understanding noether’s theorem with symplectic geometrymath.uchicago.edu › ~may › reu2017 ›...

UNDERSTANDING NOETHER’S THEOREM WITH

SYMPLECTIC GEOMETRY

CHARLES HUDGINS

Abstract. Informally, Noether’s theorem states that to every continuous

symmetry of a physical system there corresponds a conserved quantity. De-spite being one of the most celebrated results in mathematical physics, it is

seldom stated, let alone proved, with any mathematical precision. This paper

will show that Noether’s theorem, like all great theorems, is trivial to formu-late and prove once the proper mathematical theory is developed, which, in

this case, will be the theory of symplectic manifolds.

Contents

1. Introduction 12. Hamiltonian Physics Overview 23. From Manifolds to Flows — A Way of Thinking About Physical Systems 53.1. Manifolds 53.2. Vector Bundles 63.3. The Tangent Bundle 73.4. Tensors 133.5. Flows 154. Differentiation on Manifolds 175. Symplectic Geometry 225.1. A Review of Exterior Forms and Differential Forms 225.2. Symplectic Geometry 266. Noether’s Theorem 287. Conclusion 30Acknowledgments 31References 31

1. Introduction

Noether’s Theorem underlies much of modern theoretical physics and furnishesa deep connection between symmetries of physical systems and conserved quan-tities. That momentum and energy are conserved is often taken as physical law.Noether’s theorem tells us that their conservation is actually a consequence of spacetranslational symmetry and time translational symmetry respectively.

Yet, in spite of its auspicious statement and implications, Noether’s theoremis seldom proved for physics undergraduates and is instead touted as a frightfullyinteresting, yet seldom applicable fact. The aim of this paper is to demystify

Date: November 19, 2017.1

2 CHARLES HUDGINS

Noether’s theorem by developing the mathematical language necessary to stateand prove it precisely. This language, I will argue, is that of symplectic geometry.

I have attempted to write this paper so that any physicist or sufficiently curiousmathematician who has studied undergraduate analysis (and thus been exposed todifferential forms at least once before, but possibly not in the context of manifolds)will be able to follow the exposition and understand all the terms without usingany outside resources. That said, all of the listed sources are highly recommendedand the reader is encouraged to reference them to gain a deeper understanding.

Readers who have already studied manifolds are encouraged to at least read theopening paragraph of each section, as this is where most of the motivation andphysical intuition is presented.

2. Hamiltonian Physics Overview

A fundamental assumption of the Hamiltonian formulation of physics is that thetotal energy of a physical system may be written as a function of the 2n coordinatesq1, . . . , qn and p1, . . . , pn. The qi coordinates are known as the position coordinates,the pi coordinates are known as the momentum coordinates, and all 2n coordinatestaken together (qi, pi) are known as the canonical coordinates for the system. Thespace in which these 2n coordinates live is known as phase space. A point in phasespace is known as a state, since the phase space coordinates of a point uniquelyspecify the state of the physical system.

When the energy is written this way it is known as the Hamiltonian, denotedH. That is1,

(2.1) H = H(q1, . . . , qn, p1, . . . , pn)

Given the Hamiltonian for a physical system, Hamilton’s equations determinethe time-evolution of the system. They are:Hamilton’s Equations

pi = −∂H∂qi

(2.2)

qi =∂H

∂pi(2.3)

for i = 1, . . . , n [1, Section 8.1]2.To see this formalism in action, we will examine the problem of the simple

harmonic oscillator, which is of fundamental importance in all fields of physics.

Example 2.4. (The Simple Harmonic Oscillator) The Hamiltonian of a simpleharmonic oscillator is given by:

H(q, p) =p2

2m+

1

2mω2q2

where p is the momentum of the particle in question, m is its mass, ω is the angularfrequency of oscillation, and q is the displacement of the particle from its equilibriumposition.

1We will not consider the case of a Hamiltonian which depends explicitly on time. That is,∂H∂t

= 0.2In physics, given a function f which depends on time t, the derivative of f with respect to t

is often denoted f(t).

UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 3

Applying Hamilton’s equations, we find:

p = −mω2q

q =p

m

Newton’s second law states F = p where F is the force on an object. Thus, thefirst of the two equations furnished by Hamilton’s equations tells us F = −kq isthe force on the oscillating particle. This relation is known as Hooke’s law.

The second equation tells us how to calculate the momentum of the particle fromthe rate of change of position and vice versa. This is equivalent to the familiarrelation p = mv, which holds universally in classical (read: nonrelativistic, non-quantum) mechanics.

Differentiating the second equation with respect to time and substituting thefirst, one obtains:

(2.5) q = ω2q

which is the familiar second order differential equation for simple harmonic oscilla-tion with angular frequency ω.

This example shows that the Hamiltonian formalism agrees with the predictionsof Newton’s second law. Indeed, the Hamiltonian formalism may be ”derived” fromNewton’s second law in the special case of classical, nonrelativistic systems and vice-versa. Thus the Hamiltonian formalism is equivalent to Newton’s second law for acertain class of systems. That is not to say, however, that the two formulations areon equal footing when it comes to theoretical niceties.

There are two particular theoretical advantages of the Hamiltonian formalismthat will be central to this paper. The first is that Hamilton’s equations lay bare thesymplectic geometry of physical systems (that peculiar minus sign in Hamilton’sequations is not an accident). The second is that the Hamiltonian formulation lendsitself nicely to studying conservation laws. This should not come as a surprise: somerudimentary conservation laws are baked right into Hamilton’s equations. We cansee that if H doesn’t depend explicitly on pi, then qi is conserved, and vice versa.

About the first of these advantages we cannot say much more without firstdeveloping a good deal of theory. We may at least notice, however, that Hamilton’sequations may be more suggestively written in matrix form as:

(2.6)

q1

...qnp1

...pn

=

(0 In−In 0

)·

∂H∂q1...∂H∂qn∂H∂p1...∂H∂pn

where In is the n× n identity matrix.

As for the second advantage, we can already say much more (here we follow thedevelopment in [2, Lecture 15]). Suppose we have some function f : R2n × R→ Rof phase space (identified with R2n) and time (identified with R). Let’s compute

4 CHARLES HUDGINS

dfdt .

3

df

dt=∂f

∂qi

dqidt

+∂f

∂pi

dpidt

+∂f

∂t=∂f

∂qiqi +

∂f

∂pipi +

∂f

∂t

Substituting Hamilton’s equations, we have:

df

dt=∂f

∂qi

∂H

∂pi− ∂f

∂pi

∂H

∂qi+∂f

∂t

This motivates the definition:

Definition 2.7. (Poisson Brackets in the Hamiltonian Formalism) If f, g are twofunctions of phase space and time which map into R, we define the Poisson bracketof f and g, denoted f, g, by:

(2.8) f, g =∂f

∂qi

∂g

∂pi− ∂f

∂pi

∂g

∂qi

So, returning to our computation, we have:

(2.9)df

dt= f,H+

∂f

∂t

This yields the following proposition:

Proposition 2.10. (Our First Conservation Law) If f is a function of phase spacewhich maps into R and does not depend explicitly on time, then:

(2.11) f(q1(t), . . . , qn(t), p1(t), . . . , pn(t)) = constant ⇐⇒ f,H = 0

In words, f is conserved precisely when its Poisson bracket with the Hamiltonianvanishes. For a physical interpretation, we may think of such functions f as ob-servables. Evaluating at a point in phase space, i.e. a state of our physical system,yields the value of the observable at that point. A simple example would be the ith-coordinate position observable, which, when evaluated at a point in phase space,yields the i-th coordinate of the position of an object at that point. The functioncorresponding to this observable is the ith-coordinate projection map. The propo-sition gives a necessary and sufficient condition for determining when an observableis constant as a function of time.

