On the Structure of Classical Mechanics

Thomas William Barrett∗†

February 21, 2013

Abstract

Jill North (North, 2009) has recently argued that Hamiltonian me-
chanics ascribes less structure to the world than Lagrangian mechanics
does. I will argue that North’s argument is not sound. In doing so,
I will present some obstacles that must be navigated by anyone in-
terested in comparing the amounts of structure that different physical
theories ascribe to the world.

1 Introduction

Different spacetime theories ascribe different amounts of structure to space-
time. Aristotelian spacetime singles out a preferred spatial location as the
center of the universe. Absolute Newtonian spacetime does not single out a
preferred spatial location, but it does single out a preferred inertial frame as
the rest frame. Neo-Newtonian (or Galilean) spacetime does not single out a
preferred inertial frame, but it does single out a set of inertial frames as the
non-accelerating frames. The list could go on. There is an intuitive sense
in which each of the theories in the list ascribes less structure to spacetime
than do its predecessors.1

∗thomaswb@princeton.edu
†I am particularly grateful to Hans Halvorson for comments on multiple drafts, for

many clear discussions, and for giving me many of the basic ideas in Section 3. Thanks
also to John Manchak and Jeff Barrett for comments on an earlier draft of this paper, and
to Noel Swanson for helpful discussions about classical mechanics.

1See (Earman, 1989, Ch. 2).

1



Philosophers and physicists have devoted much attention to the amounts
and kinds of structure that spacetime theories ascribe to the world, but some-
what less to the amounts and kinds of structure that other physical theories
ascribe to the world. In a recent paper, however, Jill North (North, 2009)
undertakes the ambitious task of comparing the structure of the Hamiltonian
and Lagrangian formulations of classical mechanics. Among other things,2

North argues for the following conclusion:

(LS) Hamiltonian mechanics ascribes less structure to the world than La-
grangian mechanics does.

There are, of course, general philosophical issues which turn on such
questions of comparative structure. Identifying the structure that physical
theories ascribe to the world could play an important role in addressing
questions of theoretical equivalence. If two theories ascribe different struc-
tures to the world — if, for example, one theory singles out a spatial location
as the center of the universe while another does not — then these two the-
ories could not be equivalent reformulations of one another. They tell us
different things about the structure of the world. Structural comparisons
can also play an important role in parsimony arguments. All other things
being equal, Ockhamist principles urge us to prefer the theory which is more
structurally parsimonious.

If (LS) were true, therefore, one might use it to argue that Hamiltonian
and Lagrangian mechanics are not theoretically equivalent and, furthermore,
that we should prefer the Hamiltonian formulation over the Lagrangian for-
mulation.3 Or one might use (LS) to argue for something even stronger:
Hamiltonian mechanics provides us with a better description of the funda-
mental and objective features of the world than Lagrangian mechanics does.
But of course, one must demonstrate that (LS) is actually true before draw-
ing any of these philosophical conclusions. North’s argument for (LS) runs
as follows:

2(LS) appears as a premise in an argument that the two theories are not equivalent,
and in an argument that Hamiltonian mechanics is more fundamental than Lagrangian
mechanics (North, 2009, p. 76).

3As remarked in the previous footnote, North uses (LS) to argue for such conclusions.

2



(P1) Lagrangian mechanics ascribes metric structure to the world.

(P2) Hamiltonian mechanics ascribes symplectic structure to the world.

(P3) Symplectic structure is less structure than metric structure.

∴ (LS)

The primary goal of this paper will be to show that this argument for
(LS) is not sound. I will argue that North’s arguments fail to demonstrate
that (P1) and (P2) are true. Then I will argue that even if one were to
accept that (P1) and (P2) were true, (LS) would still not follow since it
is not at all clear that (P3) is true. The reader might be inclined to take
these remarks as cautionary: Much care must be taken in identifying and
comparing the amounts of structure that different physical theories impute
on the world. Some cases are harder than others. And as we will see, the
case of Lagrangian and Hamiltonian mechanics is one of the hard ones.

2 Hamiltonian mechanics imputes less structure

than Lagrangian mechanics

The first order of business is to recapitulate North’s argument in support of
(LS). I will first present a couple of arguments for (P1), then an argument
for (P2), and then finally conclude with North’s argument for (P3).

2.1 Lagrangian mechanics is Riemannian

Lagrangian mechanics and Hamiltonian mechanics are geometric formula-
tions of classical mechanics. They are formulated in the language of differ-
ential geometry. The Lagrangian approach focuses on position and velocity,
while the Hamiltonian approach focuses on position and momentum.

Consider a system of n particles. The positions of these particles can
be represented by a point q in a 3n-dimensional differentiable manifold Q,
called configuration space. Lagrangian mechanics has as its statespace the
6n-dimensional tangent bundle of Q, which we denote by T∗Q. A point

3



(q,v) ∈ T∗Q encodes the positions and velocities of all n particles in the
system.

In order to determine how the system evolves over time, one must specify
a Lagrangian. The Lagrangian of a system is a smooth function L : T∗Q→ R
which encodes some of the system’s energy properties.4 In most applications
of Lagrangian mechanics, the Lagrangian of a system is defined as the sys-
tem’s kinetic energy minus its potential energy. The intuition behind it is
something like the following:

Kinetic energy measures how much is ‘happening’ — how much
our system is moving around. Potential energy measures how
much could happen, but isn’t yet [. . . ]. So, the Lagrangian mea-
sures something we could vaguely refer to as the ‘activity’ or
‘liveliness’ of a system: the higher the kinetic energy the more
lively the system, the higher the potential energy the less lively.
(Baez, 2005, p. 7)

This intuitive interpretation of the Lagrangian helps to motivate the dynam-
ics of the theory: Systems like to evolve along paths in T∗Q which minimize
the integral of the Lagrangian — they like to minimize their total ‘activity’
or ‘liveliness’.5

One might at first try to make the following naive argument that La-
grangian mechanics ascribes Riemannian metric structure to the world.

