of (t?) and other quantities to do more
than supply exactly what is needed for the corresponding direc-
tional change in (H). The reversal in sense of the inequality
that we arrived at, declares in effect an unavoidable preponder-
ance of weight moment consistently with the other given values,
and its sufficiency to quicken the motion in (ft) that is supposed
to exist already. It is an easily deduced consequence therefore
as regards the axis (C) that it will continue its departure from
the upward vertical until conditions alter. The imaginary
range of equation (278) is one signal that the combination of
the accompanying spin about (Z) with the actual horizontal
component of (H) is within that region unequal to monopolizing
the full force-moment active. The quantitative elaboration of
these leading ideas produces the accepted results in every detail.
GENERALIZED MOMENTA AND FORCES.
134. At the date of their original announcement, Lagrange's
coordinates and the equations of motion that employed them
180 Fundamental Equations of Dynamics
were contrived in the service of what would now be called
mechanics proper, for the imperious reason that the longer list
of energy transformations which dynamics distinctively em-
braces had not yet been discovered and drawn into the funda-
mental quantitative connections. The terms coordinate, con-
figuration, velocity and momentum were enlarged by Lagrange
from usage as he found it no doubt, but his broader scheme did
not break the alliance with geometrical ideas for its kinematics.
His parameters were ultimately based on combinations of
lengths and position angles, though kept unspecialized by sup-
pressing or deferring the analysis of them into the plainer geo-
metrical elements. The energy too was introduced primarily in
its kinetic form, that and momentum deriving their dynamical
quality from those inertia factors that are in their nature either
directly given as mass, or else as literal as moments of inertia
that emerge from a mass-summation. 1
Lagrange's equations will be found akin to Euler's in two
respects: first they are normally intended for treating as a unit
some body or system of bodies; and secondly, they are after a
fashion of their own indifferent toward a substitution of one
system for another, provided that determinate equivalencies are
observed, as we have seen Euler's equations to be under invari-
ance of (A, B, C) in magnitude. This likeness extends far
enough to coordinate the two plans and to make the latter when
duly stated a special result of Lagrange's broader handling.
The demonstration offered by Lagrange himself is founded on
d'Alembert's principle; and this interconnection of the two
phases of the same idea, and of each with Hamilton's different
formulation of it, lends to the establishment of the equations of
motion an air of logical redundancy. This was the subject of a
passing remark in our Introduction; and it might be recalled too
that the noticeable swing away from the first vogue of d'Alem-
1 See Note 30.
The Main Coordinate Systems 181
bert's statement centers upon a recent discovery of more compre-
hensive adaptability in the alternative forms devised by Lagrange
and by Hamilton to a range of energy transformations that was
unsuspected when either of the latter was first accepted. By
the light of what is developing further in that quarter the esti-
mate of their fruitfulness will continue to be decided.
Because these are the origins it seems advisable to let the
treatment here conform to them, instead of making a short path
to the newest reading. There is ground to expect that the fuller
realization of meaning in the extension of method and of its
valid possibilities will have its best source in a reasoned apprecia-
tion of where the latent power resided and how it was implanted.
We hold one reliable clew already, wherever it proves true that a
mechanism, construing the word not too remotely from direct
perceptions, can be seen to give in its fluxes of energy and momen-
tum a quantitative equivalent for those fluxes under less restricted
conditions of transformation.
135. On working outwards to occupy a broader field, and
passing at points the limits earlier drawn, some elements of new
definition or specification are involved, which the circumstances
lead toward supplying in part positively, in part by noting the
barriers that remain. And we shall relinquish the attempt to
finish each topic in a systematic progressive order, wherever it
promises better success to proceed less rigidly; coming back to
add a stroke and explain or define what was at first only sketched.
When it is said that any set of coordinates must determine a
configuration completely, the plain idea is that they do for a
system what we expect of the standard frame (O, XYZ), the
coordinates being enumerated for as many joints or articulations
as removal of ambiguity makes necessary. If the coordinate
set is thus equivalent to (xyz), the same idea may be conveyed
by declaring each general coordinate to be a definite function of
the set (xyz). In normal usage we do not abandon the relation
182 Fundamental Equations of Dynamics
upheld for other coordinate systems, that the values expressed
with their aid are standard frame values of the quantities dealt
with, but we seek that aid through any convenient functions of
(xyz) and not merely through lines and angles. Such pre-
liminary conception of a coordinate denoted by (K) prepares the
way for a definition of the corresponding velocity as (K), meaning
the total time-derivative of the magnitude of (K), the question
about vector quality being left open, an equal number of veloci-
ties and of coordinates being matched each to each.
Passing next to momentum we are again confronted with a
definition that pairs each velocity with its own momentum
quantity. Let (q) denote one of these momenta belonging to
the velocity (K); then the defining equation is written, if (E)
is still the total kinetic energy of the system to be studied,
q-ff. (297)
And another fixed point in the scheme now being presented is
that (E) shall be a homogeneous quadratic function of all the
velocities (K). To this specification other things must be made
to bend should that become necessary, which is a matter for due
inquiry. But meanwhile one evident consequence of it can be
read from the last equation, regarding the constitution of the
momenta (q) ; they cannot be other than linear functions of the
velocities (K) and homogeneous. Refer however to the closing
remark of section 141.
136. Putting together what has been said, one feature in the
relation of coordinates to configuration is caused to stand in
relief: they must determine it in a form free from all reference
to velocities in order that (E) may take on the assigned type.
Let us add as being naturally required, that the members of a
coordinate set must be mutually independent, and proceed to
speak of their connection with the so-called degrees of freedom
The Main Coordinate Systems 183
that a system of bodies possesses. Consideration of simplest
instances, like that of a ball carried on the last in a numerous
set of rods jointed together, shows that a large number of speci-
fying elements or coordinates may be actually employed in
designating configuration, even in one plane. But we know also
that two rectangular or two polar coordinates only are required
in this case; and the prevailing distinction seems to follow the
line thus indicated, making degrees of freedom equal in number
to the minimum group of coordinates requisite in describing a
configuration, classing the excess in the number really used as
superfluous coordinates. This disposes of the matter well enough,
leaving for special examination only such interlocking of two
coordinates into related changes as happens when a ball rolls
(without sliding) on a table; and that finer point need not detain
us. In these terms, a rigid solid has available not more than
six degrees of freedom, three of which might call for coordinates
locating its center of mass, with the remaining three covered
by the Euler angles, for example. And we may borrow from
regular procedure in that case, as known through repeated
discussion, that an equation of motion is associated with each
degree of freedom. That normal arrangement continues with
evident good reason, though our treatment is shaped according
to Lagrange's proposals, which do not change the objective in
essence, but only the mode of reaching it.
137. To complete the plan, therefore, into which accelerations
do not enter directly, there is need to specify its forces; here the
determining thought has its root in the energy relations, running
in the course that we shall next lay out, whose first stage has no
novelty, but merely holds to the equivalence in work established
for any resultant force. The right to substitute one force (R)
for all the distributed effective force elements depends upon its
equality with them in respect to total work and impulse. The
same thought, in other words, declares equal capacity for setting
13
184 Fundamental Equations of Dynamics
up the total flux of kinetic energy and momentum in relation to
the system of bodies, the separation of force and couple moment
or of translation and rotation being a detail and without final in-
fluence. It is inherent, moreover, in the determination of any
such resultant through vector sums or through algebraic sums
that a set of components may be variously assigned to the same
resultant. The ground that Lagrange traversed led him to a
variation only on previous forms in expressing this essential
energetic equivalence of the resultant force. The fact indeed
that he set out from the equilibrium principle of virtual work due
to d'Alembert should obviate any surprise on meeting the
defining equation for his generalized forces.
With each degree of freedom which makes flux of kinetic
energy possible, associate its force (F); sum the work during
elements of displacement in all the coordinates (K) and express
its necessary equality to the same work given in terms of the
usual forces parallel to (X, Y, Z). The equation is
S(FdK) = S/ m (dR-ds)
= S/ m [dR (x) dx + dR (y) dy + dR (z) dz], (298)
which yields by a transformation that embodies through the
partial derivative notation the supposition of independence that
goes with the coordinates, for each force an expression
F = 2/ a
[dR (x) |+dR (y) |+dR (z) |]. (299)
Holding to this statement any force (F) can be defined in magni-
tude by the work per unit of displacement in its coordinate;
and the narrowing assumption does not appear that (F) and (K)
are colinear, provided a convention can be observed that gives
the work its real sign as determined by gain or loss to the system's
kinetic energy. It is this relation which Lagrange's equations
enlarge by including the other energies of dynamics.
The Main Coordinate Systems
185
We continue by introducing necessarily equivalent expressions
for a change in configuration,
(300)
in which the summation extends to all the coordinates (K).
Then in the fluxion notation
from which follow for each coordinate singly the important
equalities
ak
ak
ak
Taking the term from the first integral of equation (299), it can
be given the form, by using the last results
dR I - i ( dQ I) - dQ <-> it (I) ; (303a)
and similarly from the remaining integrals,
dy d / 5y\
a^ = dt V dQ(y) ^ ) ~
d /5y
dt
^ HO
(z) a^ = dt V dQ(z) ai ~ dQ( ' } dt
(303b)
To recast the last factors in these three equations we write
jl
dt
dt
(304)
186 Fundamental Equations of Dynamics
whose justification is somewhat a matter of mathematical con-
science. The order of the two differentiations may boldly be
inverted as a legitimate operation; or whatever hazard may be
felt in that can be guarded against by rigorous proofs that are
accessible. Incorporating the last forms and summing equations
(303), the force finds expression as
-SUE;-S> (3os)
in application to each one of the coordinates, and the whole
development is then open to further comment or illustration.
138. This exposition of Lagrange's equations, and of the con-
cepts upon which their statement rests, has been kept apart
purposely from the infusion of vectorial ideas, in order to set
forth as clearly as may be done that possibility upon which
their larger usefulness in great measure depends, and of which
insistent mention was made in the first chapter. Some care
seems needed to break up the misleading connotations of words
like velocity and momentum, that in their first and perhaps
most literal sense imply each an orienting vector. And the
emancipation of thought in this regard has been hindered doubt-
less by the unsuggestive practice of pointing out as examples of
this method of attack solely those where velocities and momenta
and forces offer themselves habitually as vectors like those
which our material has been including hitherto. If the trend
of any demonstration equivalent to the foregoing be watched,
however, it is seen to hinge essentially upon an enumeration of a
sum of terms in the total energy of all forms that are considered,
and analyzing them as products that conform to a type. This
contains always as a factor the time rate of one in a group of
quantities by whose means the changes in that energy content are
The Main Coordinate Systems 187
adequately determined. The success of the analysis therefore
depends, broadly speaking, upon the isolation of suitable factors
in the physics of the energy forms to specify the energy configur-
ation and to provide the necessary velocities. And in that direc-
tion it is interesting to note the part really played by the (XYZ)
velocities and momenta as they lead to the vital connections in
equation (305). They are scarcely more than a scaffolding, an aid
in building but removed from the structure built, impressing
effectively only one character upon the result that its scheme of
values shall be quantitatively a possible set in that mechanical
phantom or model which is mirrored in the case treated. On their
face, Lagrange's equations might seem to stand in parallel with
tangential ordinary forces only, since the latter are alone con-
cerned in work. But we shall show that this limitation does not
in fact exist, and that the pattern set by the (XYZ) axes when
they include for their projections constraints as well, is stamped
upon these other combinations, which may be caused also to
reveal normal forces that may be active (see section 141). As a
counterpart to this relation it is to be observed how the (XYZ)
axes fit everywhere into a plan of algebraic products through their
three coexistent and practically scalar operations; and how for
the element of scalar mass equations (1, II) are always free
alternatives, whatever restrictions subsequent steps may impose,
as for instance equation (67) has recorded.