We shall see that the Poisson bracket measures a purely geometrical property ofthe physical system. So, according to Proposition 2.10, we may study the conser-vation laws of a physical system by looking at its geometrical properties in phasespace.

Finally, one can go even further with the conservation properties that arise fromthe Hamiltonian formalism. Upon transforming the Hamiltonian to the Lagrangian,one can even prove a rudimentary form of Noether’s theorem. This form, however,lacks clear physical meaning and applies only to a very limited class of systems.For details on this, see [3].

In this paper, we seek a formulation of Noether’s theorem that arises naturallyfrom the geometrical structure of the physical system, which will be far more generaland hopefully more illuminating than some clever calculus trickery. To this end,we must develop symplectic geometry.

3Here we are using the Einstein summation convention, which prescribes that a repeated indexin a term should be summed over.


3. From Manifolds to Flows — A Way of Thinking About PhysicalSystems

3.1. Manifolds. Why manifolds? In the Hamiltonian formalism above, it sufficedto think of physical systems with n position coordinates as existing in a phasespace represented by R2n. What if, however, there are constraints on the positioncoordinates? A particle traveling along a 3-dimensional track may be parametrizedby a single position coordinate (namely its distance from the start of the track)and dealt with using the Hamiltonian formalism, but not without some loss ofinformation, such as the magnitude of the constraining force felt by the particleat any given time. If, for some reason, we wish to parametrize the particle byits position in R3, we will find it difficult to use the Hamiltonian formalism. Themanifolds perspective can systematically eliminate this difficulty, albeit at the costof more complicated mathematical machinery.

Quantum mechanics furnishes one of the most striking examples of the necessityof a manifold perspective in general physical systems. In finite dimensional quantummechanical systems, states are taken to be those elements of a given Hilbert space4

which have magnitude 1. States which only differ from each other by multiplicationby some complex scalar are physically identical. This is a long-winded way of sayingthat physically distinct quantum mechanical states are elements of CPn, where nis the dimension of the Hilbert space of the system. For more on this, see [4].

A more abstract reason for preferring manifolds is that manifolds are coordinate-free. A manifold, as we shall see, comes with a collection of maps that give localcoordinates, but the object itself is independent of the choice of coordinates. Thisproperty is desirable since any physical system may be parametrized any numberof ways.

Thus, going forward, we will think of our states as points on a manifold M .We will now briefly develop the theory of manifolds following the presentation

in [5]. We will develop only that which is necessary to understand the statementand proof of Noether’s theorem given in this paper. That said, [5] is a wonderfulresource for physicists curious about the rigorous math underlying much of physicaltheory.

Definition 3.1. A local manifold, U ⊂ E, is an open subset of a finite-dimensional,real vector space.

Definition 3.2. A local chart on a set S is a bijection ϕ from a subset U ⊆ S toa local manifold. We will often write (U,ϕ) to explicitly indicate the domain of ϕ.

Definition 3.3. An atlas A on a set S is a collection of local charts (Ui, ϕi)such that:

• (M1)⋃i Ui = S.

• (M2) If Ui∩Uj 6= ∅ for two charts (Ui, ϕi) and (Uj , ϕj), then ϕi(Ui∩Uj) =

Uji must be an open set, and the overlap map ψji = (ϕj ϕ−1i )|Uji must

be a C∞ diffeomorphism (i.e. has a C∞ inverse) onto its image.

Two atlases Ai and Aj are considered equivalent if Ai∪Aj is also an atlas. Thisis just the statement that two atlases are equivalent precisely when their charts arecompatible in the sense of property (M2). This defines an equivalence relation on

4If this is unfamiliar, you may think of it as a complex vector space with an inner-product.This picture isn’t exactly correct, but it will suffice for our purposes

6 CHARLES HUDGINS

atlases of a set S, and thus we may consider equivalence classes of atlases, whichwill be denoted [A ], where A is a representative of the equivalence class.

Definition 3.4. A manifold M = (S, [A ]) is a set S together with an equivalenceclass of atlases on S, [A ]. We will often identify M with the underlying set S.

Observe that M has a topology induced by its atlas. A ⊂ M is open if ∀a ∈ A∃U , the domain of a local chart, such that a ∈ U ⊂ A.

Example 3.5. A local manifold V ⊂ F is a manifold, where we take the atlasA = (U,ϕ)|U ⊂ V open, and ϕ = idU.

To introduce an important concept, we will now consider maps between mani-folds.

Definition 3.6. Suppose f : M → N is a map where M and N are manifolds.suppose (U,ϕ) is a local chart on M and (V, ψ) is a local chart on N with f(U) ⊂ V .Then the local representative of f with respect to ϕ and ψ is fϕψ = ψ fϕ−1 :ϕ(U) 7→ ψ(f(U)).

Suppose that for each x ∈M and each local chart (V, ψ) with f(x) ∈ V there isa chart (U,ϕ) of M with x ∈ U and f(U) ⊂ V . Then we say that f is Cr preciselywhen every local representative of f is Cr.

Why bother taking a local representative? Manifolds themselves are rather ab-stract objects and studying their behavior directly is difficult. However, by defini-tion, they are locally diffeomorphic to finite-dimensional, real vector spaces, whichare very well behaved and understood. The local representative of f lets us think off , at least locally, as a map between open subsets of finite dimensional real vectorspaces, which means, for example, that we may think about taking derivatives of fby taking derivatives of fϕψ.

Going forward, a local representative of a map will always involve composingand precomposing by local charts such that the resulting map is a map betweenopen subsets of finite-dimensional, real vector spaces.

3.2. Vector Bundles. If manifolds are the structures that will house our states,then vector bundles are the structures that will determine how states may evolvein time. That is, they will give physical law to our manifold of physical states. Theprecise sense in which this is true will not be clear until our discussion of integralcurves. For now, we will simply introduce the relevant definitions.

Definition 3.7. Suppose E and F are finite-dimensional, real vector spaces andU ⊂ E is an open subset. Then U × F is called a local vector bundle.

A local vector bundle mapping is a map ϕ : U × F → U ′ × F ′ between localvector bundles which is C∞ and of the form ϕ(u, f) = (ϕ1(u), ϕ2(u) · f), withϕ2(u) ∈ L(F, F ′) for each u ∈ U .5 If we additionally require that ϕ1 is a C∞

diffeomorphism, and ϕ2 is an isomorphism at each u ∈ U , then we say that ϕ is alocal vector bundle isomorphism .

Definition 3.8. Given a set S and a subset U ⊂ S, a local bundle chart ϕ : U →U ′ × F ′ is a bijection. A vector bundle atlas on S is a collection B = (Ui, ϕi)such that:

• (VBA 1)⋃i Ui = S.

5By L(F, F ′) we mean the set of linear maps ϕ : F → F ′.


• (VBA 2) Given two local bundle charts (Ui, ϕi) and (Uj , ϕj) such thatUi ∩Uj 6= ∅, Uji is a local vector bundle and the overlap map ψji is a localvector bundle isomorphism (both defined as in (M2)).

We define the equivalence relation on vector bundle atlases just as in the mani-folds case.

Definition 3.9. A vector bundle E = (S, [B]) is a set S together with an equiva-lence class of vector bundle atlases on S.