Argument 1 (Naive Argument). In applications of Lagrangian mechanics
we need to define things like lengths of curves or angles in Q. If we do
not have a metric gq on Q, then we cannot define such things. Therefore,
Lagrangian mechanics requires a metric gq on its configuration space Q.

4And the system then evolves according to the Euler-Lagrange equations

d

dt

∂L

∂q̇i
=

∂L

∂qi
,

where qi and q̇i are position and velocity coordinates, respectively, on T∗Q. It turns out
that the dynamics of Lagrangian mechanics — the way that the theory states that systems
will evolve — is not important in North’s argument.

5And this behavior is guaranteed by the Euler-Lagrange equations in the previous
footnote.

4



North, however, does not have such an argument in mind. This argument
clearly does not serve her purposes. Remember that we are trying to argue
for (LS); we are trying to argue that Hamiltonian mechanics ascribes less
structure to the world than does Lagrangian mechanics. Argument 1 does
not help us to argue for (LS). In applications of Hamiltonian mechanics
one will also need to define things like lengths of curves and angles in Q.
Argument 1 will therefore imply that Hamiltonian mechanics requires a
metric on its configuration space Q also. And this is more than we want
to argue for. If both Hamiltonian and Lagrangian mechanics posit metric
structure, then it will be hard to argue that Hamiltonian mechanics posits
less structure than Lagrangian mechanics does. Argument 1 will not do.
North’s argument is more like the following:

Argument 2 (North’s Argument). In most applications of Lagrangian me-
chanics, the Lagrangian of a system is defined as the system’s kinetic energy
minus its potential energy. In fact, in many cases the Lagrangian is defined
by

L(q,v) =
1
2
gq(v,v) −V (q) ,

where V : Q → R is a smooth potential. And this definition presupposes
a metric gq : TqQ× TqQ → R on the configuration space Q. Therefore,
Lagrangian mechanics requires a metric gq on its configuration space Q.

It is in this sense, then, that Lagrangian mechanics ascribes Riemannian
metric structure to the world. In specifying a Lagrangian for a system,
one often requires that configuration space Q has metric structure. Having
presented North’s argument for (P1), we now turn to (P2).

2.2 Hamiltonian mechanics is Symplectic

Consider again an n particle system. Hamiltonian mechanics has as its
statespace the 6n-dimensional cotangent bundle of configuration space Q,
denoted by T∗Q. A point (q,ω) ∈ T∗Q encodes the positions and momenta
of all n particles in the system.6 As in the Lagrangian approach, we now

6One might wonder why momentum is represented by a covector rather than a vector.
After all, velocity is most naturally thought of as a vector, and momentum is simply the

5



need to specify some energy conditions to be able to say how systems evolve
over time. We do this by specifying a smooth function H : T∗Q→ R, called
the Hamiltonian, which encodes the total energy of the system in question.7

The argument for (P2) is simple: The cotangent bundle T∗Q actually has
a natural symplectic structure associated with it. This symplectic structure
is constructed by first noticing the following fact.

Fact 1. There is a canonical covector field θ(q,ω) on T
∗Q, called the Liouville

(or Poincaré, or tautological) 1-form.

The construction of the Liouville 1-form is straightforward. The cotan-
gent bundle T∗Q comes equipped with a smooth projection π : T∗Q → Q
defined by

π(q,ω) = q .

Given a point (q,ω) ∈ T∗Q, the projection π induces a linear pullback
π∗ : T∗q Q → T∗(q,ω)T

∗Q. Now to each point (q,ω) ∈ T∗Q, we want to
assign a covector — that is, an element of T∗

(q,ω)
T∗Q. So given such a point

(q,ω) ∈ T∗Q we define
θ(q,ω) = π

∗(ω) .

The covector field θ(q,ω) on T
∗Q is the Liouville 1-form.

Of course, we do not yet have a symplectic form on T∗Q; we only have a
covector field. But it is not hard to get a symplectic form out of θ(q,ω). We
simply take its exterior derivative. We define

Ω = dθ(q,ω) .

And as one can easily verify, Ω is a symplectic form on T∗Q. Ω is the
canonical symplectic form on the cotangent bundle.

product of mass — a scalar — and velocity. Momentum is viewed as a covector, however,
because of how it is defined in Lagrangian mechanics: p := ∂L

∂q̇
, which most naturally lives

in the cotangent space. See (Tao, 2008) for a nice discussion.
7The dynamics of Hamiltonian mechanics are given by Hamilton’s equations of motion:

q̇
i

=
∂H

∂pi
; ṗ

i
= −

∂H

∂qi
.

See (Baez, 2005, Ch. 4).

6



It is in precisely this sense, therefore, that Hamiltonian mechanics as-
cribes symplectic structure to the world. The statespace of Hamiltonian
mechanics has a natural symplectic structure. This structure essentially
‘comes for free’ since there is a natural symplectic form Ω associated with
the cotangent bundle of any manifold.

2.3 Riemannian > Symplectic

We have argued that Lagrangian mechanics imputes Riemannian metric
structure, while Hamiltonian mechanics imputes symplectic structure. Now
all that remains in this recapitulation of North’s argument is to argue for
(P3), that Riemannian metric structure is, in some sense, more structure
than symplectic structure.

It is clear what North’s argument is for (P3). A symplectic form Ω on
a 2n-dimensional manifold determines a volume form Ωn = Ω ∧ . . .∧ Ω on
the manifold, the n-fold wedge product of Ω with itself. So the statespace
of Hamiltonian mechanics has volume structure, while the statespace of
Lagrangian mechanics has metric structure. North reminds us that while
distance determines volume,8 volume does not determine distance.