139. Having laid some preliminary emphasis upon the extent
to which they may exceed in scope other coordinate systems, it
will be advisable to carry the comparison with Lagrange's plans
into the region of overlapping, and make this last system prove
itself capable of bringing out correct consequences there too, when
orientation is reestablished. The cross relations have many
lessons that are of value; and some are jdelded by a review of
the polar coordinates that we shall put first. Borrowing from
section 106 the expression for kinetic energy of a particle, and
using fluxion notation for brevity,
188
Fundamental Equations of Dynamics
E = |m[f 2 + rM 2 + r 2 sin 2
(306)
The Lagrange coordinates must be independent and sufficient
to give configuration in (XYZ) ; and (r, #, ijr) meet this require-
ment. But the velocities must correspondingly be (r, #, ^).
The details work out into the forms, (dE/dt/0 being zero,
dE
dT
d
dt
= mr:
dE dE
- = mr 2 #: T-T = mr 2 sm 2
jT I TT. I = m ( 2r
dE
r-
or
sn * cos
dE
- =m(r 2 sin # cos
ou
(307)
A general agreement is at once manifest when these terms are
grouped and compared with equations (208) ; but it is a striking
difference that the forces (F( d >) and (F^)), associated with
those two coordinates, must now be recognized as moments of
the forces denoted previously by (R( x '>) and (R( y ')), for rotation-
axes characterized plainly through the respective lever arms.
This is a necessary concomitant of making velocities out of
($, \l/). The regrouping of terms also is instructive in betraying
that loss of distinction for the orientation changes here as well
which algebra usually evinces.
140. For a second example, let us make in the Lagrange form
a restatement of section 89, utilizing equations (154) as a starting-
point, and adapting them to a particle, as the desirably simple
case. If (x', y', z') are selected as three coordinates, the con-
figuration in (XYZ) is not determinate by them alone, but in
the plan followed the position angles for the axes (X'Y'Z') must
be known also; and of these as many as are independent can be
The Main Coordinate Systems
189
added to make the required list of coordinates, of which all but
three will then be superfluous in a sense already explained, and
not to be reckoned among the degrees of freedom. The purpose
of illustration can be attained sufficiently if we consider the
uniplanar conditions, both for the particle which is then supposed
to be restricted to the (XY) plane, and for the relative con-
figuration of (X'Y'Z'), where we assume (Z) and (Z') permanently
coincident. Hence for the kinetic energy of (m) the expression
is in understandable terms
E = |(x 2 + y 2 )m = |m[x' 2 + y' 2 + (x' 2 + y' 2 ) 7 2
- 2x'y'7 + 2x'y'7L (308)
the coordinates being now (x', y', 7) and the velocities (x', y', 7) ;
the last velocity is an algebraic derivative, (Z) being the fixed
axis for (Y). Again the details are, when this homogeneous
quadratic function of the velocities is differentiated,
dE dE
- = m(x' - y'-y); > = m(y' + x'-y);
r = m(7(x' 2 + y' 2 ) -x'y' + x'y');
, = m(7 2 x'
, = m(7 2 y' - 7x0;
6iE
= 0.
(309)
After forming the time-derivatives of the first three in the group
and substituting values, we obtain for the three forces of the
coordinates,
F (x ') = m(x' - 7'y' - 2 7 y' - 7 2 x');
F (y ' } = m(y' + 7x' + 2 7 x' - T 2 y'); |- (310)
F (v) = x'F (y ') - y'F (x) -.
1 90 Fundamental Equations of Dynamics
The third coordinate advertises that it is superfluous, in that its
force value, whose form is readily verifiable as a moment, only
confirms what is otherwise ascertained about the remaining
forces.
141. In their adaptation to the present class of cases, some
truths can be picked out that furnish clews for the lines of more
extended use. First, referring to equations (155) and collating
them with equations (302, 304), the latter are seen to be far-
reaching analogues of changes that build upon the line of the
quantity at the epoch, and of those others that depend upon a
change of slope; they are correlated respectively with changing
tensor and orientation of a vector. While a partial derivative
like (dx/dK) may appear as a direction cosine within the purely
geometrical conditions, it is a more inclusive reduction factor else-
where. It is also open to observation in the last two illustrations
that the generalized momenta become for those applications the
orthogonal projections upon a distinguishable line, either of the
momentum or of the moment of momentum in the standard
frame. Differences of distribution for the same total projection
between various pairs of groups is no more than part of the
mathematical machinery, and it is especially to be expected
where sets of partial derivatives occur whose variables have been
changed. Note that
dE dE
presuppose: the first, that all coordinates are held stationary*
and all velocities but that one; and the second, that only the
one coordinate is allowed to change, and none of the velocities.
Comparisons with other sets of partials in our developments
should prove helpful, as it will be to find answer for the question
whether the Lagrange plan, when it deals with forces like (R),
affiliates more closely with the mode of equation (112) or with
that of equation (233).
The Main Coordinate Systems 191
Related to the second example here and to the ideas about
superfluous coordinates, is another point of view that has like-
ness with the method of section 82. The standard frame coordi-
nates, as expressed in equations (150), can be discriminatingly
dependent upon time, indirectly through (x', y', z') and directly
through the direction cosines. Their exact differentials will then
appear as
with two companions, the last term in each comprising the group
that arise by differentiating the direction cosines if we have re-
garded (xyz) as given in a functional form like
x = f(x', y', z', t), (313)
and the superfluous elements are spoken of and dealt with as
due to variations of the geometrical relations with time. The
distinction that such changes of direction are assigned and not
brought about by physical action is consistent with what has
been seen above the absence of those additional force speci-
fications that would be introduced through them otherwise.
The exercise of preference in selecting the elements to be drawn
off thus into their own time function, however, need not be always
the plainest of matters. And where an accompanying verbal
usage is accepted that denies the title coordinate to position
variables not ranked among degrees of freedom, the kinetic
energy ceases to be a homogeneous quadratic function of the
(remaining) legalized velocities. Of course these comments
hold good for extension to the generalized energy configuration.
142. Retaining the energy value and imposing upon equations
(310) the conditions that (7) and the origin shall be so regulated as
to keep (V(y')) at zero permanently, they conform to the tangent
and normal resolution of force for those uniplanar restrictions;
and in space curves there is the same correspondence between
192 Fundamental Equations of Dynamics
the general case and the one duly specialized. The test of the
latter form being of some length and of no difficulty, and because
it shows finally only an equivalent for section 115, we pass it
with mention merely and proceed to examine Euler's equations
for instructive connections with those of Lagrange.
We can quote two equally valid expressions for rotational
energy of a rigid solid for which (A = B), when mounted as in
section 127:
(4* sin #) 2 )A + %(<p + cos r?) 2 C. (314)
In the former, no total time-derivatives can be detected of quan-
tities determining configuration, but only those projections of a
given (to) appear which presuppose knowledge of the configura-
tion, and which could be rated partial derivatives of (y) accord-
ing to the explanation of equation (185) as related to section 79.
This fact has been noticed in several connections since the subject
of position angles was opened (see sections 93 and 98), and it
explains why the direct expression by means of the Euler angles
is not entirely superseded by using (<>(), W(b), "(c))- The co-
ordinates are then (ifc, #, $), the velocities (\j/, d, <p) in the
fluxion notation, and we foresee that our previous force-moments
will now figure as forces. It is plain that
ff-g-O; F ( ,,-F U) =0 ; (315)
the latter pair of values expressing the controlling constancies of
the moment of momentum in this problem, or of the momenta
(tU)> Qc*)) m the present terminology. These values when
worked out, and those that complete the expression
J -jl?
Fw,-^,,)-^,
are all in recognizable identity with what was obtained elsewhere.
The Main Coordinate Systems 193
143. The action of the gjToscope has been seen capable of
diverting energy from one coordinate to another as a perhaps
secondary consequence of maintaining change of direction in a
moment of momentum that is of constant magnitude. And it is
easy to multiply instances, wherever the inertia factors (moments
and products of inertia) can be variable, that a change in value
for kinetic energy is demanded under constancy of the other
quantity, this being entailed if the rotation factor alters. Thus a
symmetrically shrinking homogeneous sphere has constant (H)
under the influence of gravitational self-attractive forces between
its parts, but the rotational energy grows as an expression of
work done in the shortening lines of stress. In symbols, for
rotation about a diameter,
1 / H \ 2 H 2
H = I (D) ; E = i<o 2 I (D) =-(-) I (D) =s r - > (317)
2 \I(D)/ 2J -(D)
with the denominator growing continually smaller. What is here
illustrated is more widely possible to happen among the analogous
factors of energy, where its different forms are interconnected in
the same system, so that the energy may be transferred and
redistributed among the Lagrange coordinates though some of the
corresponding momenta remain unaltered. Neither is it remote
from the mental attitude already alluded to, in approaching the
study of a physical system through certain external and accessible
bearings of it while a margin is left for less definite inference, to
base tentative conclusions about concealed constant momenta
upon observable indirect effects on energy. It is some prepara-
tion for those fields of usefulness to follow out the relations in the
next sequence of ideas, which may be carried through first for
directed momenta and finally be restated more broadly.
We shall suppose a system with four generalized coordinates,
three (^, #, <p) what we have termed accessible, and details about
the fourth (T) to be subjects for inference, as we may say. The
194 Fundamental Equations of Dynamics
latter has then naturally no force assigned to it for direct con-
nection with changes of energy, and is adapted to the thought
expressed above, by having its momentum assumed a constant
magnitude. Accordingly these conditions are written
F (T) = 0; Q( T > = constant. (318)
Add the supposition as conforming reasonably to the limitations
upon knowledge, that no known relations contain (r) itself.
Then since
d r)E
each term in the second member vanishes separately or is a blank.
144. The momentum (q( T )) being actually present can modify
the phenomena; that is the effects of other forces and the energy
reactions. It is to be asked : How will the statements be recast,
if we detect (q( T >) as though distributed in parts added to the other
momenta, to which the phenomena are being exclusively ascribed?
This moves in the direction of suspending direct inquiry into (T),
so the method is frequently described as allowing ignoration of
coordinates. 1 Expressing this resolution of (q( T )) with the aid of
the direction cosines (1, m, n), and adding its components to the
other momenta as indicated, the total orthogonal projections on
the lines will indicate
= q (*> + lq<r>; -^ = q u> + mq (T) ;
3E
(320)
^ == q
The coordinates (^, r?, <) need not be themselves orthogonal, but
the parts (q') and (qco) are.
The adjudged energy (E) would then have to satisfy the general
relation growing by implication out of the real scalar product for
rotation
i See Note 31.