The zero section of E is defined by:

E0 = e0 ∈ E | ∃(U,ϕ) ∈ B ∈ [B] such that e0 ∈ U and ϕ(e0) = (u′, 0)Given e0 ∈ E0, and (U,ϕ) with ϕ(U) = U ′×F ′, where ϕ(e0) = (u′, 0), we define

Ee0,ϕ = ϕ−1(u′ × F ′) with the obvious vector space structure induced by ϕ−1.A map f : E → E′ is a vector bundle mapping if for each e ∈ E and each local

chart (V, ψ) with f(e) ∈ V there is a chart (U,ϕ) of E with e ∈ U and f(U) ⊂ Vso that the local representative fϕψ is a local vector bundle mapping.

Note, that, in particular, a vector bundle E is a manifold, as any vector bundleatlas is also an atlas.

The following proposition shows that E0 is a manifold and that there is a welldefined projection operator that maps from E into E0. The proof is omitted, butfor a complete proof, see [5, p. 38].

Proposition 3.10. (1) Ee0,ϕ1= Ee0,ϕ2

, that is, Ee0,ϕ does not depend on thechoice of local chart.

(2) If e ∈ E, there exists a unique e0 ∈ E0, such that e ∈ Ee0,ϕ ((1) shows thatthe choice of chart here is arbitrary).

(3) E0 is a manifold.(4) In light of (1) and (2), the projection map π : E → E0; e 7→ e0 is a C∞

surjection.

In words, this proposition says that we may think of E as a manifold E0 with afinite-dimensional, real vector space Ee0,ϕ ”attached” at each e0 ∈ E0.

This characterization is such a useful way to think about vector bundles that,often, one describes a vector bundle by its projection map and zero section. Thatis, instead of saying, ”let E be a vector bundle,” one might say, ”let π : E → B bea vector bundle” (where π is the projection map and B is the zero section of E).

Having defined the projection map, now is a convenient time to introduce theconcept of a section, which will be instrumental later.

Definition 3.11. Let π : E → B be a vector bundle. A Cr section of π is a mapι : B → E of class Cr such that, for each b ∈ B, π(ι(b)) = b. We denote the set ofall Cr sections of E by Γr(E) or Γr(π).

The requirement that π(i(b)) = b is equivalent to the requirement that i(b) ∈π−1(b). We call π−1(b) the fiber over b, which is, in words, the finite-dimensional,real vector space attached at b. In other words, a section maps elements in the zerosection of a vector bundle to vectors attached at that point in the zero section.

3.3. The Tangent Bundle. We have seen that any vector bundle has a built-inmanifold in the form of its zero section. Is the converse true? Is every manifoldnaturally the zero section of a vector bundle?

8 CHARLES HUDGINS

We shall see, by studying the tangent bundle, that the answer is emphatically yes.This is good news because it means there is a natural way to equip our state spacewith physical laws, in the sense of our discussion at the beginning of Section 3.2.

Our strategy in defining the tangent bundle will be to construct something thatsquares with our notion of ”tangent” and then to check that the object is in facta vector bundle. In particular, we want the tangent bundle of a manifold to be anobject which has the set of all vectors tangent to the manifold at a particular pointattached at that point. This is the same strategy employed in [5].

Recall that the velocity vector of a particle in motion is tangent to the path ofthe particle. This is why we are interested in the tangent bundle, and it is thisintuitive notion of tangency which we will abstract to the case of manifolds in orderto define the tangent bundle.

Definition 3.12. Given a manifold M and a point on the manifold m ∈ M , acurve at m is a C1 map c : I →M , where I is an open interval containing 0, withc(0) = m.

If c1 and c2 are curves at m ∈M , and (U,ϕ) is a local chart of M with m ∈ U ,then c1 and c2 are said to be tangent at m with respect to ϕ if ϕ c1 is tangent toϕ c2 at 0, that is D(ϕ c1)(0) = D(ϕ c2)(0).

The above definition would be quite poor if the tangency of two curves dependenton the choice of local chart. The next proposition assures us that this is not thecase.

Proposition 3.13. Suppose c1 and c2 are curves at m ∈ M , and (U1, ϕ1) and(U2, ϕ2) are local charts of M with m ∈ U1 ∩ U2. Then c1 and c2 are tangent atm ∈M with respect to ϕ1 iff c1 and c2 are tangent at m ∈M with respect to ϕ2.

Proof. It suffices to prove one direction, as the other direction will follow by theexact same reasoning after replacing each 1 with 2 and each 2 with 1. Hence,we suppose c1 and c2 are tangent at m ∈ M with respect to ϕ1, i.e. supposeD(ϕ1 c1)(0) = D(ϕ1 c2)(0). Then, making prominent use of the chain rule:

D(ϕ2 c1)(0) = D(ϕ2 ϕ−11 ϕ1 c1)(0)

= D(ϕ2 ϕ−11 )((ϕ1 c1)(0)) ·D(ϕ1 c1)(0)

= D(ϕ2 ϕ−11 )((ϕ1 c1)(0)) ·D(ϕ1 c2)(0)

= D(ϕ2 ϕ−11 ϕ1 c2)(0)

= D(ϕ2 c2)(0)

which shows c1 and c2 are tangent at m ∈ M with respect to ϕ2, completing theproof.

Thus, we may speak of c1 and c2 being tangent at m ∈ M without reference toany particular local chart.

Given a manifold M , the tangency of curves at m ∈ M is reflexive, symmetric,and transitive; in other words, tangency defines an equivalence relation on curvesat m ∈ M . An equivalence class of curves at m defined by the tangency relationwill be denoted by [c]m, where c is a representative of the equivalence class. Thus,we may make the following definition:


Definition 3.14. Given a manifold M , the tangent space of M at m ∈M , denotedTmM , is given by:

(3.15) TmM = [c]m | c is a curve at m

We define the tangent bundle of A ⊂ M , denoted TA, by⋃m∈A TmM . In the

case A = M , we say TM the tangent bundle of M .We define the tangent bundle projection of M by τM : TM →M ; [c]m 7→ m.

It might seem that the next task would be to check that TM is a vector bundlein a natural way. This, however, cannot be accomplished before thinking abouthow to extend maps between manifolds to maps between tangent bundles.

Definition 3.16. If f : M → N is of class C1, define the tangent of f , Tf : TM →TN , by:

(3.17) Tf([c]m) = [f c]f(m)

Note that, since f is C1, f c is a curve through N at f(m), which ensuresTf([c]m) = [f c]f(m) ∈ TN . Thus, the map Tf does indeed have the target-spaceclaimed in the definition.

Since equivalence classes are involved in the definition of Tf , well-definednessmust be checked, i.e. it must be shown that Tf([c]m) does not depend on the choiceof equivalence class representative.

Proposition 3.18. Suppose c1 : I1 → M and c2 : I2 → M are curves which aretangent at m ∈ M . Suppose, moreover, that f : M → N is a C1 map betweenmanifolds. Then f c1 and f c2 are tangent at f(m) ∈ N .

In other words, [c1]m = [c2]m =⇒ [f c1]f(m) = [f c2]f(m).

Proof. Take a local chart (U,ϕ) of M such that m ∈ U and a local chart (V, ψ)such that f(U) ⊂ V . Then, by assumption, D(ϕ c1)(0) = D(ϕ c2)(0). Observethat f c1 and f c2 are curves at f(m) ∈ N . We compute:

D(ψ f c1)(0) = D(ψ f ϕ−1 ϕ c1)(0)

= D(ψ f ϕ−1)((ϕ c1)(0)) ·D(ϕ c1)(0)

= D(ψ f ϕ−1)((ϕ c1)(0)) ·D(ϕ c2)(0)

= D(ψ f ϕ−1 ϕ c2)(0)

= D(ψ f c2)(0)

which shows f c1 and f c2 are tangent at f(m) ∈ N .