[. . . ] there is a clear sense in which a space with a metric struc-
ture has more structure than one with just a volume element.
Metric structure comes with, or determines, or presupposes, a
volume structure, but not the other way around. [. . . ] Intu-
itively, knowing the distances between the points in a space will
give you the volumes of the regions, but the volumes will not
determine the distances. Metric structure adds a further level of
structure. (North, 2009, p. 74–75)

Since the statespace of Lagrangian mechanics has full-blown metric struc-
ture, while the statespace of Hamiltonian mechanics only has volume struc-
ture, North concludes that Hamiltonian mechanics imputes less structure

8A metric does not determine a volume form because a volume form comes with an
orientation and a metric does not give you that. For example, one can put a metric on the
Möbius strip, but one cannot put a volume form on it. A metric, however, does determine
something very close to a volume form, a volume element (which is equal to the absolute
value of a volume form). A volume element can be put on any manifold, orientable or not.

7



on the world than does Lagrangian mechanics. This completes North’s ar-
gument for (LS).

3 Hamiltonian mechanics does not impute less struc-

ture than Lagrangian mechanics

The goal of this section will be to show that North’s argument for (LS),
as presented in the previous section, is not sound. I will first argue that,
in general, Lagrangian mechanics does not posit metric structure. Then I
will show that whenever the Lagrangian statespace has metric structure, the
Hamiltonian statespace has more structure than mere symplectic structure.
Finally, I will conclude by arguing that there is a compelling sense in which
metric structure is not more structure than symplectic structure.

3.1 Lagrangian mechanics has less than metric structure

The Lagrangian statespace does not always have metric structure. In gen-
eral, specifying a Lagrangian for a system neither presupposes nor gives rise
to a metric on configuration space. But before arguing for this, let me as-
suage another potential worry that one might have with North’s argument
for (LS).

3.1.1 A Potential Worry

One might have noticed an asymmetry in the above arguments. The argu-
ment for (P1) concludes that specifying a Lagrangian requires a metric on
the configuration space Q. The argument for (P2) concludes that there is
a natural symplectic form on the cotangent bundle T∗Q. We are therefore
comparing metric structure on Q to symplectic structure on T∗Q. This
might strike one as problematic. But this worry can be assuaged, at least
in part, by recognizing the following fact.

Fact 2. A metric gq on Q can be ‘pulled back’ onto T∗Q using the smooth
projection π.

8



Suppose that, as North argues, Lagrangian mechanics requires a met-
ric gq on Q. The projection π : T∗Q → Q induces a linear pushforward
from the tangent spaces of T∗Q to the tangent spaces of Q denoted by
π∗ : T(q,v)T∗Q → TqQ, for each (q,v) ∈ T∗Q. The metric gq on Q and the
map π∗ naturally determine a metric g′(q,v) on T∗Q defined by

g′(q,v)(x,y) = gq(π∗(x),π∗(y)) ,

where x,y ∈ T(q,v)T∗Q for some (q,v) ∈ T∗Q.
This gives us at least a semblance of symmetry in the two arguments. In

Lagrangian mechanics, we have metric structure on T∗Q,9 while in Hamil-
tonian mechanics we have symplectic structure on T∗Q. This asymmetry
between the argument on the Lagrangian side and the argument on the
Hamiltonian side does not, therefore, impinge on North’s conclusion.

3.1.2 General Lagrangians

What does impinge on North’s conclusion, however, is that in general the
Lagrangian statespace does not actually have metric structure. In general,
(P1) is not true. As North rightly points out (North, 2009, p. 73), in the
completely general case the Lagrangian does not make reference to a metric
on Q. The Lagrangian is merely an arbitrary smooth scalar function L :
T∗Q→ R. It need not be of the form

L(q,v) =
1
2
gq(v,v) −V (q) ,

with gq a metric on Q and V : Q → R a smooth potential. In general,
specifying a Lagrangian L for a system simply does not require a metric gq
on configuration space.

One might try to argue, however, that although specifying a Lagrangian
L : T∗Q→ R neither presupposes nor requires a metric, it in some sense gives
rise to or determines or implicitly defines a metric. A potential argument
for (P1) along these lines is suggested by the following two facts.

9And, of course, we still have metric structure on Q.

9



Fact 3. Given a point q ∈Q, a smooth function L : T∗Q→ R gives rise to
a map LqL : TqQ→ T

∗
q Q defined by:

LqL(v) ·w =
d

dt
L(q,v + tw)

∣∣∣∣
t=0

,

for each v,w ∈ TqQ, where · denotes the application of a covector to a
vector. The map LL : T∗Q→ T∗Q defined by (q,v) 7→ (q,L

q
L(v)) is usually

called the Legendre transformation associated with L.10

The Legendre transformation allows one to ‘translate’ between the La-
grangian framework and the Hamiltonian framework. Intuitively, the covec-
tor LqL(v) encodes the momenta of the particles in a system given that they
have velocity v ∈ TqQ and the system has Lagrangian L.

As following fact shows, certain maps between a vector space and its
dual naturally give rise to additional structure on the vector space.

Fact 4. Let V be a finite dimensional real vector space, and V ∗ its dual.
Let f : V → V ∗ be a vector space isomorphism. Then f gives rise to a
nondegenerate bilinear form g : V ×V → R on V defined by

g(x,y) = f(x) ·y ,

where · again denotes the application of a covector to a vector.11

Fact 4 is illustrated in the following example. This example demonstrates
that in some cases, an isomorphism between a vector space and its dual
actually induces an inner product on the vector space.

Example 1. Let V be an n-dimensional real vector space, and V ∗ its dual.
10For a nice geometrical discussion of the Legendre transformation see (Mac Lane, 1970).