The Main Coordinate Systems 195
E = K"-H), (321)
the possible non-linearity of any velocity (k) and its momentum
(q) being here also recognized; this yields the form
+ <p(q' U) + nq (r) )]. (322)
Introducing (Q) in this connection to denote the constant magni-
tude (q(T>), the forces derivable from the supposed energy will
appear as containing the terms
_
dt
_d
dt
d_
dt
dt
= . n
^
/dE\ d /
\dtf / dt V
dE
(0 )
dt
dm
+ Q dt~ ;
(323)
The quantity of energy (E (0 )) represents what would be present
if (Q) were non-existent, and the last terms in the equations
register the modification due to the introduction of (Q) on the
supposed basis, namely through its resolved parts that maintain
the directions of the momenta (q^), q(*), qu>)- Their indi-
cated connection with changes of direction relative to (\f/, &, <p)
momenta should not pass unnoticed. To conform with the above
values, the energy (E 7 ) allowed for in excess of (E (0 )) must be
E' = MQ) + *(mQ) + *(nQ); (324)
and in order to fill out consistently the scheme begun with equa-
tions (323) we must continue in the expressions of force with
aTP .aTj 1 ->TTI/
aJcj oLi(o)
dE(o) dE
'
(325)
d<p
196 Fundamental Equations of Dynamics
But we find
aE'
aE'
aE'
ai .am an
ai . am an
ai .am
(326)
Hence the aggregate departures from the forces that would be
indicated by (E (0 )) alone can be seen in
aE d
'(0)
n I ~ i dl , dm . an
I ^-V I Jx I T f\ / 1 A.1 I -r f\_/
aE
(0 )
.am an
(327)
a.E (0 )
dt\ a^>
dn / . ai .am
d_/aE\ _ aE __d /
dt V 3d )~ d& ~ dt V
_
dt \d<p
145. But the energy really introduced by the momentum (Q),
like the other portion E( ) of the energy is expressible by a
homogeneous quadratic function of the velocities which it is
permissible at any one epoch to put into the form
E (Q) = $K(f + \i + nn*'+ n) 2 , (328)
(K) being a function of coordinates only, and the value being in
The Main Coordinate Systems 197
other respects fixed by necessary relations for partial derivatives
of E (Q ). Thus
-Tn
= K(r + 1^ + mi? + n<p) = Q [by definition];
or
[by equations (320)].
Further we have, since (E (Q) ) involves coordinates through both
factors,
- _i- A _u ^ i . fQQm
~\ 17 ~| (0 I . lOOv/ J
and the second part is recognizable through equations (327,
329). In order to adapt the remaining part to the present
connection, first put equation (328) into the legitimate form
next shown, and then express its partial derivative for a coordi-
nate, subject to our condition that (Q) is a constant magnitude.
The results are
F 1Q2 -
(Q) = 2 K '
(331)
3E (Q) !Q 2 dK l^K i
... = o^TT = ~o rr( T + + mt? + np), 2
Oy i\.~ (71^ Oy
and the last member is identified as the negative of the corre-
sponding quantity in equation (330) . Its appearance in the final
forms is intimately related to a diversion of energy that persists,
though the action of (Q) is veiled otherwise. Utilizing all these
detailed relations justifies the equality, where the notation for
the last term in the first member indicates the condition observed,
and for (dl/dt) we have inserted the value
198
Fundamental Equations of Dynamics
d/3E^\
dt\ dj /
dl 31 . 31 . , d\
TI = T7'/'+T^ ?? +V~^>
at ay ov o<p
+ Q
dn\
) <P
d\f/ /
h
dm\ "I pE (Q) ] _d/dE\
^/ J L ^ JQ dt \d^7
dE
S- (332a)
+ r^|(Q)i .fir* (E(0) +
a
~3^ (E(0) +E ^ } =
To which the companions added after cyclic interchanges are
- _ __
dt \ a^ J ~ . d&
/dm _an\ . 1 , rE
V d<p dd J * \ + L 5* Q dt
aE p^ E (Q)i d r ^
"a7" ~^~J Q ~dtL^ ( (0) "
- (E(o) + E (Q) ) =
TT I -". I
dt \ o<f> /
dE
(0)
, _
It?
+
d<f>
~ (E
<o)
(332b)
146. It is plain from these forms how the actual values of the
last members but one for the energy changes in the system may
The Main Coordinate Systems 199
be preserved and an account of them be given under various
other interpretations that are in a sense fictitious. Or they are
put in a fashion that uses knowledge up to its borders, with safe
non-committal beyond them. What is here exemplified for one
coordinate ignored, can be extended of course to many by a
similar procedure. And when acceptance of reduction factors
has widened the range outside that covered by the geometrical
direction cosines, intricacies of energy connections are made
resolvable in many general ways.
It may happen that some contributions to the total group of
forces acting on a system are comprised under a potential energy
function; and it is in the nature of those relations that such
forces are independent of velocities. If therefore there is any
gain in doing so, the active forces may be held asunder in two
groups, one containing all the forces derivable from any potential
energy functions (<). Then in any coordinate (K) the new model
of Lagrange's equations is only formally varied when it is written
- (E - $), (333)
since ($) is inoperative in the first term, and in the second it
only transposes one group of the forces. But this type offers
the significant feature that a course of events to which the first
member can be the key, is exhibited as depending upon the
momentary outstanding difference between two quantities
measurable as energy. And with the door opened as usual to
seemingly vital analogies among energy forms, much is being
done in these days to increase the command of dynamical state-
ment for the most inclusive rules or principles deciphered among
physical sequences of transformed energy. It did not seem, there-
fore, that the objects of the chapter on the side of stimulating
suggestion would be attained unless we were brought to this gate-
way into a larger field. But then too we must be content with
14
200 Fundamental Equations of Dynamics
that much of accomplishment, leaving the other forms of La-
grange's equations, beside this second one as they are usually
counted, to the systematic continuations of which there is no lack.
The exploitation of the concept called kinetic potential, whose
roots can be traced in the difference (E $), and its alternative
origin as a deduction from Hamilton's principle of stationary
action, are the groundwork of much modern dynamical thought. 1
1 See Note 32.
NOTES TO CHAPTERS I-IV.
A
Note 1 (page 2). To be aware of are an initial trend through
the drift impressed by the nature of the material, as well as an
active later movement with its propaganda. Regarding the
first of these headings it is discussible whether the opinion alluded
to in section 3 is fully representative of Newton's own stand-
point, or whether that tendency to one-sided development was
due to adherents whose acceptance of ideas was narrower than
the scheme of his proposal. So much can be done by way of
expanding or contracting the thought lying behind a condensed
formulation in Latin that we tread on insecure ground in at-
tempting a decision. Safest it seems to allow in Newton's
plan at least potential provision inclusive of all that two succeed-
ing centuries could reasonably urge on this score. Adding per-
haps, what expert judges would have us not overlook, that a
comprehensive power-equation is laid down in the scholium to
the third law. Read in English thus: "If the Activity of an
agent be measured by its amount and its velocity conjointly;
and if, similarly, the Counter-activity of the resistance be meas-
ured by the velocities of its several parts and their several
amounts conjointly, whether these arise from friction, cohesion,
weight or acceleration; Activity and Counter-activity, in all
combinations of machines, will be equal and opposite" (Thomson
and Tait, Natural Philosophy (1879), Part I, page 247). The
genius of Heaviside for directest dynamical thinking approves
this scholium as capable of covering the fluxes and transforma-
tions of energy that more recent dynamics introduces (Electro-
magnetic Theory, III, pages 178-80).
In the movement toward basing the derivation of other con-
cepts upon energy, Tait put forward an early denial of primary
201
202 Fundamental Equations of Dynamics
quality to force in a lecture before the British Association (1876).
The habits of thought in these respects, however, are interwoven
with a widespread campaign extending over the main issues of
epistemology (Erkenntnistheorie) that enlivened the period 1895-
1905, some of whose other aspects are touched upon subsequently
(see notes 4 and 5). The party there whose watchword was
"Phenomenology" made common cause with energetics as a
properly neutral mode of statement, in opposition to theoretical
physics or more justly to overweight in speculation. These
matters of broad sweep are only to be hinted here; they are fully
in evidence throughout the journals of that date. Yet we may
admit mention of two books, one showing how energetics
counterpoises ' and supplements other aspects of dynamics, and
the second exhibiting by contrast exaggerations into which
zealous advocates were led. The titles are: Helm, die Energetik
(1898); Ostwald, die Naturphilosophie (1902).
Note 2 (page 4). The spirit of this paragraph finds confirma-
tion in recent judicial utterances, as regards both appreciation
of the new movement and prudent reserve in passing judgment.
Consult Silberstein, The Theory of Relativity, for a lucid account
of the Lorentz-Einstein method that estimates its gains with
candor and acumen. The workable value in the opened vein
of possibilities will be extracted progressively, as its logic is
brought to bear upon questions involving previous sequences
and their origins. Poincare expresses this plainly in his summing
up: "Aujourd'hui certains physiciens veulent adopter une con-
vention nouvelle ... plus commode, voila tout. . . . Ceux qui
ne sont pas de cet avis peuvent le'gitimement conserver 1'an-
cienne. . . . Je crois. entre nous, que c'est ce qu'ils feront encore
longtemps" (Dernieres Pensees, page 54). Clarification and
settlement here seem delayed by an observable tendency to
expound the central ideas of relativity in an entanglement with
much irrelevant mathematics that is describable also as tran-
Notes to Chapters I-IV 203
scendental. This blurs essentials and will obstruct the final
rating of the novel features among the resources of physics.
It is foreign to such alliance, and hence perhaps one influence
toward dissolving it, that the modified handling of simultaneous-
ness traces its lineage so directly to experimental evidence, and
the effort to state its results with unforced symmetry. Yet on
that side, too, there might arise need of corrective, if perchance
the conclusion were entertained seriously, that any newly as-
sumed attitude releases us from that bondage to idealized
concepts and simplifying approximations which sections 12 and
13 indicate. We should be compelled to reject every inference
that some system invented to replace Newtonian dynamics can
be other than differently conceptual and approximate. What
alternative concepts to employ will always remain as a choice
determined on practical grounds. It would be breaking with
the canons of sound scientific doctrine to displace one series of
working ideas by another whose improved adaptation to universal
service is at best to be classed among open questions. Though
symmetry in equations is desirable, it is not to be secured at all
costs. In order to turn the balance conclusively, insight must
first be attained that goes far enough in excluding illusion from
the corresponding dynamics. The characteristic formulas of
relativity draw their suggestion from groups of phenomena that
spread over limited area as compared with the explored range
of physics. Their analysis beyond the kinematical stage, more-
over, is too obscure and intricate as yet to afford mandatory
reasons, or even trustworthy guidance, for much reshaping of our
fundamental equations. See note 11 below, in continuance of
this thought.
Note 3 (page 6). The reference is to Maxwell's Treatise on
Electricity and Magnetism, II, Chapters V and VI of Part IV.
He records (1873) the stimulus received from the Natural
Philosophy by Thomson and Tait (1867), and from the revival
204 Fundamental Equations of Dynamics
of dynamical advance inspired by "that stiff but thoroughgoing
work" (Heaviside). It continues to offer an unexhausted mine
to a later generation. In its second edition (1879) the present
topic by added material and recasting points rather plainly
toward mutual reaction between Maxwell and its authors. It
is true that their expanded treatment does not explicitly occupy
his larger field, though their gyroscopic illustrations run easily,
as can be seen, into a generalized scheme of cyclic systems. In
that direction Ebert, with Chapters XX-XXII of his Magne-
tische Kraftf elder (1897), has made a junction by elaborating
into dilution the results of Hertz and Helmholtz. Others like
Gray prolong directly the line of Maxwell's initiative (Absolute
Measurements in Electricity and Magnetism, II, Part I, Chapter
IV (1893)).