This T operation has some nice properties, which we now check.6

Proposition 3.19. Suppose f : M → N and g : K → M are C1 maps betweenmanifolds. Then:

(1) f g : K → N is C1 and T (f g) = Tf Tg;(2) T (idM ) = idTM ;(3) If f : M → N is a diffeomorphism, then Tf : TM → TN is a bijection

with Tf−1 = T (f−1).

6In fact, these properties make T a functor on the category of C1 maps between manifolds

10 CHARLES HUDGINS

Proof. (1) The first part of this statement just amounts to the composite map-ping theorem and choosing appropriate local charts and inserting a cleverform of the identity, a trick which, by now, should be familiar. As for thesecond part, one computes:

T (f g)([c]m) = [(f g) c](fg)(m)

= [f (g c)]f(g(m))

= Tf([g c]g(m))

= Tf(Tg([c]m))

= (Tf Tg)([c]m)

(2) T (idM )([c]m) = [id c]id(m) = [c]m = idTM ([c]m)

(3) Tf T (f−1) = T (f f−1) = T (idN ) = idTN , and T (f−1) Tf = T (f−1 f) = T (idM ) = idTM .

With maps between tangent bundles understood, we now seek a more concreteway of thinking about TM , at least locally. This will also furnish a concrete wayof thinking about Tf .

Definition 3.20. Suppose U ⊂ E is a local manifold. We define cu,e : I → E by:

cu,e(t) = u+ te

where u ∈ U and e ∈ E. Note that cu,e is a curve at u.

Proposition 3.21. Suppose U ⊂ E is a local manifold and [c]u ∈ TuU . Then thereexists a unique e ∈ E such that [cu,e]u = [c]u.

Proof. This amounts to showing that there exists a unique e ∈ E such that cu,e istangent to some c ∈ [c]u at u.

In other words, fixing a c ∈ [c]u, we must show that there exists a unique e ∈ Esuch that D(cu,e)(0) = Dc(0). We have D(cu,e)(t) = e, so D(cu,e)(0) = e. Thus, ife = Dc(0), then D(cu,e)(0) = Dc(0), which shows existence. Uniqueness follows bythe uniqueness of Dc(0) and D(cu,e)(0).

As a corollary to this proposition, we may define a useful map which will facilitatethinking about tangent bundles, as promised.

Corollary 3.22. Suppose U ⊂ E is a local manifold. Define i : U × E → TU byi(u, e) = [cu,e]u. Then i is a bijection. Moreover, E = i−1(TuU).

Proof. This is just a reformulation of the preceding proposition.

We now obtain a formula on local manifolds for the conjugation of Tf by i,i−1 Tf i, which is often taken as the definition of Tf on local manifolds (forexample, this definition is given in [5]).

Proposition 3.23. Suppose U ⊂ E and U ′ ⊂ E′ are local manifolds, and supposef : U → U ′ is a C1 mapping between manifolds. Then

(3.24) (i−1 Tf i)(u, e) = (f(u), Df(u) · e)


Proof. We compute:

(i−1 Tf i)(u, e) = (i−1 Tf)([cu,e]u)

= i−1([f cu,e]f(u))

Now, by definition, i−1([f cu,e]f(u)) is the unique point (u′, e′) ∈ U ′ × E′ suchthat [cu′,e′ ]u′ = [f cu,e]f(u). We must have u′ = f(u), or else the two curves arenot even at the same point. We must also require D(cu′,e′)(0) = D(f cu,e)(0).Observe:

D(f cu,e)(t) = Df(cu,e(t)) ·D(cu,e)(t) = Df(u+ te) · eThus,

e′ = D(cu′,e′)(0)

= D(f cu,e)(0)

= Df(u) · e

Therefore, i−1([f cu,e]f(u)) = (f(u), Df(u) · e), which completes the proof.

In light of this fact, we have the following proposition.

Proposition 3.25. Suppose f : U ⊂ E → U ′ ⊂ E′ is a diffeomorphism of localmanifolds. Then i−1Tf i : U×E → U ′×E′ is a local vector bundle isomorphism.

Proof. Proposition 3.23 shows that i−1 Tf i is a local vector bundle mapping.Proposition 3.19 shows that Tf is a bijection, which, since i is a bijection, ensuresthat i−1 Tf i is a bijective local vector bundle mapping, in particular, it ensuresthat (i−1 Tf i)2(u) = Df(u) is an isomorphism at each u ∈ U . Finally, we haveseen that (i−1 Tf i)1 = f , which is a diffeomorphism since f is a diffeomorphism.Therefore, i−1 Tf i is a local vector bundle isomorphism.

We are now ready to construct a natural atlas on TM from the atlas on M .

Proposition 3.26. Suppose M is a manifold that admits atlas A = (Ui, ϕi).Then TM is a vector bundle which admits a natural vector bundle atlas B =(TUi, i

−1 Tϕi).

Proof. Since TUi =⋃m∈Ui TmM , (VBA 1) follows from (M1). We now check (VBA

2). Suppose TUi ∩ TUj 6= ∅ for two charts (TUi, i−1 Tϕi) and (TUj , i

−1 Tϕj).Because i sets up a bijection between tangent spaces of local manifolds and localvector bundles, we are guaranteed that Uij;TM = i−1 Tϕi(TUi ∩ TUj) is a localvector bundle, and i−1 Tϕi is indeed a local bundle chart.

Additionally, it follows that Ui ∩Uj 6= ∅, so we may form the overlap map ψji;M(the M subscript denotes the fact that this is the overlap map as defined for atlasesof manifolds), which is a diffeomorphism onto its image (which is a local manifold,by (M2)). Thus, by Proposition 3.25, i−1 Tψji;M i is a local vector bundleisomorphism. But,

i−1 Tψji;M i = i−1 T (ϕi ϕ−1j ) i

= i−1 Tϕi (Tϕj)−1 i

= (i−1 Tϕi) (i−1 Tϕj)−1

= ψji;TM

12 CHARLES HUDGINS

(the TM subscript denotes the fact that this is the overlap map as defined forvector bundle atlases). Thus, the overlap map is a local vector bundle isomorphism.Therefore, the proposed atlas satisfies (VBA 2).

We have shown that TM is a vector bundle in a way that arises naturally fromthe manifold structure on M . Finally, we check that M may be thought of asthe zero section of (TM)0 in the sense that M and (TM)0 are diffeomorphic withrespect to the projection map.

Proposition 3.27. The map τM |(TM)0 : (TM)0 →M is a diffeomorphism.