We will restrict our attention here to ‘nice’ Lagrangians — ones whose Legendre trans-
formations LL are actually diffeomorphisms between the tangent and cotangent bundles.
Such Lagrangians are typically called hyperregular (Abraham and Marsden, 1978, §3.6).

11Although the notation g is suggestive, one should note that for an arbitrary isomor-
phism f the associated bilinear form g need not be an inner product on V .

10



Let {ei} be a basis for V . For each i ∈{1, . . . ,n} let ei ∈ V ∗ be defined by

ei ·ej =

{
1 if i = j
0 otherwise

.

Now define the isomorphism f by f(ei) = ei and extending linearly. By Fact
4, f induces a nondegenerate bilinear form g on V . As one can easily verify,
in this case g is the inner product which makes the basis {ei} orthonormal.

Fact 3 and Fact 4 together suggest a potential argument for (P1). If at
each point q ∈Q, the map LqL : TqQ→ T

∗
qQ is a vector space isomorphism,

then by Fact 4, LqL would naturally give rise to a nondegenerate bilinear form
on TqQ. And if for each q ∈ Q, L

q
L is like the isomorphism f in Example

1,12 then L would naturally induce an inner product on each of the tangent
spaces of Q — that is, a metric on Q. It would be in this sense, then, that
we could say that specifying a Lagrangian L gave rise to a metric on Q.

Unfortunately, however, this argument for (P1) does not work. As the
following example illustrates, the map LqL is not always a vector space iso-
morphism.

Example 2. Let Q = R. Define the Lagrangian L : T∗R → R by

L(x,v) = v4 ,

for each (x,v) ∈ T∗R. Given x ∈ R, LxL : TxR → T
∗
x R is given by

LxL(v) ·w =
d

dt
L(x,v + tw)

∣∣∣∣
t=0

=
d

dt
(v + tw)4

∣∣∣∣
t=0

= (4v3)(w) ,

for each v,w ∈ TxR. So we conclude that LxL(v) = 4v
3. The map LL is a

diffeomorphism, but for every x ∈ R, LxL is not linear and thus not a vector
space isomorphism.

This attempt to rescue premise (P1) therefore does not work. The maps
LqL : TqQ→ T

∗
q Q induced by the Lagrangian L are not always vector space

12By which I mean, if the nondegenerate bilinear form induced by LqL were symmetric
like g in Example 1.

11



isomorphisms. Therefore, one cannot use Fact 4 to conclude that a La-
grangian gives rise to nondegenerate bilinear forms on the tangent spaces of
Q. In general, specifying a Lagrangian does not presuppose a metric on con-
figuration space. And Example 2 shows that specifying a Lagrangian also
does not give rise to a metric on Q, at least in the sense described above.
It is thus exceedingly hard to see any sense in which (P1) is true.

3.2 Hamiltonian mechanics (often) has more than symplectic

structure

One might respond to these concerns about (P1), however, simply by stipu-
lating that the scope of Lagrangian mechanics should be restricted to only
include systems with Lagrangians which do presuppose a metric on Q. One
might argue, for example, that systems with strange Lagrangians — like the
one in Example 2 — are not physically reasonable systems. After all, many
common systems that one deals with in Lagrangian mechanics do have La-
grangians of the form L(q,v) = 1

2
gq(v,v) − V (q). One might argue that it

is in this sense, therefore, that (P1) is true: Most of the physically reason-
able systems dealt with in Lagrangian mechanics have Lagrangians which
presuppose a metric on configuration space.

In this section, however, I will argue that even if one restricts the scope
of Lagrangian mechanics to deal only with systems which presuppose a met-
ric on Q, North’s argument for (LS) still does not go through. There is a
strong sense in which whenever Lagrangian mechanics posits metric struc-
ture, Hamiltonian mechanics does the same. I will first argue that whenever
there is a metric on the Lagrangian side of things, the metric naturally in-
duces a dual structure on the Hamiltonian side of things. Then I will show
that many simple Hamiltonians presuppose this dual structure in exactly
the same way that many simple Lagrangians presuppose a metric.

3.2.1 A Dual Structure on the Hamiltonian Side

Suppose that, as North has argued, in Lagrangian mechanics the configura-
tion space Q has a metric gq. The following fact is then crucial in comparing
the structures of the Lagrangian and Hamiltonian statespaces.

12



Fact 5. Given a point q ∈Q, the metric gq induces a vector space isomor-
phism Fqg : TqQ→ T∗qQ, defined for each v ∈ TqQ by

Fqg (v) = gq(v, ) ,

where the covector gq(v, ) ∈ T∗q Q acts on arbitrary vectors x ∈ TqQ by
x 7→ gq(v,x). The map Fg : T∗Q→ T∗Q defined by Fg : (q,v) 7→ (q,F

q
g (v))

is a diffeomorphism, usually called the musical isomorphism.

The isomorphism Fg lets us essentially ‘move’ the metric structure gq
from the Lagrangian side of things onto the Hamiltonian side of things. The
metric gq allows one to determine lengths of and angles between vectors —
which are supposed to encode velocities — at points of Q. And it turns out
that gq naturally gives rise to a corresponding structure g∗q which allows one
to determine lengths of and angles between covectors — which are supposed
to encode momenta — at points of Q.

The metric gq is a smooth assignment of inner products to the tangent
spaces of Q; g∗q is a smooth assignment of inner products to the cotangent
spaces of Q. It is easy to define how g∗q acts on pairs of covectors using
the fact that for each q ∈Q, Fqg is an isomorphism and thus invertible and
linear:

g∗q (α,β) = gq((F
q
g )
−1(α), (Fqg )

−1(β)) , (1)

for every α,β ∈ T∗qQ.13 Whenever there is a metric gq on the Lagrangian side
of things, therefore, a dual structure g∗q naturally arises on the Hamiltonian
side of things. If (P1) is true and the Lagrangian statespace has metric
structure, then the Hamiltonian statespace has more structure than mere
symplectic structure.