It is not premature to remark, in anticipation of notes 30
and 31, and with bearing upon the current presentations of
Lagrange's equations, how guardedly the vectorial connections
of their original scope are relaxing. We may suppose that the
freedom to cut loose in this respect has been for a time masked
by the cartesian (XYZ) forms, whose effective reduction to
quasi-scalar expressions has had an influence elsewhere, as pointed
out in section 91, toward indifference about such distinctions
that fails to regard them as vital.
Note 4 (page 9). What is appropriate here in preparing for
intelligent command of stock resources must not go far beyond
claiming for these inquiries a continued relation to the organic
structure of dynamics of which their perennial life is one con-
vincing proof. Some study of their literature cannot be dis-
pensed with, from which differently shaded opinions will be
drawn, to be sure, that will yet unite in agreement on the final
importance of the answers. To recommend this as one region
for deliberate thinking is the purpose at this place, leaving
opinions to shape themselves individually. The concession how
Notes to Chapters I-IV 205
fully routine belonging to execution can go its way unhampered
by deeper questions should be permitted to repeat itself without
undermining finally the need incumbent upon us to discuss them.
Section 16 alludes to some temporary grounds for unconcern,
others are supplied by the sufficiency of a fixed earth's surface
for staging so many investigations of physics, and in various
directions a fortunate postponement is tolerated. But testimony
is broadcast how steadfastly some settlement is nevertheless
held in view, for the experimental bearings of it even, when freed
from all metaphysical residue. For exemplifying reference take
Larmor's comment (Aether and Matter, page 273) and Helm's
pertinent remark (Energetik, page 216).
There were several leaders in the public sifting of these theories.
Prominent among them Mach, who has gone on record in his
Science of Mechanics, Chapter II, and elsewhere.* The possi-
bility of the so-called Newtonian transformation having been
put on a secure basis, that headed unconstrainedly toward using
an origin at the center of mass of the solar system and directions
determined by the stars for a natural reference-frame. Espe-
cially for what are rightfully classed as internal energies of the
system this would be capable of high precision in presenting
through accelerations relative to it, for the bodies with which
we deal, the physical forces active among them or upon them
(see section 52, and note 17). It is a live question of the passing
time whether that habit of mind had better be upset, or can be
superseded with definite net gain.
Note 5 (page 17). The assertion is hardly contest able, that
quantitative physics deals with an idealized and simplified
skeleton built of concepts, so soon as its content exceeds the
rules that are empirical by intention and form. The supports
found for outstanding argument are then two: first, uncom-
* This is the briefer title of the English translation, the original title being
"Die Mechanik in ihrer Entwickelung historisch-kritisch dargestellt."
206 Fundamental Equations of Dynamics
promising denial that the goal can be aught else than empirical
rules, ingenuity being restricted to embodying best in them the
ascertained data; or secondly, in questioning doubt how the
boundary-line runs among special cases. Troubles of the latter
origin involve no radical divergencies, since they are everywhere
inherent in such a separation of two classes, both being acknowl-
edged to exist. Positions like the first mentioned would be a
fetter upon growth through their exclusive blindness to patent
and historic facts, were not a saving clause inserted in extremist
tenets by human readiness to lapse into inconsistency for good
cause. To illustrate how the main contention spoken of would
cramp effort, we find place for a quotation, which however is
content to set two standards in opposition: "Die Fourier 'sche
Theorie der Warmeleitung kann als eine Mustertheorie bezeichnet
werden. Dieselbe . . . griindet sich auf eine beobachtbare
Tatsache nach welcher die Ausgleichungsgeschwindigkeiten
[kleiner] Temperaturdifferenzen diesen Differenzen selbst pro-
portional sind. Eine solche Tatsache kann zwar durch feinere
Beobachtungen genauer festgestellt werden, sie kann aber mit
andern Tatsachen nicht in Widerspruch treten. . . . Wahrend
eine Hypothese wie jene der kinetischen Gastheorie . . . jeden
Augenblick des Widerspruchs gewartig sein muss" (Mach,
Prinzipien der Warme, page 115). We know that the goal here
implied for theory is only the starting-post for it in the doctrine
of another school of thinking; but must abstain from even
outlining the argument.
The important concern for dynamics here turns plainly upon
the question of aligning it in method with the rest of mathematical
physics, or of excepting it from partnership in a search for con-
fessedly empirical rules. In point of fact, this one undeniably
fruitful wielding of idealized conditions has been a bulwark of
defense for universal procedure. No interested student can
afford to neglect Poincare's pronounced judgment in this field,
Notes to Chapters I-1V 207
to be found especially under the four book-titles: La science et
1'hypothese; La valeur de la Science; Science et Methode;
Dernieres Pense"es. The first three are most compactly accessible
in one volume of English translation headed The Foundations of
Science (1913); the fourth not included in that collection is of
recent date (1913) and presents much that is of value. Far from
putting these matters aside as completed, latest developments
have renewed and intensified their lively discussion. As repre-
sentative in one direction we name the work of Robb : A Theory
of Time and Space (1914); and on another line a paper by N.
Campbell (1910), The Principles of Dynamics (Philosophical
Magazine, XIX, page 168). These will sufficiently lay out a
track for further pursuit, in connection with notes 1, 4 and 6.
Note 6 (page 24). There is much more here than the kine-
matical colorlessness that precedes the introduction of dynamical
elements. Attention is being directed to that stage of inclusive
preparedness in the fundamental equations that is one permanent
attribute of "Analytic mechanics," in so far as its forms of
statement are made equally ready to contain various specialized
data. Workers in the subject really avail themselves of this
privilege to delay in particularizing. Lorentz for example does
not attempt to settle in advance which reference-frames meet the
conditions attached to the primary relations for the electro-
magnetic field. He lays the decision aside temporarily with the
passing remark that the equations remain valid so long as they
accord with the value (c = 3 X 10 10 cm./sec.) for light-speed in
free space. So a top's local behavior relatively to the earth's
surface follows equations of motion in common with the gyro-
scopic compass up to a certain divergence-point, though the
former ignores the earth's rotation, and the latter may be said
to reveal it. In a group of parallel cases the differences center
upon replacing gravitation by weight; which illustrates how
essentially the standards of desirable or attainable precision
enter into adapting broader analytic expressions.
208 Fundamental Equations of Dynamics
Note 7 (page 26). A number of points touching the fuller in-
corporation of vectors into physical purposes must become more
definite presently, as the novelty of their use subsides. Con-
ventions that have been transferred from mathematical defini-
tions, or that have been added tacitly, will be opened to needed
revisions first by being made explicit. The text will be found to
adopt this feature of sound policy at several places, none of which
should be slurred. Care to delimit equivalences legitimately in
relation to physical conclusions is one leading idea as regards
substitutions that approves itself to be a needed refinement
upon the looser term equality. For accelerating the center of
mass of a system forces have the quality of free vectors, because
their position is without effect upon equivalence in this respect.
Yet when we discuss motion relative to the center of mass,
forces fall away from that equivalence, being then dependent
upon position for their effect, and consistently they cease to be
free vectors. Such instances compel us to qualify classifications
and permissible substitutions.
Similar deliberateness in borrowing from mathematics is en-
couraged in section 68, with its suggested distinction between
triangle and parallelogram as graphs of a vector sum; and in
section 74, where an element of parallel shift enters to round out
the variableness of a vector quantity.
The idea of vector-angle used in equation (2) has not yet
found its way into textbooks. Its introduction is an almost
self-evident detail of any systematic vector algebra, to supply
the missing member of the series in which angular velocity and
acceleration were long since recognized. How that proves help-
ful is elaborated in section 92 and its sequel. The simple step
of completing with natural orienting unit-vectors the established
ratio (ds/r) for magnitude of angle seems to be announced first
in the Physical Review (N. S.), I, page 56 (1913). In section
46 the text opens from this side a new meaning for the rotation-
Notes to Chapters I-IV 209
vector that fits usefully in several ways, though it is, of course,
nothing but that second interpretation possible for every vector
product which happens to have been overlooked here. We
( must ascribe the oversight to a continuance of the earlier exclu-
sive habit of using only the projection of (r) that is perpendicular
to (o), and not the corresponding projection of the latter vector.
Notice how the rotation- vector can be given another role if we
rewrite equation (44) in the form
v = - (r X <o),
reading the second member as the negative moment of (w)
distributed locally at each (dm) . This has important connections
with the uses of vector potential, and the association of the curl
operator with the latter.
Note 8 (page 31). Later research has come to the aid of
mathematical demands or convenience on this side, by detecting
real transitions with however sharp gradient behind most first
'assumptions of discontinuous break. In proportion as facts of
that character gather they soften the impression of artifice in
making phenomena amenable to treatment by allowing for quick
gradations, and incline modern physics away from recognizing
discontinuous change except upon compulsion. See Lorentz,
The Theory of Electrons (1909), page 11. This accounts prob-
ably for some psychology alongside the mathematical needs
mentioned in section 26, of which we might admit an admixture
in the satisfaction, when identity preserved or at least quantity
conserved is attributable anywhere without too strained devices.
Poincare's shrewd remark is to this effect: "Physicists can be
relied upon to find something else whose total remains invariant,
should energy leave them in the lurch." And is there not some
shade of disappointment in conceding our failure to trace indi-
vidual elements of energy by Poynting's theorem, as well as the
paths of flux? Compare Lorentz, The Theory of Electrons,
210 Fundamental Equations of Dynamics
page 25; Heaviside, Electromagnetic Theory, I, page 75
(1893).
Note 9 (page 33). To follow lines that are accommodated to
some directive idea of constancy gives in many ways a natural
order. About this we should acknowledge though, how inevi-
tably our assigning conceptually common or constant values takes
its suggestion from what are means or averages in their experi-
mental basis. Neither must the truth be forgotten with which
section 69 closes. The enlargement in application through free
use of mass-averages, time-means, and the like can be instanced
for the immediate connection from sections 20, 21 and 31. But
it confronts us without any special search everywhere in physics,
when we remember that the point at which values are admitted
to be "local" is in practice solely a matter of scale; they are
finally representative of mean values to a certain order of
precision (compare section 42). Less familiar but perhaps just
as significant is that reading of the curl and the divergence locally
in a vector field which sees in them the specification of an arti-
ficial symmetry which rests upon mean values, and replaces
legitimately for certain ends the actual field-distribution. See
the Physical Review, XXXIV, page 359 (1912); Boussinesq,
Note sur le potentiel spherique, pages 319-329, in his Application
des Potentiels a 1'etude de I'Equilibre et du Mouvement des
Solides elastiques (1885).
Note 10 (page 36). Every such element that is force-moment
presents a local resultant, similar to those met in section 19
through being normal to the individual plane of its factors. As
vector products these local resultants are all open to the same
sort of double reading as is brought up for the rotation-vector
in note 7 and completed in note 16. The process of mass-
summation for a system then continues associated with the
resultant elements (dH) or (dM), combining each group as a
vector sum to a total, resultant of determinate orientation and
Notes to Chapters I-IV 211
tensor. The fraction of this last resultant effective or available
in relation to any particular axis of unit-vector (ai) is ascertain-
able by one final projection, representable respectively by
H (a) = ai(H-ai); M (a) = ai(M-ai).