Proof. We have:

(TM)0 = [c]m ∈ TM | ∃(TU, i−1 Tϕ) such that

[c]m ∈ TU and (i−1 Tϕ)([c]m) = (u′, 0)

Take [c]m ∈ (TM)0 and a local bundle chart (TU, i−1 Tϕ) such that [c]m ∈ TU .We take the local representative of τM |(TM)0 with respect to ϕ and i−1Tϕ, whichis given by

(τM |(TM)0)ϕ = ϕ (τM |(TM)0) (i−1 Tϕ)−1

= ϕ (τM |(TM)0) Tϕ−1 i

Note that the domain of definition of (τM |(TM)0)ϕ, which must be some subsetof U ′ × E′ = (i−1 Tϕ)(TU), is determined precisely by the requirement that(Tϕ−1 i)(u′, e′) ∈ (TM)0. That is, we require, for some chart (TV, i−1 Tψ) with(Tϕ−1 i)(u′, e′) ∈ TV that there exists (v′, 0) ∈ V ′ × F ′ = (i−1 Tψ)(V ) suchthat:

((i−1 Tψ) (Tϕ−1 i))(u′, e′) = (v′, 0)

The function on the right hand side above is just an overlap map on TM , andthus a local vector bundle isomorphism. Therefore, we must have e′ = 0 by theinjectivity of the linear part of a local vector bundle isomorphism. We have:

(i−1 Tψ) (Tϕ−1 i) = i−1 T (ψ ϕ−1) i= (ψ ϕ−1, D(ψ ϕ−1))

Since ψ ϕ−1 : U ′ → V ′ is guaranteed to be a diffeomorphism of local manifoldsby (M2), we may require that, for all u′ ∈ U ′, (ψ ϕ−1)(u′) = v′ for some v′ ∈ V ′.Thus, we see that for all u′ ∈ U ′

((i−1 Tψ) (Tϕ−1 i))(u′, 0) = (v′, 0)

may be satisfied for some v′ ∈ V ′.Therefore, the domain of (τM |(TM)0)ϕ is precisely U ′ × 0. Moreover:

(τM |(TM)0)ϕ(u′, 0) = (ϕ (τM |(TM)0) Tϕ−1 i)(u′, 0)

= (ϕ (τM |(TM)0) Tϕ−1)([cu′,e′ ]u′)

= (ϕ (τM |(TM)0))([ϕ−1 cu′,e′ ]ϕ−1(u′)

)= ϕ(ϕ−1(u′))

= u′

This shows that (τM |(TM)0)ϕ : U ′ × 0 → U ′ : (u′, 0) 7→ u′ is a bijection. Be-cause this map is just the projection onto the first coordinate with constant second


coordinate, it is actually a diffeomorphism. Therefore τM |(TM)0 : (TM)0 →M isa diffeomorphism, since our choice of chart was arbitrary.

To summarize, we found that, for each manifold M , there is a vector bundleTM that is induced naturally by the structure of M . Moreover, we constructedthis vector bundle TM such that the fiber over a pointm ∈M (defined by τ−1

M (m) =TmM) was the set of ”ways” a curve could be tangent at the point m. Then, westudied the map i, which showed that the sets TmM have a natural vector spacestructure, and allowed us to give TM a vector bundle structure that was a naturalconsequence of the manifold structure of M . Finally, we saw that M is realized inTM as the zero section.

This concludes our development of tangent bundles, but, before moving on, itwill be important to establish some simplifying conventions. We defined TM asa set of equivalence classes of curves to be sure that it was indeed ”tangent” toM in a way that extended our intuition. This definition is, however, cumbersome.The map i is the crucial element that lets us work with tangent spaces and mapsbetween them as though we were working on a local tangent bundle (at least locally,that is). Working with TM in this way is so preferable that, going forward, we willexclusively think of it this way. As such, the map i will be suppressed. If at anytime this becomes confusing, refer back to this section to see where the i’s shouldbe if they were all written out explicitly.

3.4. Tensors. We briefly recall the definition of tensors and establish some con-ventions. The goal of this section is to develop the notion of a vector field, but wewill gain generality and notational uniformity at the cost of almost no additionaldifficulty by considering tensors in general.

Definition 3.28. Suppose E is a finite-dimensional, real vector space. E∗ will

denote the set of linear maps from E to R, denoted L(E,R). A tensor of type

(rs

)is a multi-linear map t from r copies of E∗ and s copies of E to R. The set of suchtensors with the natural real vector space structure is denoted T rs (E).

The tensor product of two tensors t1 ∈ T r1s1 (E) and t2 ∈ T r2s2 (E), denoted t1 ⊗ t2is given by:

(t1 ⊗ t2)(α1, . . . , αr1 , β1, . . . , βr2 , e1, . . . , es1 , f1, . . . , fs2)

= t1(α1, . . . , αr1 , e1, . . . , es1) · t2(β1, . . . , βr2 , f1, . . . , fs2)

We will make the identifications E = T 10 (E) and E∗ = T 0

1 (E).

If these identifications are not obvious, it helps to choose a basis for E andthink of T 1

0 (E) as the set of column matrices acting on row matrices by rightmatrix multiplication. Think of T 0

1 (E) as the set of row matrices acting on columnmatrices by left matrix multiplication.

We have the following proposition which tells us how to construct a basis forT rs (E) given a basis of E. We omit the proof. See [5] for details.

Proposition 3.29. Suppose dimE = n and let e = (e1, . . . en) be an ordered basisof E. Let e∗ = (α1, . . . , αn) be the dual basis, i.e. the basis of E∗ which satisfies

αj(ei) = δji .Then T rs (E) has dimension nr+s and admits a basis:

(3.30) ei1 ⊗ · · · ⊗ eir ⊗ αj1 ⊗ · · · ⊗ αjs | ik, jk ∈ 1, . . . , n

14 CHARLES HUDGINS

where ei(αj) ≡ αj(ei).

We define an action of L(E,F ) on T rs (E).

Definition 3.31. Suppose ϕ ∈ L(E,F ) is an isomorphism. We define T rs ϕ = ϕrs ∈T rs (F ) by:

(3.32) ϕrst(β1, . . . , βr, f1, . . . , fs) = t(ϕ∗β1, . . . ϕ∗βr, ϕ−1(f1), . . . ϕ−1(fs))

where ϕ∗β · e ≡ β(ϕ(e)), βk ∈ F ∗, and fk ∈ F .

We find that T rs has the same nice properties as T from before7. We omit theproof as it is a rather tedious, but nevertheless straightforward computation.

Proposition 3.33. Take isomorphisms ϕ ∈ L(F,G) and ψ ∈ L(E,F ).

(1) (ϕ ψ)rs = ϕrs ψrs(2) (idE)rs = idT rs (E)

(3) ϕrs : T rs (E)→ T rs (F ) is an isomorphism.

We generalize to maps between local vector bundles:

Definition 3.34. Suppose ϕ : U × F → U ′ × F ′ is a local vector bundle mappingsuch that, for each u ∈ U , ϕ2(u) is an isomorphism. Then we define ϕrs : U ×T rs (F )→ U ′ × T rs (F ′) by:

(3.35) ϕrs(u, t) = (ϕ1(u), (ϕ2(u))rst)

This definition comes with a proposition that is easily verified, but not illumi-nating to prove here.

Proposition 3.36. If ϕ : U × F → U ′ × F ′ is a local vector bundle map andϕ2(u) is an isomorphism for each u ∈ U , then ϕrs is a local vector bundle map and(ϕ2(u))rs is an isomorphism for each u ∈ U . Moreover, if ϕ is a local vector bundleisomorphism, then so is ϕrs.

With this, we may make the following definition:

Definition 3.37. Suppose π : E → B is a vector bundle. Let Eb = π−1(b). Wedefine T rs (E) =

⋃b∈B T

rs (Eb), and we define πrs(e) = b ⇐⇒ e ∈ T rs (Eb).

Remark 3.38. If (U,ϕ) is a local bundle chart of E, then (T rs (U), ϕrs) is also a localbundle chart, which will, by construction, preserve the linear structure on eachfiber.

The local charts (T rs (U), ϕrs) give rise to a natural vector bundle atlas on T rs (E),turning T rs (E) into a vector bundle.