13Two remarks are worth being made here. First, as anyone familiar with geometry
will have already recognized, in coordinates the components of gq are usually written gij;
the components of g∗q are usually written g

ij. And second, it is important to note that
the metrical structure g∗q is not just some arbitrary assignment of inner products to the
cotangent spaces of Q. Given a metric gq , g∗q arises in a perfectly natural way. This is the
sense in which I say gq determines, or gives rise to, g

∗
q .

13



3.2.2 Simple Hamiltonians

And there is, in fact, a second way to see this. Notice that North’s argument
for (LS) does not compare the structures of Lagrangian and Hamiltonian
mechanics on level footing. Argument 2 demonstrates that Lagrangian me-
chanics has metric structure by considering both the statespace T∗Q and the
Lagrangian L : T∗Q → R defined on that statespace. The argument that
Hamiltonian mechanics has symplectic structure, however, only relies on the
Hamiltonian statespace T∗Q. It does not take into account the Hamiltonian
H : T∗Q→ R defined on the statespace.

This might strike one as strange. Lagrangians and Hamiltonians serve
similar purposes in Lagrangian and Hamiltonian mechanics. They encode
certain energy conditions of the systems in question and, in conjunction
with the dynamical laws, dictate how these systems will evolve over time.
If in determining the structure of the Lagrangian statespace, one considers
the Lagrangian L, then in determining the structure of the Hamiltonian
statespace one should consider the Hamiltonian H. When this asymmetry
is remedied, one can see that insofar as Lagrangian mechanics imputes the
structure gq, Hamiltonian mechanics imputes the structure g∗q in exactly the
same way.

It will be convenient to first introduce some terminology. A simple me-
chanical system, defined as follows, is precisely the kind of Lagrangian sys-
tem which is considered above in Argument 2 for (P1).

Definition 1 (Simple Mechanical System). A simple mechanical system is
a quadruple (Q,gq,V,L), where

• Q is a differentiable manifold, the configuration space for the system.

• gq : TqQ×TqQ→ R is a Riemannian metric on Q.

• V : Q→ R is a smooth potential on Q.

• L : T∗Q→ R is the Lagrangian of the system, and is defined by

L(q,v) =
1
2
gq(v,v) −V (q) .

14



Given a Lagrangian system, there is a way to translate it into a Hamil-
tonian system.14 The key is to turn the Lagrangian L : T∗Q → R into a
Hamiltonian H : T∗Q→ R. Recall that the Hamiltonian H for a system is
supposed to encode the system’s total energy. And of course, there is a way
to define the total energy of a system in Lagrangian mechanics.

Definition 2 (Total Energy). Given a system with Lagrangian L : T∗Q→
R, the total energy EL : T∗Q→ R of the system is defined by

EL(q,v) = L
q
L(v) ·v −L(q,v) ,

for each (q,v) ∈ T∗Q.15

As was remarked earlier, the Legendre transformation LL : T∗Q→ T∗Q
is a ‘translation’ between Lagrangian and Hamiltonian mechanics. The
Hamiltonian H : T∗Q→ R associated with a given Lagrangian L : T∗Q→ R
is simply defined as the translation of the total energy EL:

H = EL ◦L−1L .

The Hamiltonian H associated with a simple mechanical system requires
or presupposes g∗q in exactly the same way that the Lagrangian L associated
with a simple mechanical system requires or presupposes gq. This fact can
be seen in two simple steps. First notice that for simple mechanical systems,
the Legendre transformation associated with L is the same as the musical
isomorphism associated with gq.

Lemma 1. Let (Q,gq,V,L) be a simple mechanical system. Then LL = Fg.
14In general, the Lagrangian L of the system must be sufficiently ‘nice’ for this transla-

tion to work. More precisely, the Legendre transformation LL must be a diffeomorphism.
Recall that we are restricting attention to such ‘nice’ Lagrangians here.

15See (Abraham and Marsden, 1978, §3.5).

15



Proof. At each point q ∈Q, the map LqL is given by

LqL(v) ·w =
d

dt
L(q,v + tw)

∣∣∣∣
t=0

=
d

dt

(1
2
gq(v + tw,v + tw) −V (q)

)∣∣∣∣
t=0

=
d

dt

(1
2
gq(v,v) + tgq(v,w) +

t2

2
gq(w,w) −V (q)

)∣∣∣∣
t=0

= gq(v,w) = F
q
g (v) ·w .

Since this holds for arbitrary v,w ∈ TqQ, it must be that LL = Fg.

Then notice that the form of the Hamiltonian for a simple mechanical
system is perfectly dual to the form of the system’s Lagrangian.

Proposition 1. Let (Q,gq,V,L) be a simple mechanical system. The Hamil-
tonian H : T∗Q→ R associated with this system is given by

H(q,ω) =
1
2
g∗q (ω,ω) + V (q) ,

where g∗q : T
∗
qQ×T∗qQ→ R is defined as in equation (1).

Proof. Notice that for a simple mechanical system, the energy EL : T∗Q→ R
is given by

EL(q,v) = L
q
L(v) ·v −L(q,v) = F

q
g (v) ·v −L(q,v) =

1
2
gq(v,v) + V (q) ,

where the second equality follows from Lemma 1, and the third from Def-
inition 1 and Fact 5. The Hamiltonian H associated with the system is
therefore given by

H(q,ω) = EL ◦L−1L (q,ω)

= EL ◦F−1g (q,ω)

=
1
2
gq((F

q
g )
−1(ω), (Fqg )

−1(ω)) + V (q)

=
1
2
g∗q (ω,ω) + V (q),

16



where the last equality follows from the definition of g∗q in equation (1).