The departure from the cartesian scheme consists especially in
reserving projection for the closing operation, to be executed only
when the demand for it enters. There is the common inversion
of order between mass-summation and projection, on passing
over to vector algebra.
Note 11 (page 43). There is a considerable region opened to
plain sailing among developments like those of sections 32-35,
whenever the observed material justifies our major premiss that
inertia occurs as a variable quantity. But whatever general
bearings may be obtained thus, we do not of necessity touch
the source of the inferred variableness, and much less do we reach
a halting-place about it in default of supplementary evidence.
The emphasis of the text is focussed upon the truth of this remark
which is of wide application, the electronic case being included
among others. Consequently there is a warning implied to avoid
a pitfall: ascribing prematurely the appearing variableness to
one type of source among several of which experience has made
us aware, and thereby affecting the conclusions with fallacy.
The conscious fictions that cluster round the idea of effective
mass should make us wary of deceptive illusions there whose
enigma has not been resolved. The capacity of an unincluded
(or undetected) force to compel indirect recognition of itself
in the inertia-coefficient is well-known. And a long line of
suggestive connections with processes of continuously repeated
impact have their root in an old problem. This is the trans-
mission of elastic deformation through a bar struck at one- end
(see Clebsch, Theorie d'Elasticite des corps solides, translated
by. St. Venant, page 480a, Note finale du 60). A possible
212 Fundamental Equations of Dynamics
modification of that treatment for impact has been set forth
repeatedly, in the attempt to cover wider conditions of converting
and storing energy within a system, under some form of structure
or arrangement. Heaviside especially has achieved instructive
results under that heading. The cogency of the logic in trans-
ferring demonstrated consequences of this nature to electrons
hinges on the query in how far the convective energy of electro-
magnetic inertia is adequately analogous to the kinetic energy of
(ponderable) mass. At this date it would plainly beg the larger
question to assert unreservedly that both these forms of energy
are literally the same.
Note 12 (page 44). In the closing chapter of his Kritische
Geschichte der allgemeinen Prinzipien der Mechanik (1877)
Diihring urges the sound advice not to stop short of first-hand
contact with the notable contributions that mark epochs of
advance. The case of d'Alembert's discovery enforces the
wisdom of that counsel, because a tradition echoing an imperfect
apprehension of the principle has leaned toward perverting the
gist of it from the meaning that the leaders in dynamics state
clearly, whose essential thought sections 38-41 aim to restore.
Compare them with the analysis of the principle in Mach's
Science of Mechanics and in Helm's Energetik. One source of
confusion can be located in the transposition that yields the
forms of equation (38). This point is alluded to at the end of
section 41; and the idea is expanded with elementary detail in
Science, XXVIII, page 154. Some obstacles to ready under-
standing are due no doubt to a certain crabbed brevity of the
nascent formulation in d'Alembert's Traite de Dynamique
(1758), found in Chapter I of Part II. A German translation of
this classic is provided among Ostwald's Klassiker der exakten
Wissenschaften (Number 106).
Note 13 (page 51). The influence of the energetic view per-
vades the handling of energy flux and of the accompanying forces
Notes to Chapters I-IV 213
or stresses. The transfer-forces of the text appear for example
in Helm's exposition (Energetik, page 233 and passim). The
habit of thinking in these terms is cultivated by greater familiar-
ity with storage of energy in media, which has added the vigor
of a physically conceived process to the formal nature of potential
energy in the earliest instance of gravitation, where the mecha-
nism remains completely obscure (see section 3). It is growing
increasingly evident how the outcome of explorations among
energies intrinsic and external is capable of reduction in parallel
fashion, exhibiting the conditioned modes of revealing their
presence and the measured extent of their availability. The
lessons about cautious inference of which some scant mention
is made in the text are perhaps nowhere more impressive among
the inductions of physics, when once the safety of non-committal
attitude must be abandoned in active search for a determinate
process. We remember the remark that "An infinite number
of mechanical explanations are possible" (Poincare), especially
since we deal primarily with finite or statistical resultants;
and even plausible schemes are numerous enough to leave a
broad margin for indecision. See Lorentz, Theory of Electrons,
pages 30-32.
Poynting's original paper should not be left unread (Philo-
sophical Transactions (1884), Part II, page 343); nor touch be
lost with Heaviside's stimulating directness (e. g., Electromag-
netic Theory, I, pages 72-78). A sensible summary incorporat-
ing links with relativity is furnished by Mattioli; Nuovo Cimento
(series 6), IX, pages 255, 263 (1915).
Note 14, page 53. Geometrical conditions are always a need-
ful auxiliary in expressing constraints for the reason named in
the text. The use of indeterminate multipliers would carry
unreduced geometrical forms into the equations of motion, giving
what might be called quasi-forces. Lagrange himself offers that
analysis of their significance in his Mecanique Analytique, I,
214 Fundamental Equations of Dynamics
pages 69-73 (Bertrand's edition (1853)). Later practice runs
more nearly in the line of separating these supplementary rela-
tions from the purely dynamical truths, and using the former
admittedly as mathematical aid in eliminations looking to ends
like integrations. Thomson and Tait held it part of their
service to have brought together the fully dynamical treatment
of constrained and of free systems (Natural Philosophy, Part I,
pages 271, 302).
Note 15 (page 54). The point now reached offers occasion to
add explicit reference to Routh's encyclopedic work in two
parts: Elementary Rigid Dynamics, Advanced Rigid Dynamics;
as a storehouse to which we shall long resort for authoritative
presentation of characteristic material in this field. The design
of our text has acknowledged as one main object to foster the
study of masters such as Kelvin, Routh and a few others in
dynamics. To this end we are building a less steep approach to
the level upon which their progress moves. It cannot be said
to stand in prospect that these writers will become antiquated;
but need will arise from time to time for seeing the older system-
atic grouping in an altered perspective, in order to renew connec-
tions or symmetry that temporary stress upon some lines of
growth may have disturbed.
Note 16 (page 57). Preparation has been made by anticipa-
tion in the connection of notes 7 and 10 to accept this meaning
and office for the rotation-vector which are an enlargement upon
the usual current statement about it. That aspect is adapted to
set in higher relief its comprehensive and yet particular relation
to those individual radius-vectors upon which vector algebra
turns attention. There is some advantage gained, too, by
approaching the special rigid connection on the line that starts
with the complete freedom in equation (2), and sees the vector
(to) of common application to all radius-vectors to be an out-
growth of that rigidity.
Notes to Chapters I-IV 215
Note 17 (page 64). It is important to keep track of successive
restrictions that enter to affect the range of conclusions. Here
we must not overlook that the added condition of rigidity
influences only a general reduction in form for certain parts of
(E, H, P, M) that are seen to occur already in equations (10, 12,
54, 55) as written for any non-rigid system of constant mass.
In brief, the notion of a constituent translation with the center
of mass applies to all such systems ; and so does the independent
treatment of that translation and of the motion relative to the
center of mass, as spoken of in section 52. That point is elab-
orated for elementary purposes in my Principles of Mechanics,
Part I, pages 91-101. Including now equations (19, 20) it is
made fully evident how no new situation is introduced when we
ascribe rigidity to the body, except in the entrance of rotation.
While absorbing the residual (E, H, P, M), this type of motion
also gives concise expression to their values, in every one of
which, it will be noticed, either (a>) or (o>) appears, marking the
relation of both to the body as a whole.
Note 18 (page 70). The frequent necessity of a dynamically
active couple for an adjusted control securing kinematical con-
stancy in the vector (o>) is now an everyday lesson learned from
the directive couple of rotation about a fixed axis. The possible
divergence of (<>) and (H) furnishes the simple key which cuts
off vector constancy of both together; with habitual demand
then prevailing for some (M) a&sociated with every change in
(H). But there has been an astonishing record of tenacious
refusal to distinguish between such conditions of active control
and the conditions of equilibrium, here and in the companion
instance of radial control requisite for continuance of circular
motion. The surviving power of instinctive prepossessions has
perpetuated in unexpected quarters the ancient unclearness
lurking behind " centrifugal force and couple "; and this threatens
to endure under the full illumination of the vector view. The
15
216 Fundamental Equations of Dynamics
root of many like confusions is traceable to a failure really to
grasp the facts in the first of equations (38) , with unfaltering dis-
crimination between impressed and effective forces. That equa-
tion does not describe an actual equilibrium; neither does the
result of any transposition which yields an equation like the
second form of (77). Yet compare the presentation by authori-
ties: Klein and Sommerfeld, Theorie des Kreisels (1897), pages
141, 166, 175, 182; though no criticism applies anywhere to their
mathematical correctness.
Note 19 (page 82). This labored insistence upon the dual
aspects of all coincidences is indeed designed to remove an
ambiguity in symbolism whose currency has grown out of im-
perfect attention to them. There is usually reward for watch-
fulness on those points. But the allowableness of such detail
in the text rests more upon its initiative for developing the idea
of shift in section 79. Notice, as we proceed, how often the
unit-vectors and the tensors of vector quantities offer themselves
naturally as independent variable elements, and afford a ground
for partial differentiations of a type peculiar to vector algebra.
Note 20 (page 88). Of course forces are "bound to super-
position" only by the same tie of definition or specification
that holds velocity and acceleration also, and that is broken
when we abandon the parallelogram graph. But it is remark-
able how regularly in physics that mutual independence among
energies (and among forces that change them) is experimentally
supported, of which superposition and linear relation are mathe-
matical expression. Still it is reasonable to grant that not all
definitions devised for physical quantity have escaped a bias
from this side which will need to be allowed for or rectified.
Yet the high price paid for relinquishing that simplest rule
warrants the change of base only on clearest showing of the
balance-sheet.
By referring to "physical status" the text means to encourage
Notes to Chapters I-IV 217
that scrutiny for terms of algebraic origin whose favorable and
unfavorable outcome in particular connections it cites in several
places. To be sure, candor and detachment are called for con-
tinually in reaching judgment through the arguments by con-
vergent plausibility upon which closing of the doubtful issues
here depends (see sections 6 and 7).
Note 21 (page 93). The superficial features of what is here
named shift are detectable generally in previous accounts of
coordinate systems; and Hay ward is often credited with a com-
prehensive survey of the subject in a paper: On a direct method
of estimating velocities with respect to axes movable in space
(Cambridge Philosophical Transactions (1864), X, page 1*).
Anticipations of the controlling purpose in shift might be ex-
pected confidently, since its ramifications are now recognizable
through all that coordinate machinery of early devising without
which commonest operations of algebra would have been blocked.
But the circumstance seems exceptional that completed analysis
of its working has been postponed. The proposition presented
by equation (137) does not occur in the first editions of Routh,
and he never gives to it deserved prominence. Abraham's state-
ment of it is of course formally right, yet he describes our
(X'Y'Z') questionably as a " Rotierendes Bezugssystem " (The-
orie der Elektrizitat (1904), , I, page 34). The relations of
coincidence that make equation (124) important Routh disposes
of in one obscurely placed line: "As if the axes were fixed in
space" (Elementary Rigid Dynamics (1905), page 213). Equally
casual is Abraham (p. 115) : "Die Umrechnung [auf ein bewegtes
Bezugssystem] geschieht genau so, als ob das bewegte System in
seiner augenblicklichen Lage ruhte." This comparative blank
left place for that more systematic or conscious display which
vector algebra favors of the really operative methods. Its
*This is the date of publication. The paper itself was dated and read
(1856).