This remark, of course, must be checked, but the proof is analogous to what wasdone in the previous section with tangent bundles, and so is omitted.

We now connect what we have done with tensors to the tangent bundle.

Definition 3.39. Let M be a manifold with tangent bundle TM . We call T rs (M) ≡T rs (TM) the vector bundle of tensors of contravariant order r and covariant order

s, i.e. of type

(rs

).

We call T 01 (M) the cotangent bundle of M .

7Again, this means that T rs is a functor


Remark 3.40. Given a local chart (U,ϕ) on M , the natural chart on T rsM is(T rsU, (Tϕ)rs). This follows easily from the functorial properties of T and (·)rs afterRemark 3.38 has been verified.

We are now ready to introduce the idea of a tensor field on a manifold. Beforewe do, we should review our strategy so far. We developed the idea of the tangentbundle TM of a manifold M to give us a vector bundle with only tangent vectorsattached at each point of M . Then, given a vector bundle π : E → B, we developedthe vector bundle T rs (E) which gives us a meaningful way of attaching tensors of

type

(rs

)at each point in B, where previously only vectors were attached.

Definition 3.39 brings these two ideas together; it tells us what it means for atensor to be tangent at a point on a manifold.

Definition 3.41. A tensor field of type

(rs

)is a C∞ section of T rs (M). We denote

T rs (M) = Γ∞(T rs (M)).Let F (M) denote the set of C∞ maps from M into R.A vector field on M is an element of X (M) = T 1

0 (M).A covector field on M is an element of X ∗(M) = T 0

1 (M)

Remark 3.42. The definitions for vector fields and covector fields are motivated bythe identification discussed at the beginning of the section.

In words, a vector field is some smooth way of assigning to each point of amanifold, m ∈M , a vector in the fiber over m, TmM ≡ T 1

0 (TmM). The same goesfor the other types of tensors; however, the picture is less clear.

3.5. Flows. We are now ready to make precise the statement that vector fieldsyield the laws of our physical system.

The physical laws of a system govern how the system evolves from one state toanother as a function of time. That said, there are some evolutions which may beout-of-hand ruled unphysical. Unless our system is altered by some external force(such as performing a measurement in the case of quantum mechanics), we expectthat the state of the system evolves smoothly as a function of time. Intuitively, by”smoothly,” we mean that, in the state space, the state of the system some smallincrement of time later must be nearby and the ”vector” connecting the initial andfinal states must be tangent at the initial state. To give a crude example, consider aparticle constrained to move on the surface of a sphere. Certainly no valid evolutionof the system has the particle move away from the sphere. Rather, the particle mustmove in a direction tangent to the sphere in order to stay on the sphere.

What’s more, we expect that physical systems are deterministic. In other words,for a given point in state space, there should be just one physical path through thatpoint.

We shall see that a vector field on our state space, which is, again, taken to bea manifold, furnishes time evolutions of our system which are physical in the waysdescribed above.

First, however, we give a definition.

Definition 3.43. Given a manifold M and a vector field X ∈X (M), an integralcurve c : I →M at m ∈M is a curve at m which obeys the following equation on

16 CHARLES HUDGINS

I:

(3.44) X(c(t)) = c′(t) ≡ Tc(t, 1)

To make sense of this equation, we must look at local representatives. Take alocal chart of M , (U,ϕ), such that c(0) ∈M . We have:

cϕ(t) = (ϕ c)(t)(3.45)

(c′)ϕ(t, 1) = (Tϕ Tc)(t, 1) = T (ϕ c)(t, 1) = (cϕ)′(t)(3.46)

Xϕ(m) = (Tϕ X ϕ−1)(m)(3.47)

(X c)ϕ(t) = (Tϕ X c)(t) = (Tϕ X ϕ−1 ϕ c)(t) = (Xϕ cϕ)(t)(3.48)

Therefore, the defining equation in Definition 3.43 is equivalent to:

(3.49) Xϕ(cϕ(t)) = (cϕ)′(t)

in a local chart (U,ϕ). Ignoring base points (i.e. the first coordinate), this is justa system of first order ordinary differential equations.

In physical terms, Equation 3.44, says that the ”velocity” of the system in statespace must be tangent to a vector which itself is tangent to the state space manifold.This tangency requirement ensures the smooth (in the intuitive sense describedabove) evolution of states as a function of time.

We also want to have a notion of flow on our physical systems. Given a stateof the system, a flow outputs the state of that system after the system has evolvedforward by time t. A flow box is a structure which defines both integral curves andflows on an open subset of a manifold.

Definition 3.50. (Flow Box)8 Let M be a manifold and X ∈ X (M). A flow boxof X at m ∈M is a triple (U0, a, F ) such that:

(1) U0 ⊂M is open, m ∈ U0, and a ∈ R>0 ∪ ∞;(2) FU0 × Ia →M is of class C∞, where Ia = (−a, a);(3) if u ∈ U0, then cu : Ia →M defined by cu(λ) = F (u, λ) is an integral curve

of X at u;(4) if Fλ : U0 → M is defined by Fλ(u) = F (u, λ), then for λ ∈ Ia, Fλ(U0) is

open, and Fλ is a diffeomorphism onto its image.

Remark 3.51. One may prove, albeit after a good deal of work, that FλFµ = Fλ+µ.That is, evolving the system forward by time µ and then time λ yields the sameresult as simply evolving the system forward by time λ + µ. This is another wayof saying that the flow of states on the state space manifold induced by the vectorfield X is deterministic.

We have the following existence and uniqueness theorem, whose proof is wellbeyond the scope of this paper.

Theorem 3.52. Let M be a manifold, X ∈ X (M), and m ∈ M . Then thereexists a unique flow box of X at m ∈ M . That is, (U0, a, F ) and (U ′0, a

′, F ′) areflow boxes of X at m, then F = F ′ on U0 × Ia ∩ U ′0 ∩ Ia′ .

In summary, given a state space manifold M , a vector field X ∈X (M) inducesa unique and ”physical” flow of states on M .

8This is the definition as stated in [5].


4. Differentiation on Manifolds

Having seen how we may precisely formulate the time evolution of a physicalsystem as a flow, we must now turn to the question of conservation, i.e. howquantities change with time along these flows. The usual tool for this sort of thingin Rn is the standard multivariable derivative combined with the chain rule. In thissection, we will develop the analogous tool for manifolds.

By ”quantities”, as in our introduction to Hamiltonian mechanics, we meanfunctions which take on real values when evaluated at states in our state space. Inour new manifolds setting, then, the set of ”quantities” which may be measured onour state space is precisely the set F (M). Again, as in the Hamiltonian mechanicssection, we will call elements of F (M) observables.

Definition 4.1. Let f ∈ F (M), so that:

(4.2) Tf : TM → TR = R× Rand

(4.3) Tmf = Tf |TmM ∈ L(TmM, f(m) × R)

We define df : M → T ∗M by df(m) = P2 Tmf , where P2 denotes projectiononto the second coordinate. That is, df is the part of Tf corresponding to a lineartransformation between vector spaces, ignoring base points.

Given X ∈ X (M), the Lie derivative of f with respect to X is LXf : M → Rgiven by LXf(m) = df(m)(X(m)), where we forget the base point associated withX(m) (i.e. we just look at the vector part).

We have the following proposition, which will help us understand df :

Proposition 4.4. Take f ∈ F (M). Then df ∈ X ∗(M), and for X ∈ X (M),df(X) = P2 Tf X; that is, df(X)(m) = P2 Tmf(X(m)). Finally, we haveLXf ∈ F (M).