In light of Fact 5 and Proposition 1, it is hard to see any sense in which
Lagrangian mechanics could impute metric structure without Hamiltonian
mechanics doing the same. If Lagrangian mechanics presupposes a metric
gq on configuration space Q, then Fact 5 shows that gq naturally induces
the corresponding structure g∗q on the Hamiltonian statespace. And further-
more, Proposition 1 shows that simple Hamiltonians make reference to g∗q
in precisely the same way that simple Lagrangians make reference to gq.

Recall the two arguments for (P1) presented above. Argument 2 is, in
fact, like Argument 1. Argument 1 failed because it demonstrated too much;
it implied that both Lagrangian and Hamiltonian mechanics have metric
structure. The same goes for Argument 2. Insofar as Argument 2 demon-
strates that Lagrangian mechanics has metric structure, it demonstrates the
same of Hamiltonian mechanics. If (P1) is true, then North’s argument for
(LS) does not go through.16 The argument is self-undermining.

3.3 Comparing Lagrangian and Hamiltonian Structures

Even if there were a sense in which Lagrangian mechanics had metric struc-
ture without Hamiltonian mechanics having the same, North’s argument for
(LS) would quickly encounter another obstacle. It is not clear that symplec-
tic structure is, in fact, less structure than metric structure. According to
one plausible criterion for comparing amounts of mathematical structure,
(P3) is not true.17

3.3.1 Counting Mathematical Structure

North counts the structure that a physical theory ascribes to the world by
looking to the models of the theory and the mathematical structure of those
models. Given a mathematical object, North suggests, there is a particularly
natural way to assess the amount of structure that this object has (North,

16If (P1) is true and North’s argument for (P2) is sound, then Hamiltonian mechanics
imputes metric structure and symplectic structure on the world. And this will clearly not
serve the same purpose in North’s argument as (P2) did.

17In this section I expand upon a remark made in (Swanson and Halvorson, 2012).

17



2009, p. 87). One simply looks to the automorphisms (or symmetries) of the
object. Automorphisms are structure preserving maps from a mathemati-
cal object to itself. The idea is this: If an mathematical object has more
automorphisms, then it has less structure that the automorphisms need to
preserve; if it has fewer automorphisms, then the object has more structure
that the automorphisms need to preserve. The amount of structure that a
mathematical object has is inversely proportional to, in some sense, the size
of the object’s automorphism group.

We therefore count the structure of mathematical objects, like the states-
paces of Lagrangian and Hamiltonian mechanics, with something like the
following principle in mind:

(SYM) A mathematical object X has more structure than a mathematical
object Y if Aut(X) is ‘smaller than’ Aut(Y ).18

Of course, SYM is not useful until we spell out precisely what we mean by
‘smaller than.’ As a first attempt, North suggests that we use the dimension
of the automorphism groups as the relevant measure (North, 2009, p. 87).
Consider the following criterion for comparing structure:

(DIM) A mathematical object X has more structure than a mathematical
object Y if dim(Aut(X)) < dim(Aut(Y )).

Indeed, when we apply DIM to one particular case, it gives the intuitive
answer. According to DIM, an n-dimensional vector space V has less struc-
ture than an n-dimensional inner product space (V,〈 , 〉). This is good. An
inner product space intuitively has more structure than a mere vector space;
it has inner product structure in addition to vector space structure.

But DIM suffers from a couple of shortcomings. First of all, it is not com-
pletely general. There are many mathematical objects whose automorphism
group does not in any sense have a dimension. Groups do not in general have
a dimension.19 Furthermore, DIM gives some puzzling answers. Consider
the vector space R and the vector space R2. As one can easily verify, accord-
ing to DIM, R2 has less structure than R. This is a strange verdict. DIM,

18Where Aut(X) is the group of automorphisms of X.
19One might try to compare group orders, but as one can easily verify with the case of

V vs. (V,〈 , 〉), this criterion will not always give the intuitive verdict.

18



therefore, seems unappealing as a general criterion for comparing amounts
of mathematical structure.

The following example of vector space structure as compared to inner
product space structure, however, suggests another more promising way to
make SYM precise.

Example 3 (Vector Space vs. Inner Product Space). Let V be a vector space
and g an inner product on V . As mentioned above, there is an intuitive sense
in which (V,g) has more structure than V . It has all of the vector space
structure of V along with additional inner product structure. And note that
every automorphism of (V,g) is also an automorphism of V , but there are
some automorphisms of V which are not automorphisms of (V,g).20

The automorphism group of (V,〈 , 〉) is properly contained in the auto-
morphism group of V . This suggests the following revision of SYM:

(SYM∗) A mathematical object X has more structure than a mathematical
object Y if Aut(X) ( Aut(Y ).

It turns out that SYM∗ provides the intuitive verdict for many easy cases
of structural comparison. In addition to Example 3, the following are some
of the cases that SYM∗ answers correctly:

• A set X has less structure than a group (X, ·).

• A set X has less structure than a topological space (X,τ).

• A differentiable manifold (M,Uα,ϕα) has less structure than a Rie-
mannian manifold (M,Uα,ϕα,gp).

Furthermore, SYM∗ provides the intuitive verdict when applied to the
mathematical models of the classical spacetime theories mentioned at the
beginning of this paper. According to SYM∗, Aristotelian spacetime has
more structure than absolute Newtonian spacetime, which in turn has more

20This is easy to see. Automorphisms of V are just bijective linear maps f : V → V ,
while automorphisms of (V, g) are bijective linear maps f : V → V such that g(x, y) =
g(f (x), f (y)) for each x, y ∈ V .