218 . Fundamental Equations of Dynamics
partial novelty has set its measure at a length in the text that
may well be curtailed when their leading thought has once been
laid down.
Note 22 (page 98). Some authors cover the point by a dis-
tinction between explicit and implicit functions of time. Or
again the changing relation fairly equivalent to our shift of
(i'j'k') among (ijk) is made to introduce a partial time-derivative
(Thomson and Tait, Natural Philosophy, Part I, page 303).
It cannot escape notice what direct gain in clearness the regular
acceptance in our algebra of time-derivatives for unit-vectors
yields. The due adjustment of pace for shift, especially in
order to simplify dynamical problems in astronomy, has called
forth important discussion touching the double entry of time,
while methods of treating perturbations were becoming fully
established; and this engaged the attention of men like Donkin,
Jacobi, Hansen. There is a sequel in that region to sections
107-112; see, for instance, Cayley, Progress in Theoretical
Dynamics, British Association Report (1857).
Note 23 (page 109). The type to be remarked in equations
(154) as leading to generalizations of them is the functional
relation between each of (x', y', z') and all of both (x, y, z) and
(x, y, z). The same combinations show reciprocally when equa-
tions (150) are differentiated, and they affect characteristically
the expressions derived for kinetic energy. In equations like
(155) the first equality of partial derivatives brings out the
extent to which building up is occurring in the instantaneous
lines of (x', y', z'); and the second such equality connects
the remainder of the increment visibly with changes of slope that
are proceeding. It becomes then a simple matter to forecast
how these constituents will reproduce the result given through
a vector derivative.
Note 24 (page 118). One main objective being to specify
configurations in the standard frame, it is indispensable in the
Notes to Chapters I-IV 219
plan that some unbroken link with the latter should be main-
tained. The permanent orientation in (Z) of the angle-vector
(t|r) serves that purpose, every displacement (di{r) being im-
mediately relative to (XYZ). By the terms of section 93 dis-
placements in (#) have this one step interposed between direct
junction with (XYZ); and finally displacements in ($) are two
removes from that immediate relation. Taking other comment
from the text, it is made apparent how adequately all this
parallels the conception of displacements parallel to (X, Y, Z)
as successive, independent, and cumulatively relative. There
too, whichever the second and third displacements are, according
to the order selected, each must accept a determined initial state
due to the displacements that have preceded it. The residual
difference is inherent in the mutually supplementary qualities
of linear and angular displacements. Other parallel features
with longer-established vector schemes will repay attention;
for example the sentence just preceding equation (174) does not
mark an exceptional condition. It is of interest, too, to dwell
upon the fact implied on page 120, that (t|r, #, $) give us the
model of a coordinate-set with a changing obliquity among its
unit-vectors. It is obviously unessential, except for conven-
ience, that (i'j'k') should be orthogonal or retain any constant
relative obliquity. Some proposals have been made to include
the more general relation of direction for sets of unit- vectors ;
and the necessary modification of section 45 would be no more
than simple routine.
Note 25 (page 125). Needless to say, the revised conclusion
reached through equation (186) renounces any attempt to make
complete derivatives out of what are actually partials; but it
succeeds in assigning their proper quality to derivatives, for all
such combinations involving vectors, under a general rule stated
at the close of section 100. The root of the matter goes back to
equation (124); and the establishment of angle among vectors-
220 Fundamental Equations of Dynamics
places it in a category with them in this respect also. In what
form the omission of that element raises the difficulty may be
gathered from Klein and Sommerfeld, Theorie des Kreisels,
page 46. The truth is that a similar non-integrability of tensor
accompanies every plan of shift, except those in which a special
condition is satisfied that includes them among what may be
classed with envelope solutions (see section 116).
Note 26 (page 137). The text bears frequent testimony con-
sistently to a high appreciation for the genius and inspiration
of the earlier workers who built dynamics, among whom we may
name Coriolis. Yet we should respect our obligation also to
carry forward or to rectify the first suggestions; being taught to
expect advances in our reading attached to results especially,
whose mathematical accuracy has never been questioned. It is
that hint of possible improvement which the text here submits,
affirming the lesson of cultivating perception of physical mean-
ings upon which best modern thought concentrates, and which
is illustrated by sections 35, 57 and 104; all to be taken in
the light of repeated comment upon those clouding transfers
between the two members of equation (37) which are still too
prevalent.
Note 27 (page 141). Hansen, Sachsische Gesellschaft der
Wissenschaften, Mathematisch-physikalische Klasse, III, pages
67-71. This original statement retains value, partly still
through the material it discusses, and again through the moral
it conveys that vector methods have made these problems more
manageable. The reaction of Jacobi in some letters to Hansen
(Crelle, Journal fur reine und angewandte Mathematik, XLII,
(1851)) shows instructively the struggle toward clear and firmly
grasped thought proceeding, with strictest scrutiny of detail
in the new proposal. In the paper referred to above, Hansen's
double use of time is worked out (compare note 22), that remains
current among astronomers.
Notes to Chapters I-IV 221
Note 28 (page 155). We do not measure rightly the inherit-
ance of rigid dynamics from Euler's labors without conscious
effort to reconstruct the void that they filled once for all. Unless
his inventive intuitions had here been favored by a happy chance,
he could hardly have moulded from the first heat so many of the
forms that seem destined to hold permanent place. We can
imagine that his inspiration caught early glimpses of the relation
that equations (72) and (258) now convey; but Euler may have
been content to seize the validity of equation (257) without
proving it, as Fourier did in like case. Certain it is that the
point involved in that equivalence seemed troublesome enough
to be made the object of various special proofs, before our
general equation (137) had been attained (see Routh, Elementary
Rigid Dynamics (1882), page 212). For the historic date, the
memoir presented to the Berlin Academy is quoted (1758).
But a satisfactory survey of Euler's contributions on the topic
is best obtained through his collected works. Easier access
perhaps is had in the German translation (Wolfers, 1853); in
the volumes 3-4 entitled Theorie der Bewegung the "Centrifugal
couple" appears at page 323, and our main interest would prob-
ably concentrate on pages 207-443.
Note 29 (page 169). Klein and Sommerfeld, Uberdie Theorie
des Kreisels (1897-1910), is one instance, quoting our Preface,
how special treatises of unquestioned excellence make superfluous
an attempt to replace them. This work, and Routh's version in
the Advanced Rigid Dynamics (edition of 1905), Chapter V,
with Thomson and Tait's discussions passim in Natural Philos-
ophy, Part I, supply for gyroscopic problems the indispensable
material, exhaustive of more than their general aspects. The
aim of the text here is strictly confined to lending its announced
special emphasis to two items. One is shown to be of ramifying
importance as a singular value round which deviations from it
may be organized; the other is uniquely characteristic, and it
222 Fundamental Equations of Dynamics
proves amenable to this analysis most simply, in comparison
with other methods. Compare in verification Theorie des
Kreisels, pages 247, 316, on strong and weak tops.
Note 30 (page 180). A fuller command of generalized co-
ordinates and forces as an effective working method can be
inferred from evidence on two sides: first, more unequivocal
recognition is accorded to their finally scalar type; and secondly,
the primary demonstration of relations shows increasingly
directer insight. Dispose of the latter point by collating
Lagrange's proof (Mecanique Analytique, I; Dynamique, Sec-
tion IV); Thomson and Tait, whose change between (1867)
and (1879) is instructive; and Heaviside, Electromagnetic
Theory, III, page 178. The last-named is a climax of condensa-
tion, and thereby somewhat unfitted for the text; but it will
be quoted below for a double reason. The quantitative emanci-
pation of Lagrange's equations may be traced gradually, if we
like, beginning with equations such as (150, 151), where the
(1, m, n) coefficients are particular reduction factors conditioned
as in equation (152). Next advance to the more liberal possi-
bilities of linear vector functions illustrated by equations (86,
89), and clinch the series with Byerly's half-humorous emphasis
(Generalized Coordinates (1916), page 33). This book has the
merit of helpfully discursive approach to a large subject; and
though it seems tacitly limited to the vector conception, closing
the matter on the range that Lagrange occupied at one bound
and not gradually, proper antidote can be sought elsewhere.
See Silberstein, Vectorial Mechanics (1913), page 59; while
Ebert has been referred to in note 3, for his treatment in the
larger spirit of energetics.
We insert now the quotation from Heaviside; it illustrates
fairly the ne plus ultra in both respects. Notation of our text
is continued. Because (E) is a homogeneous quadratic function
of the velocities, Euler's theorem about homogeneous functions
enables us to .write
Notes to Chapters I-IV 223
of which the legitimate total time-derivative is
2 dE
2 df
Since (E) is "by structure" a function of velocities and co
ordinates only,
Divide the last equation by (dt) and subtract from the second,
giving
dE_
dt =
the last member expressing the energetic in variance of activity
(see equation (298)).
It would be misleading if the text pretended to do more than
give Lagrange's equations their setting of introductory connec-
tion with the other topics treated. In order to proceed safely
the results here gleaned must be followed up seriously; the
references given already indicate where to begin, and they can
be relied upon to supplement themselves as the subject opens.
Questions to be met at once are alluded to incidentally in section
136: a rationally consistent view of superfluous coordinates,
including how they may drop that character and become physical;
and the bearing of that quoted "interlocking" upon the signifi-
cance of the term holonomous. That there are more vital issues
aw r aiting analysis is suggested by Burbury (Proceedings of the
Cambridge Philosophical Society, VI, page 329) ; by such com-
ment as Heaviside's (Electromagnetic Theory, III, page 471)
upon Abraham's successful extension of Lagrange's equations;
and by the lines of inquiry to which note 32 points.
224 Fundamental Equations of Dynamics
Note 31 (page 194). This development is seen to be borrowed
from Thomson and Tait, pages 320-24. The few changes are
adapted here and there to an even keener intent to keep the
energies and momenta at the front, subordinating the investiture
with mathematics. It was thought needful to drive the entering
wedge before closing, for the sake of those continuations to which
Maxwell's example leads. The reduction factors (1, m, n) are
easily released from their trigonometrical meaning, and other
geometrical implications cancelled.
Note 32 (page 200). For the justified application of equation
(333), or of forms derivable mathematically from it, to all se-
quences of energy change, one turning-point is set by delimiting
the necessary equivalences between the mechanical readings of
(E) and (<) and the broader dynamical ones. This general idea
is pursued by Konigsberger in his papers, Uber die Prinzipien
der Mechanik (Sitzungsberichte der Berliner Akademie (1896),
pages 899; 1173); and is entertained by Whittaker in his
Analytic Dynamics (1904), Chapter X, passim. The stimulus
to this quest seems still attached to the possibility of construct-
ing a parallel in mechanical energy by using values connected
with other energy changes. One gathers this meaning from the
utterance of Larmor (Aether and Matter, page 83) and others
like it.
INDEX
The Numbers refer to Pages
Abraham, 217, 223.
Absolute measure, 5.
Absolute motion, 9, 10.
Acceleration, 83; and center of mass,
37, 61; and ideal coordinates, 144-
147; and Newton's second law, 33;
and shift, 150, 152; and tangent-
normal, 148, angular, 62; in rota-
tion, 65-6; in space curves, 151-2;
invariance of, 83, 90; mass-average
of, 37; polar components of, 134-
136; relative to center of mass, 61 ;
transfer for, 89; uniplanar, 136,
150.