Proof. If df is smooth, then df ∈X ∗(M), since df is already a section of T 01 (M),

essentially by definition. Let (U,ϕ) be an admissible chart on M so that the localrepresentative of df in the natural charts is (df)ϕ = (Tϕ)0

1df ϕ−1 : U ′ → U ′×E∗,where ϕ : U ⊂M → U ′ ⊂ E. Then (taking arbitrary u′ = ϕ(u), e ∈ E):

(df)ϕ(u′) · e = ((Tϕ)01 df ϕ−1)(u′) · e

= (Tuϕ)01 · df(u) · e

= df(u)(Tu′ϕ)−1 · e= (df(u) Tu′ϕ−1) · e= (P2 Tuf Tu′ϕ−1) · e= (P2 Tu′(f ϕ−1)) · e= D(f ϕ−1)(u′) · e= D(fϕ)(u′) · e

at which point, the composite mapping theorem establishes that (df)ϕ and hencedf is of class C∞. Therefore df ∈X ∗(M).

One checks that (P2 Tf X)(m) = P2(Tf(X(m))) = P2(Tmf(X(m))) =df(m)(X(m)) = df(X)(m) = df(m)(X(m)). This shows that LXf = df(X) =P2 Tf X.

18 CHARLES HUDGINS

With the same chart as above and taking arbitrary u′ = ϕ(u), we compute:

(LXf)ϕ(u′) = (P2 Tf X)ϕ(u′)

= ((P2 Tf X) ϕ−1)(u′)

= ((P2 Tf idTM X) ϕ−1)(u′)

= ((P2 Tf TidM X) ϕ−1)(u′)

= ((P2 Tf T (ϕ−1 ϕ) X) ϕ−1)(u′)

= (P2 Tf Tϕ−1 Tϕ X ϕ−1)(u′)

= (P2 T (f ϕ−1) (Tϕ X ϕ−1))(u′)

= ((P2 Tu′(fϕ)) ·Xϕ)(u′)

= D(fϕ)(u′) ·Xϕ(u′)

at which point, the composite mapping theorem establishes that (LXf)ϕ and henceLXf : M → R is of class C∞. Therefore, LXf ∈ F (M).

We now make precise the statement that LXf is the rate of change of the ob-servable f along the flow of a vector field X. Recall that the governing equationfor an integral curve c of a vector field X is:

X(c(t)) = c′(t)

which, in a local chart, may be expressed as (ignoring base points):

Xϕ(cϕ(t)) = (cϕ)′(t) = T (cϕ)(t, 1) = D(cϕ)(t) · 1.

Now, observe that (ignoring base points):

(f c)′(t) = (f ϕ−1 ϕ c)′(t)= (fϕ cϕ)′(t)

= T (fϕ cϕ)(t, 1)

= D(fϕ cϕ)(t) · 1= Dfϕ(cϕ(t)) ·Dcϕ(t) · 1= Dfϕ(cϕ(t)) ·Xϕ(cϕ(t)) · 1= (LXf)ϕ(cϕ(t)) · 1= (LXf ϕ−1)((ϕ c)(t)) · 1= (LXf ϕ−1 ϕ c)(t) · 1= (LXf c)(t) · 1= (LXf)(c(t)) · 1

Therefore, suppressing the multiplication by 1, we have:

(4.5) (LXf)(c(t)) = (f c)′(t)

or, equivalently:

(4.6) LXf c = (f c)′

That is, (LXf)(c(t)) is precisely the rate of change, i.e. time derivative, of theobservable f along the integral curve c of X.


The explicit expression for (LXf)ϕ we derived in Proposition 4.4 makes it clearthat LX : F (M)→ F (M) will be R-linear and obey the product rule on F (M) asa consequence of the R-linearity of D and · and the fact that D obeys the productrule. A map which is R-linear and obeys the product rule is known as a derivation,for reasons that will become apparent. For now, we state this as a proposition:

Proposition 4.7. Given a vector field X, the map LX : F (M) → F (M) is a(R-linear) derivation. That is, for f, g ∈ F (M):

(4.8) LX(fg) = (LXf)g + f(LXg)

Corollary 4.9. If cλ : M → R is given by cλ(m) = λ, i.e. cλ is a constantobservable with value λ everywhere, then LXcλ = 0.

Proof. Consider c1 defined as above, i.e. c1(m) = 1. Observe that:

c1(m) = 1 = 1 · 1 = c1(m) · c1(m)

so c1 = c1c1. Observe, moreover, that for arbitrary f ∈ F :

f(m) = 1 · f(m) = c1(m) · f(m)

so c1f = f . Then:

LXc1 = LX(c1c1) = (LXc1)c1 + c1(LXc1) = 2c1(LXc1) = 2(LXc1)

Therefore:

LXc1 = 0

Observe that λc1(m) = λ · 1 = λ = cλ(m), so cλ = λc1. Then:

LX(cλ) = LX(λc1) = λ(LXc1) = λ · 0 = 0

Remark 4.10. This corollary admits a partial converse. We found that (LXf)(c(t)) =(f c)′(t), where c is an integral curve of X. Thus, if LXf = 0, then (f c)′(t) = 0,so f c = constant . Therefore, if LXf = 0, then the value of the observable f isconstant along integral curves of X.

Finally, we note the following corollary, which has little to add in the way ofphysical meaning, but will be useful in calculations:

Corollary 4.11. If f, g ∈ F (M), then d(fg) = (df)g+f(dg), and, if c is constant,then dc = 0.

We now present an existence and uniqueness result which reveals a deep corre-spondence between derivations on F (M), as defined above, and elements of X (M).The proof requires too much additional machinery to present here, but we will statethe key insight which makes the proof work.

Theorem 4.12. If θ is an R-linear derivation on F (M), then there exists a uniquevector field X ∈ X (M) such that LX = θ. Moreover, the map σ : X (M) →DerX (M) given by X 7→ LX is an isomorphism of real vector spaces and F (M)-modules between X (M) and the set of R-linear derivations on F (M), DerX (M).

20 CHARLES HUDGINS

In the process of proving this theorem, one finds that, for a local chart (U,ϕ) ofM :

Xϕ(u′) = (u′, X1ϕ(u′), . . . , Xn

ϕ(u′)) = (u′, (LXϕ1)ϕ(u′), . . . , (LXϕ

n)ϕ(u′))

(LXf)ϕ(u′) =

n∑i=1

∂fϕ∂xi

(u′)(LXϕi)ϕ(u′)(4.13)

where ϕi is the i-th component of ϕ. This is the computation underlying theperhaps unexpected isomorphism.

The following proposition arises from an entirely straightforward computation,but will allow us to meaningfully extend our notion of LX to X (M).

Proposition 4.14. If X and Y are vector fields on M , then [LX , LY ] = LX LY −LY LX is an R-linear derivation on F (M).

Proof. Evaluate [LX , LY ] on fg for f, g ∈ F (M) and apply the product rule forLX and LY .

In light of this proposition and the above theorem, we may make the followingdefinition:

Definition 4.15. [X,Y ] = LXY ∈ X (M) is the unique vector field such thatL[X,Y ] = [LX , LY ].

Remark 4.16. The brackets [·, ·] defined above turn X (M) into a Lie algebra. Thatis, [·, ·] is bilinear, alternating ([X,X] = 0 ∀X ∈ X (M), and satisfies the Jacobiidentity, which, in this case, may be written:

(4.17) LX [Y,Z] = [LXY,Z] + [Y,LXZ]

To see this, just apply the existence and uniqueness theorem over and over again.For example: L0 = 0 and L[X,X] = [LX , LX ] = 0, so, by the theorem, 0 = [X,X].