19



structure than Neo-Newtonian (or Galilean) spacetime.21 As one can read-
ily appreciate, SYM∗ adequately captures our intuition concerning which
mathematical space has more structure at least when applied to certain easy
cases. Unfortunately, the case of metric structure and symplectic structure
is not one of these easy cases of structural comparison. According to SYM∗,
these two types of structure are incomparable.

3.3.2 Symplectic and Metric Structure are Incomparable

Recall North’s argument that metric structure is more structure than sym-
plectic structure. The argument points out the relationships that the two
structures bear to volume structure. A symplectic form on a manifold gives
rise to volume structure. It naturally determines a volume form on the
manifold. But volume structure, as North rightly points out, is less struc-
ture than metric structure. A metric gives rise to a notion of volume, but
not vice versa.22 North concludes that symplectic structure is less structure
than metric structure.

The problem with this argument is readily apparent: A symplectic form
is not the same thing as a volume form. Symplectic structure is, in a strong
sense, more structure than volume structure. If metrics and symplectic
forms are both more structure than volume structure, then one cannot de-
duce from this alone that symplectic structure is less structure than metric
structure. Metric structure and symplectic structure are, in fact, incom-
parable according to SYM∗. Metric structure is not more structure than
symplectic structure, but also symplectic structure is not more structure
than metric structure.

This can be seen by considering the following two examples. For sim-
plicity, we restrict attention to the case of inner products and symplectic
forms on vector spaces rather than metrics and symplectic forms on mani-
folds. But of course, all of this generalizes easily to the case of manifolds.
One is invited to think of the vector space V as the tangent space TpM to
a manifold M at a point p ∈M.

21The reader can easily verify this by inspecting Chapter 2 of (Earman, 1989).
22Specifically, as pointed out in footnote 9, a metric gives rise to a volume element on

the manifold.

20



Example 4 (Symplectic 6> Metric). Consider a 2-dimensional real vector
space V . Let g : V ×V → R be an inner product on V , and Ω : V ×V → R
be a symplectic form on V . Let {e1,e2} be a arbitrary basis for V . Define
f1 : V → V by

f1(e1) =
1
2
e1

f1(e2) = 2e2 ,

and extending linearly. One can easily verify that f1 preserves the symplectic
form Ω:

Ω(x,y) = Ω(f1(x),f1(y)) ,

for every x,y ∈ V . But f1 does not preserve the inner product g. This can
be seen simply by noting that g(e2,e2) 6= 4g(e2,e2) = g(f1(e2),f1(e2)).

Example 5 (Metric 6> Symplectic). Let V , g, Ω, and {e1,e2} be as in
Example 2. Define f2 : V → V by

f2(e1) = e2

f2(e2) = e1 ,

and extending linearly. One can easily verify that g(x,y) = g(f2(x),f2(y))
for all x,y ∈ V . But f2 does not preserve the symplectic form Ω, as
Ω(e1,e2) 6= −Ω(e1,e2) = Ω(f2(e1),f2(e2)).

To summarize, Example 4 presents an automorphism f1 of the structure
(V, Ω) which is not an automorphism of (V,g). The map f1 preserves the
symplectic form, but ‘breaks’ the inner product structure. So we cannot
apply SYM∗ to conclude that a symplectic vector space has more structure
than an inner product space. Example 5, on the other hand, presents an
automorphism f2 of the structure (V,g) which is not an automorphism of
(V, Ω); f2 preserves the inner product while ‘breaking’ the symplectic form.
So we cannot apply SYM∗ to conclude that an inner product space has more
structure than a symplectic vector space.

According the criterion SYM∗, therefore, neither is a metric more struc-
ture than a symplectic form nor is a symplectic form more structure than a

21



metric. The two structures are in this sense incomparable. At the very least,
this discussion demonstrates that comparing metric structure and symplec-
tic structure is not one of the easy cases of structural comparison. It is not
like comparing the structure of a vector space to that of an inner product
space, nor is it like any of the other easy cases enumerated above. Some-
thing more must be said in support of premise (P3). According to SYM∗,
(P3) is not true.23

4 Two Methodological Remarks

I hope to have demonstrated by this point that North’s argument for (LS) is
not sound. In general, Lagrangian mechanics does not presuppose nor give
rise to a metric on configuration space. This means that in general, (P1)
is not true. And even if (P1) were true, it would be hard to see a sense
in which the Lagrangian statespace could have metric structure without the
Hamiltonian statespace having the same. Furthermore, even if one conceded
that both (P1) and (P2) were true, it is not at all clear that (LS) would then
follow. Comparing metric structure and symplectic structure is not one of
the easy cases of structural comparison. According to SYM∗, (P3) is not
true. In this final section I would like to make two brief methodological
remarks which should be kept in mind by anyone trying to compare the
structure that Lagrangian and Hamiltonian mechanics ascribe to the world.

4.1 Statespace Realism

In identifying the structure that Lagrangian mechanics and Hamiltonian me-
chanics impute to the world, one might be puzzled as to why we are looking
at the structure of the statespaces of the theories. Recall the method that
was used in the arguments for (P1) and (P2) above. We tried to show that

23If someone wants to argue that metric structure is more structure than symplectic
structure, then the impetus is on them to provide a general criterion of structural com-
parison which gives this verdict. As North mentions (North, 2009, p. 87), DIM does give
this verdict. According to DIM metric structure is more structure than symplectic struc-
ture. But as we have seen, DIM is not a general criterion, and it gives some unintuitive
verdicts.

22



Lagrangian mechanics imputed metric structure by showing that its states-
pace has metric structure. And likewise, we tried to show that Hamiltonian
mechanics imputed symplectic structure by showing that its statespace has
symplectic structure. One might balk, however, at the inference from “The
statespace of theory X has structure Y ” to “Theory X ascribes structure Y
to the world.”