Activity, 35, 36, 201.
Adjustments, and equilibrium, 69,
174, 216; and force-moment, 69,
174; imaginary, 175, 177. 179; of
shift, 97.
Angle, and moment of momentum,
27.
Angle-vector, 58, 208, 219.
Angular acceleration, 62; and force-
moment, 67-8, 69. 71-2; and shift,
126-131, 160-161, 164-165; axis
of, 66; base-point for, 66; transfer
for, 124.
Angular displacement, 27, 56, 58, 79,
115.
Angular velocity, 57; base-point for,
57; transfer for, 124.
Approximation, 17, 18, 203; and
particle, 29; and rigid dynamics,
53.
Atomic energy, 51.
Average acceleration, 37.
Axes, principal, 73, 157, 162.
Axis, of angular acceleration, 66; of
rotation- vector, 58.
Base-point, for angular acceleration,
66; for angular velocity, 57; for
force-moment, 36; for moment of
momentum, 26-27.
Bodies, system of, 1C.
Body, 16; continuous mass of, 16;
homogeneous, 31.
Boussinesq, 210.
Burbury, 223.
Byerly, 222.
Campbell, 207.
Cartesian coordinates, and funda-
mental quantities, 113; and shift,
107-111; scalar character of, 112,
204.
Cayley, 218.
Center of mass, 28; acceleration
relative to, 61; and mean accelera-
tion, 37; and energy, 55, 60, 64;
and force-moment, 60, 63; and
impressed -force, 48-49, 63; and in-
variance, 83; and moment of mo-
mentum, 55, 60, 83; and momen-
tum, 29, 63; and particle, 37-38,
63; and power, 60, 63; and pure
rotation, 64-65, 75; and rigid solids,
55, 60-64; and total force, 37, 48.
225
226
Fundamental Equations of Dynamics
The Numbers refer to Pages
63; and translation, 37-38, 63, 215;
and velocity, 29, 55, 57; rotation
about, 55, 57, 60-63; velocity rela-
tive to, 55, 57.
Centimeter-gram-second system, 21.
Centrifugal couple, 68, 215.
Centrifugal force, compound, 137,
220.
Clebsch, 211.
Coincidence, dual nature of, 82, 91,
216.
Comparison-frame, 78; and accelera-
tion, 89; and shift, 94, 96, 97, 104;
and velocity, 82-88; notation for,
78; velocity of, 85-88.
Comparisons, timeless. 81.
Compound centrifugal force, 137,
220.
Concepts, physical, 8, 19.
Conditions, geometrical, 52, 213.
Configuration, 78-79, 181-182, 187.
Configuration angle, 79-80; and shift,
123, 124, 126-131; derivatives of,
117-123.
Configuration angles, Euler's, 114,
219; and rotation-vector, 117-123;
partial derivatives of, 125, 219.
Connections, internal, 49-50, 55;
transmit force, 50.
Conservation of energy, 4.
Conservative system, 5, 7.
Constancy, of mass, 25; simplifica-
tion by. 33, 54.
Constraints, 3, 47, 68, 140; and La-
grange equations, 187; and pure ro-
tation, 68, 75; and rigid solids, 53,
55, 62.
Continuity, 34, 209; of density, 31;
of mass, 16.
Convection, of energy, 45; of momen-
tum, 45.
Conversions of energy, 45, 52.
Coordinates, and configuration, 181-
182, 187; Euler's, 114; generalized,
179; ideal, 141-147; ignoration of,
194-200, 224; oblique, 116, 219;
polar, 130-135; standard frame,
112; superfluous, 183, 190, 191,
223; tangent-normal, 147; and
shift, 97.
Coriolis, 137, 220.
Couple, 36, 63, 68; centrifugal, 68,
215; directive, 215.
D'Alembert, 2, 7, 8, 50, 53, 212; and
equation of motion, 53; and La-
grange, 180, 184; and Newton's
third law, 50.
D'Alembert's principle, 50; and im-
pulse, 50.
Defining equalities, 23, 44, 48.
Definitions: activity, 35; angular
acceleration, 62; angular velocity,
57; body, 16; constraints, 47; cen-
ter of mass, 28; effective force, 48;
force, 34, 36-38; force-moment,
35, 36; impressed force, 48; inertia,
6; kinetic energy, 22, 25; mean
vector, 28; moment of momentum,
22, 27; momentum, 22; power, 35,
36; rotation- vector, 57; system of
bodies, 16; translation, 27.
Degrees of freedom, 178, 182; and
equations of motion, 183.
Density, and volume-integral, 30, 31;
continuity of, 31.
Derivatives, of tensors, 93, 96, 102,
154; partial and total, 128, 218,
219.
Descriptive vectors, 137, 141.
Differentiation, of mass-summa-
tions, 32-33.
Index
227
The Numbers refer to Pages
Direction-cosine generalized, 190,
222.
Directive, forces and power, 140:
moment, 68, 215.
Discover}-, of principles, 18.
Discrimination, among time-func-
tions, 98, 218, 220.
Displacement, angular, 27, 56, 58,
79, 115; by rotation, 56, 58.
Distributed vectors: force, 34; mo-
mentum, 26; transfer-forces, 4546.
Divergence: angular acceleration and
force-moment, 70, 74; moment of
momentum and rotation vector,
67, 70, 215.
Donkin, 218.
Driving point, 50.
Duhring, 212.
Dynamical equations, Euler's, 155-
166, 180, 192; Lagrange's, 179-200.
Dynamical systems, 16.
Dynamics, and kinematics, 9, 12, 13,
72, 165-166; and Lagrange equa
tions, 7, 180; and mathematics, 1,
36-37, 113, 137, 139, 174-175, 2C8,
216, 220; and mechanics, 16; fic-
tions in, 8, 36-37; of precession,
171-174; stability of, 2, 8, 15.
Ebert, 204, 222.
Effective force, 48, 216.
Electromagnetic, energy, 6, 43, 212;
inertia, 40, 212.
Energetics, 2, 3, 6, 202.
Energy,. configuration, 187; conserva-
tion of, 4; conversions of, 45, 52;
electromagnetic, 6, 43, 212; flux of,
44, 181, 183, 212; internal, 64;
molecular and atomic, 51; over-
emphasis on, 3; potential, 4, 199;
storage of, 6, 7.
Energy-changes, and momentum,
193, 194; fictitious, 199.
Energy factors, and Lagrange equa-
tions, 186-187.
Energy forms, and mechanisms, 181.
Energy transfer, and Lagrange equa-
tions, 181.
Envelope solutions, 220.
Equation of condition, precession,
170.
Equation of impulse, 44, 46.
Equation of motion, 44, 48; and
degrees of freedom, 183; character
of, 53, revision of, 52.
Equation of work, 44, 46.
Equations, and identities, 23, 44, 48;
Euler's, 155-166, 180, 192; La-
grange's, 8, 180, 222.
Equilibrium, and adjustment, 69,
174, 216; fictitious, 50.
Equivalence, 208, 224; of particle, 28,
64.
Euler, 2, 34, 112, 114, 147, 155, 221;
configuration angles, 114; dynami-
cal equations, 155-166, 180, 192;
geometrical equations, 117.
Euler equations, and moments of
inertia, 162, 180; and principal
axes, 157, 160, 162-163; and rota-
tion, 155; and shift, 160-161, 164-
165; apply to rigid body, 155;
generalized form, 162-165; and
Lagrange, 192.
Experiment, and impressed force, 52.
Fictions, in dynamics, 8; in energy-
changes, 199; in force, 41.
Fictitious, equilibrium, 50; transla-
tion, 28.
Flux, of energy and momentum, 44,
181, 183, 212.
228
Fundamental Equations of Dynamics
The Numbers refer to Pages
Force, activity of, 35, 36, 201 ; a dis-
tributed vector, 34, a fundamental
quantity, 21-22; and fluxes, 45; and
momentum change, 32 and vari-
able mass, 38, 40, 42; effective, 48,
216; fictitious, 41; generalized, 7,
184, 223; gyroscopic, 137; im-
pressed, 48, 216; ponderomotive,
13; supplemented by force-moment,
36, 63; transmitted, 50.
Force elements, and rotation, 66, 168;
and transfer-forces, 46.
Force-moment, 34-36, 210; a funda-
mental quantity, 21; and angular
acceleration, 68, 69, 71-72; and
center of mass, 60, 63; and pre-
cession, 171-172; and principal
axes, 74; and rigid solids, 60, 61;
and rotation, 66-68, 71, 74; and
rotation-axis, 167-168; and rota-
tion-vector, 68; and shift, 106, 161;
and tangent-normal, 166-167; a
resultant, 35-36, 210; directive, 68;
disturbing precession, 174; supple-
ments force, 36, 63.
Forces, and degrees of freedom, 184;
directional, 140; equivalent through
work, 184; gene/alized, 183-186;
lost, 50.
Fourier. 221.
Free vectors, and shift, 100-104.
Fundamental groups, relation of, 22,
43-44.
Fundamental quantities. 21, 63, 65,
113, 140, 154; and invariance, 83;
and reference-frames, 23, 24, trans-
fer for, 24.
Gauss, 5.
Generalized, coordinates, 180, 182;
Euler's equations, 162-165; forces,
7, 183-186; momentum, 7, 182,
190; velocity, 182, 186-187, 191.
Geometrical, conditions, 52, 213;
equations, 117, 191.
Gravitation, and energy, 51.
Gray, 204.
Gyroscope, 139, 163, 169, 193, 207,
221; diverts energy, 179; weak or
strong, 177.
Gyroscopic, forces, 137.
Hamilton, 2, 8, 180, 200.
Hansen, 141, 218, 220.
Hayward. 217.
Heaviside, 201, 204, 210, 212, 222,
223.
Helm, 202, 205, 212, 213.
Holonomous, 223.
Homogeneous, body, 31; functions,
182, 223.
Huyghens, 4.
Ideal coordinates, 141-147; and ac-
celeration, 144; and polar, 142;
and shift, 143, 147; and tangent-
normal, 151-152; and velocity, 142.
Identities, and equalities, 23, 44, 48.
Identity and continuity, 34.
Ignoration, of coordinates, 194-200,
224.
Ignored force, and variable mass,
40-41.
Imaginary, precession, 175, 177, 179.
Impact, 42, 212.
Impressed force, 44, 48, 216; and
center of mass, 48-49; and rigid
solids, 55, 62-63; and rotation,
62-63; and translation, 62-63; ex-
perimentally known, 52.
Independence, of coordinates, 182;
of rotation and translation. 63.
Index
229
The Numbers refer to Pages
Indeterminate multipliers, 53, 213.
Individuality, of masses and points,
34, 81.
Inertia, 6. 7, 16; variable, 211.
Integration, and shift, 126, 153-154.
Internal, actions and energy, 42, 51-
52, 55, 64, 205.
Interpretation, mechanical, 12, 13,
43, 187.
Invariance, and center of mass, 83;
of acceleration, 83, 90; of funda-
mental quantities, 83; of moments
of inertia, 160; of radius- vector,
80; of velocity, 82, 90.
Invariant, frame-groups, 84.
Inverse square, law of, 5.
Jacobi, 218, 220.
Kinematics, and dynamics, 9, 12, 13,
54, 72, 165; and transfer, 77.
