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) + (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 = (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 = ) 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 (-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 (-H); Total quantities (67) P = to M ; - for pure rota- (68) H = M; tion. (69) 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') + (), 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 () retains direction (). 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(-r)dm. (75) Next form for comparison the product x [to(r-r) r(-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() 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 (

), 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# \ , / ) 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 () 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 (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

) 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

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 + %(

(), W(b), "(c))- The co- ordinates are then (ifc, #, $), the velocities (\j/, d, 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 (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 (^, #, = 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; -^ = 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 + ), 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/, &, TTI/ aJcj oLi(o) dE(o) dE ' (325) d

dn / . ai .am d_/aE\ _ aE __d / dt V 3d )~ d& ~ dt V _ dt \d

at ay ov o

/ dE (0) , _ It? + d ~ (E ) 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. UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. 3UN 1 1 19M NOV 2 ROTO AU6 1 8 1965 nrr o K NOV 1 5 196-8 f IB OCT 1 9 1967 OCT20&ECO 1 DEC 2 3-1968 DEC 't : gfl* Form L9-116m-8,'62(D1237s8)444 A 000165700 6 UBRARY LOS