Consider once again the case of spacetime theories. The sense in which
spacetime theories ascribe structure to the world is fairly clear. For example,
in absolute Newtonian spacetime there is a mathematical structure which
allows one to identify spatial points over time.24 Neo-Newtonian spacetime
does not have this structure. These two theories therefore tell us different
things about the structure of spacetime.

The sense in which the mathematical structure of a theory’s statespace
ascribes structure to the world is not as clear as in the spacetime case. One
might claim, I think reasonably, that a statespace in classical mechanics
merely provides one with a convenient way of encoding the dynamical prop-
erties of bodies which are located in spacetime — the positions, velocities,
and momenta of these bodies, for example. If this is the case, then it is hard
to see how the mathematical structure of Lagrangian or Hamiltonian states-
pace imputes any structure whatsoever on the world. Certain statespace
structures might make the description of what is happening in spacetime
easier, but that does not mean that the world, in some sense, actually has
the mathematical structure that these statespaces employ.

This serves to point out the extent to which this discussion of the struc-
tures that Lagrangian and Hamiltonian mechanics ascribe to the world relies
on some form of statespace realism.25 If one thinks, as a statespace real-
ist does, that statespace exists as a concrete thing,’ then the mathematical
structure of statespace ascribes structure to the world in exactly the same
way that the structure of spacetime theories ascribes structure to the world.
This is not the place for a critical evaluation of statespace realism. But it
is important to mention that any argument seeking to demonstrate (LS) by

24Specifically, this structure is a unit timelike vector field which is covariantly constant
(Earman, 1989, Ch. 2).

25North points this out as well (North, 2009, p. 80–81).

23



appealing to the mathematical structure of the Lagrangian and Hamiltonian
statespaces will rely on some form of statespace realism.26

4.2 Physical Structure is not Mathematical Structure

Recall that in North’s argument for (LS) the following method was used to
count the structure that a physical theory imputes: We take the physical
theory, look at its models, and count the mathematical structure of the
models. In general, using this sort of method to count the structure that
a physical theory ascribes to the world requires great care. The following
example illustrates a case where this method goes awry.

Example 6 (Special Relativity). Special relativity can be formulated in
two equivalent ways: One can use a metric g with signature (+,−,−,−) or
a metric g′ with signature (−, +, +, +).27 The only difference between these
two formulations of special relativity is a sign convention. In the former,
one defines timelike vectors v as those vectors that satisfy g(v,v) > 0; in the
latter, one defines timelike vectors v as those vectors that satisfy g′(v,v) < 0.
One does not want to say that these two formulations of special relativity
impute different structure on the world. But crucially, the models of these
two theories are non-isomorphic.28

Example 6 has the following moral: In assessing the structure that a
physical theory ascribes to the world, one cannot always just look to the
mathematical structure of the models of the theory.29 This tactic would
lead one to conclude that the two equivalent formulations of special relativity
impute different, non-isomorphic structures to the world.30 And this would

26This is not to say, however, that any argument in support of (LS) must rely on
statespace realism.

27Formally, it can formulated using a 4-dimensional (1,3) metric affine space, or a 4-
dimensional (3,1) metric affine space. See (Malament, 2009) for the definition of a metric
affine space.

28This is simply because two metric affine spaces of the same (finite) dimension are
isomorphic if and only if they have the same signature (Malament, 2009, Prop. 2.3.4).

29Mathematical models for a physical theory might involve unnecessary mathematical
structure. This is another (distinct) reason to be cautious when reading off physical
structure from the mathematical structure of models.

30This is not to say that one who uses this tactic is committed to the claim that one
formulation imputes more structure than the other.

24



be an undesirable conclusion, to say the least. The difference between the
two formulations is only a sign convention. That should not in any way
affect the structure that the theory ascribes to the world.

The moral of Example 6 extends well beyond the case of Lagrangian
and Hamiltonian mechanics. Anyone interested in assessing the structure
that physical theories ascribe to the world should keep this in mind. One
cannot read off the structure that a physical theory ascribes to the world
simply from the mathematical structure of its models. One needs to re-
member the physical significance that certain bits of mathematics had —
for example, which vectors were defined as timelike vectors. It is naive to
think that we can forget the physical significance that certain bits of mathe-
matics employed by a physical theory have, and then still correctly identify
the structure that the physical theory ascribes to the world. Such a method
entails remarkably unintuitive conclusions, like the one in Example 6.

5 Conclusion

I hope to have shown that there are some significant obstacles that must be
navigated by anyone interested in comparing the amount of structure that
different physical theories ascribe to the world. The first type of obstacle
arises in the task of identifying the kind of structure that each theory im-
putes, the second type in the task of comparing these kinds of structure.
Unfortunately, in the case of Hamiltonian and Lagrangian mechanics, both
types of obstacle have proven to be particularly troublesome. And I have
argued here that neither has yet been successfully navigated.∗

References

Abraham, Ralph, and Jerrold E. Marsden. (1978). Foundations of Mechan-
ics. Reading, MA: Benjamin/Cummings Pub. Print.

Baez, John. (2005). “Lectures on Classical Mechanics.” Manuscript.
∗This material is based upon work supported by the National Science Foundation under

Grant No. DGE 1148900.

25



Earman, John. (1989). World Enough and Space-time: Absolute versus Re-
lational Theories of Space and Time. Cambridge, MA: MIT. Print.

Mac Lane, Saunders. (1970). “Hamiltonian Mechanics and Geometry.” The
American Mathematical Monthly 77.6: 570–586.

Malament, David B. (2009). “Geometry and Spacetime.” Web.

North, Jill. (2009). “The ‘Structure’ of Physics: A Case Study.” Journal of
Philosophy 106: 57–88.

Swanson, Noel, and Hans Halvorson. (2012). “On North’s ‘The Structure of
Physics’”. Manuscript.

Tao, Terence. (2008). “Phase Space.” Web.

26