Kinetic energy, a flux, 44; a funda-
mental quantity, 21; analogues of,
7; and generalized velocity, 182,
191; and gravitation, 51; and inter-
nal actions, 51-52; and particle, 28,
29; and principal axes, 74, 157; and
rigid solids, 55, 60; and rotation,
63, 65, 71, 74; and translation, 28,
63: a scalar product, 22, 25; con-
vection of, 45; diversion of, 64,
179; supplements mean values, 29,
30.
Kinetic potential, 200, 224.
Klein and Sommerfeld, 216, 220, 221.
Konigsberger, 224.
Lagrange, 2, 7, 8, 180, 184, 213, 222.
Lagrange equations, 180-200, 204,
223; and energy factors, 186-187;
and Euler's, 180, 192; and polar
components, 188; and standard
frame, 187; and tangent-normal,
187; are scalar, 184, 186, 222; in-
clude constraints, 187.
Larmor, 205, 224.
Latency, of momentum and energy,
7, 45.
Law, of inverse square, 5; of inertia-
change, 40.
Laws, of motion, 4, 32-33, 50, 201.
Localized vectors, 22, 26, 36; and
shift, 104-106.
Lorentz, 202, 207, 209, 213.
Lost forces, 50.
Mach, 205, 206, 212.
Maclaurin, 112.
Mass, and volume-integral, 30, 31;
as quotient, 3, 40; constancy of, 25;
continuity of, 16; generalized, 6;
variable, 38.
Mass average, and precision, 49; of
acceleration, 37; of velocity, 29.
Mass constancy, 33.
Mass-summation, 22; differentiated,
24, 32-33.
Mathematics, and dynamics, 36-37,
113, 137, 139, 174-175, 208, 216,
220; simplifies, 17.
Mattioli, 213.
Maxwell, 6, 203, 224.
Mean values, 210; residues from, 30,
36, 60-63.
Mean vector, 28.
Mechanical models, 12, 13, 43, 181,
187, 213, 224.
Molecular energy, 51.
Moment of momentum, 22, 27, 210;
a fundamental quantity, 21; a
localized vector, 22, 27; and par-
ticle, 29; and precession, 171; and
principal axes, 73; and rigid solids,
230
Fundamental Equations of Dynamics
The Numbers refer to Pages
55, 60, 73, 156; and rotation- vector,
27, 68, 69; and shift, 106; and trans-
lation, 28; and volume-integral,
30, 31; supplements mean values,
29, 30.
Moments of inertia, and Euler equa-
tions, 162, 180; in variance of, 160.
162.
Momentum, 22; a distributed vector,
26; a flux, 44; a fundamental
quantity, 21; and center of mass,
29, 63; and generalized velocity,
182; and translation, 28; and vari-
able energy, 193-194; and volume-
integral, 30-31; convection of, 45;
generalized, 182, 190; invented by
Newton, 4; latency of, 7, 45; recti-
fied, 153; transformed, 45.
Momentum change, and force, 32,
36, 46.
Motion, absolute, 9, 10; equation of,
48; relative to center of mass, 28,
38, 215; second law of, 32-33;
third law of, 50, 201.
Multipliers, indeterminate, 53, 213.
Newton, 4, 9, 32-33, 50, 201.
Notation, comparison-frame, 78; prin-
cipal axes, 157-158; standard
frame, 77-78.
Oblique coordinates, 116, 219.
Orthogonal axes, adopted, 23.
Ostwald, 202, 212.
Parameters, Lagrange's, 180.
Partial derivatives, 91-96, 125, 149,
185, 190. 216, 218, 219.
Particle, 28; and center of mass, 37-
38; and energy, 29; and moment of
momentum, 29; and polar com-
ponents, 139-140: and rigid solid,
54; and tangent-normal, 154; equi-
valence of, 64.
Phenomenology, 202.
Poincare\ 81, 202, 206, 213.
Points, individualized, 81; motion of,
81-82.
Polar components, 140; and ideal co-
ordinates, 142; and Lagrange equa-
tions, 188; and pure rotation, 138-
139; and superposition, 136; and
tangent-normal, 148, 149; uni-
planar, 136.
Polar coordinates, 130-135.
Polar velocity, and shift, 133.
Ponderomotive force, 5, 13.
Position coordinates, auxiliary, 81.
Potential, 5; energy, 4, 7, 51, 199;
kinetic, 200, 224.
Power 35, 36; a fundamental quan-
tity, 21; and center of mass, 60, 63;
and directive action, 68, 140; and
rigid solid, 60, 61, 63; and shift,
140-141, 152-153; and variable
mass, 39, 42.
Power equation, 201.
Poynting, 213.
Precession, 169-174; condition for,
170; imaginary, 175, 177, 179.
Precision, 207, 210; and mass average,
49.
Principal axes, 73, 157, 160, 162-163;
and energy, 74; and Euler equa-
tions, 157, 160, 162; and force-
moment, 74; and moment of mo-
mentum, 73; notation for, 157-
158.
Principle, d'Alembert's, 50; Hamil*
ton's, 8; of vis viva, 4.
Principles, discovery of, 18; stability
of, 8.
Index
231
The numbers refer to Pages
Projection, of angle- vector, 58, 115-
116.
Proximate reference, 11.
Pure rotation, 59; and center of mass,
64-65, 75; and constraints, 75; and
polar components, 138-139.
Quantity of motion, 32.
Radius- vector, in variance of, 80;
mean, 28: paitial derivative of, 91-
96; prominence of, 27, 36, 209, 210,
214; typical character of, 99.
Reduction factor, 190, 218, 224.
Reference-axes, orthogonal, 23.
Reference-frame, conceived fixed, 23;
postponed choice of, 24, 207; proxi-
mate, 11; transfer for, 76; ulti-
mate, 9, 10, 11, 88.
Reference-frames, configuration of,
78-79; invariant groups of, 84.
Regular precession, 169-174; and
force-moment, 171-172; and mo-
ment of momentum, 171; dynamics
of, 171-174; imaginary, 175, 177,
179.
Relativity, 4, 11, 202, 213.
Representative particle, 28.
Resolution, tangent-normal, 40, 147-
154.
Resultant elements, force, 34; force-
moment, 36, 210; moment of mo-
mentum, 27, 210; momentum, 22.
Revision, of physical equations, 52.
Rigid dynamics, and Euler equa-
tions, 155, 221; approximate, 53.
Rigidity, 214, 215; and internal
energy, 55; of ultimate parts, 54.
Rigid solid, 53; and center of mass,
55; and Euler equations, 155; and
force-moment, 60, 61; and im-
pressed force, 55, 62-63; and mo-
ment of momentum, 55, 60, 73,
156; and particle, 54; and power,
60, 61, 63; and rotation, 55, 57, 58,
63, 215; angular velocity of, 57;
general motion of, 63; structure of.
53, 55, 62.
Robb, 207.
Rotation, 55-57, 215, and accelera-
tion, 65-66; and center of mass, 55,
57, 60; and energy, 63, 65, 71, 74:
and Euler equations, 155; and
force-moment, 66-67, 68, 71, 74;
and impressed force, 62-63; and
uniplanar motion, 72, 167; and
velocity, 57, 59; of rigid solid, 55,
57, 58, 63.
Rotational stability, 169, 175-179;
condition for, 176, 178.
Rotation-axis, and force, 66, 168;
and force-moment, 167-168.
Rotations, superposition of, 59.
Rotation-vector, 57, 208, 214; and
configuration angles, 117-123; and
force-moment, 68; and shift, 123,
124, 126-131; and standard frame,
58-59; divergence from moment of
momentum, 27, 67-70.
Routh, 214, 217, 221.
Scalar equations: cartesian, 107-111;
Lagrange's, 184, 186; standard
frame, 112, 187.
Shift, 94, 97, 216, 217, 218; and ac-
celeration, 150, 152; and angular
acceleration, 126-131, 160, 164;
and cartesian axes, 107-111; and
Euler equations, 160-161, 164; and
force-moment, 106, 161; and free
vectors, 100-104; and ideal co-
ordinates, 143, 147; and integra-
232
Fundamental Equations of Dynamics
The numbers refer to Pages
tion, 126, 153-154; and localized
vectors, 104-106; and moment of
momentum, 106; and motion com-
pared, 96-97, 104; and polar ac-
celerations, 134; and polar veloci-
ties, 133.
Shift rate, 97-98; and power, 140,
152-153; and rotation-vector, 123,
124, 126-131.
Silberstein, 202, 222.
Simplifications, in dynamics, 17, 54,
205.
Space curves, acceleration in, 151-
152.
Stability, condition for, 176, 178; of
principles, 2, 8, 15; rotational, 169,
175-179.
Standard frame, and fundamental
quantities, 113; and Lagrange
equations, 187; and rotation-vec-
tor, 58-59; arbitrary choice of, 78;
as coordinate system, 112; nota-
tion for, 77-78.
Storage of energy, 6, 7, 64.
Summation, continuous or discrete,
23.
Superfluous coordinates, 183, 190,
191, 223.
Superposition, 59, 88, 216; failure of,
136; of rotation and translation,
63.
System, conservative, 5, 7; dynamical,
16; internal connections of, 49-50;
of bodies, 16.
Tait, 201.
Tangent-normal, 40, 147; and ac-
celeration, 148; and force-moment,
166; and fundamental quantities,
154; and ideal coordinates, 151-
152; and Lagrange equations, 187;
and polar components, 148. 149;
and velocity, 147; as prototype,
149.
Tensors, derivatives of, 93, 96, 102,
154; groups of, 92.
Thomson and Tait, 201, 203, 214,
218, 221, 222, 224.
Time-derivative, of geometrical equa-
tions, 191.
Time functions, two classes of, 98,
218, 220.
Timeless comparisons, 81.
Total and partial derivatives, 91-96,
125, 128, 219.
Total force, 34; and center of mass,
37.
Transfer: angular acceleration, 124;
angular velocity, 124; reference-
frame, 24, 76, 77.
Transfer- force, 45; a distributed vec-
tor, 47; and local resultants, 46; as
constraints, 47.
Transformation, of momentum, 45.
Translation, 27, 28; and center of
mass, 37-38, 63, 215; and energy,
28, 63, and impressed force, 62-63;
and rigid solid, 55, 63; and rotation,
63, 215.
Transmission of force, 50.
Ultimate, reference, 9, 10, 11, 205;
rigidity, 54.
Uniplanar, acceleration, 136, 150;
rotation, 72, 167.
Variable mass, 38-42, 211; and ig-
nored force, 40-41 ; and summation,
24-25.
Vector algebra, 13, 208.
Vectors, descriptive, 137, 141; dis-
Index
233
tributed, 26, 34, 45, shift for, 100-
106.
Velocity, 82; and center of mass, 29,
55, 57, and ideal coordinates, 142-
144; and rotation, 57, 59; and tan-
gent-normal, 147, 152; angular, 57;
generalized, 182, 186-187, 223; in-
variance of, 83, 90; mass-average
of, 29; partial derivative of, 149;
polar components of, 132, 133, 136;
relative to center of mass, 55, 57;
transfer for, 85-88; virtual, 50.
Virtual, velocity, 50; work, 7, 50.
Vis viva, principle of, 4.
Volume integrals, 30, 31.
Whittaker, 224.
Work, virtual, 7, 50.
Work equivalence, and force, 46, 183-
184, 223.
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