Remark 4.18. We compute [X,Y ] in a local chart (U,ϕ), making use of equa-tions (4.13).

[X,Y ]iϕ(u′) = (L[X,Y ]ϕi)(u′)

= ([LX , LY ]ϕi)ϕ(u′)

= ((LX LY − LY LX)ϕi)ϕ(u′)

= (LX(LY ϕi)− LY (LXϕ

i))ϕ(u′)

= (LX(LY ϕi))ϕ(u′)− (LY (LXϕ

i))ϕ(u′)

= (LXYiϕ)ϕ(u′)− (LYX

iϕ)ϕ(u′)

= (D(Y iϕ)(u′) ·Xϕ(u′)−D(Xiϕ)(u′) · Yϕ(u′))

=∂Y iϕ∂xj

(u′) ·Xjϕ(u′)−

∂Xiϕ

∂xj(u) · Y jϕ (u′)

That is:

(4.19) [X,Y ]iϕ =∂Y iϕ∂xj

·Xjϕ −

∂Xiϕ

∂xj· Y jϕ

Finally, we see that LX obeys the Leibniz rule on F (M)⊗X (M).


Proposition 4.20. For X ∈ X (M), LX is an R-linear derivation on F (M) ⊗X (M). That is:

(4.21) LX(fY ) = (LXf)Y + f(LXY )

Proof. For g ∈ F (M), we have:

L[X,fY ]g = LX(Lfyg)− LfY (LXg)

= LX(fLY g)− f(LY (LXg))

= (LXf)(LY g) + f(LX(LY g))− f(LY (LXg))

= (LXf)(LY g) + fL[X,Y ]g

Hence

(4.22) L[X,fY ] = (LXf)LY + fL[X,Y ]

Thus, by Theorem 4.12:

(4.23) [X, fY ] = (LXf)Y + f [X,Y ]

Therefore:

(4.24) LXfY = (LXf)Y + fLXY

which shows LX is a derivation on (F (M),X (M)) (we already know LX is R-linearon both sets).

The next theorem, which we shall not prove, as its proof is beyond the scope ofthis paper, justifies calling an R-linear map which obeys the product rule a deriva-tion. It tells us that, with the right assumptions, LX is the notion of derivative (i.e.the only object satisfying properties we conventionally associate with derivatives)

Definition 4.25. A differential operator on T (M) is an operator D such that:(DO 1) D(t1 ⊗ t2) = Dt1 ⊗ t2 + t1 ⊗Dt2(DO 2) D is natural with respect to restrictions, i.e. it is a local operator.(DO 3) For t ∈ T r

s (M), α1, . . . , αr ∈X ∗(M), X1, . . . , Xs ∈X (M), we have:

D(t(α1, . . . , αr, X1, . . . , Xs)) = (Dt)(α1, . . . , Xs)

=

r∑j=1

t(α1, . . . , Dαj , . . . , αr, X1, . . . , Xs)

=

r∑j=1

t(α1, . . . , αr, X1, . . . , DXk, . . . , Xs)

Theorem 4.26. (Willmore) Suppose U ⊂ M is open, and we have maps EU :F (U) → F (U) and FU : X (U) → X (U), which are natural with respect torestriction, i.e. are local, and are R-linear derivations. That is:

(1) EU (fg) = (EUf)g + f(EUg) and(2) FU (fX) = (EUf)X + f(FUX).

Then there is a unique differential operator D which coincides with EU on F (U)and FU on X (U).

22 CHARLES HUDGINS

For EU = LX and FU = LX , we denote this unique differential operator LX . Itsatisfies the following property, which is sometimes taken as the definition of LX[5]. If F ∗λ t ≡ (TF−λ)rs t Fλ, then:

(4.27)d

dλF ∗λ t = F ∗λLXt

As an obvious corollary, since F ∗0 = id:

(4.28) LXt = 0 ⇐⇒ t = F ∗λ t

5. Symplectic Geometry

Of course, not just any flow of a system is physical: masses don’t fall up, heliumfilled balloons don’t fall down, etc. By adding additional geometric structure to ourphysical systems, however, we will obtain flows that are as physical as we could wantthem to be. In particular, these flows will locally reproduce Hamilton’s equations.

The first hint of geometry is our reformulation of Hamilton’s equations in Equa-tion 2.6. To see this explicitly, we will need to briefly develop the theory of exteriorforms and differential forms. For a more thorough and highly accessible treatment,please see [6].

5.1. A Review of Exterior Forms and Differential Forms.

Definition 5.1. For a finite dimensional, real vector space E with dim(E) = n,the alternation mapping A : T 0

k (E)→ T 0k (E) is defined by:

(5.2) At(e1, . . . ek) =1

k!

∑σ∈Sk

(sign σ)t(eσ(1), . . . , eσ(k))

where Sk is the set of permutations on 1, . . . , k and (sign σ) is 1 if σ is an evenpermutation and −1 otherwise.

We define A(T 0k (E)) = Ωk(E), which we call the set of exterior k-forms on E.

This is the set of k-multilinear, alternating maps on E.

Given α ∈ T 0k (E) and β ∈ T 0

l (E), define α∧β = (k+l)!k!l! A(α⊗β) ∈ Ωk+l(E). The

∧ operator is known as the wedge product.For ϕ ∈ L(E,F ) and α ∈ T 0

k (E), define ϕ∗ : T 0k (E)→ T 0

k (F ) by ϕ∗α(e1, . . . , ek) =α(ϕ(e1), . . . , ϕ(ek)).

Remark 5.3. It is a fact that A A = A, i.e. A is a projection from T 0k (E) onto

Ωk(E).

The wedge product has the following properties:

Proposition 5.4. For α ∈ T 0k (E), β ∈ T 0

l (E), and γ ∈ T 0m(E) and ϕ ∈ L(E,F )

we have:

(1) α∧ ∵= Aα ∧ βα ∧Aβ;(2) ∧ is bilinear;(3) α ∧ β = (−1)klβ ∧ α;(4) α ∧ (β ∧ γ) = (α ∧ β) ∧ γ(5) ϕ∗(α ∧ β) = ϕ∗α ∧ ϕ∗β

Recall that we have an explicit basis for Ωk(E) in terms of the dual basis.


Proposition 5.5. Let n = dimE. If ei is a basis for E and αj is the dual basis,then

(5.6) αi1 ∧ · · · ∧ αik | 1 ≤ i1 < · · · < ik ≤ n

is a basis for Ωk(E).

In the same spirit as Definition 3.37, we have:

Definition 5.7. If π : E → B is a vector bundle, and A ⊂ E, then:

(5.8) ωk(E)|A =⋃b∈A

Ωk(Eb)

If A = E, denote ωk(E)|E = ωk(E). Define ωk(π) : ωk(E) → B by ωk(π)(t) =b ⇐⇒ t ∈ Ωk(Eb).

Theorem 5.9. If (Ui, ϕi) is a vector bundle atlas of π, then (ωk(E)|Ui, ϕi∗) isa vector bundle atlas of ωk(π), where ϕi∗ is defined in the same manner as ϕrs isdefined in Definition 3.34.

Proof. This is a special case of Proposition 3.36.

We are now ready for the definition of a differential form in the context ofmanifolds.

Definition 5.10. A differential k-form on a manifold M is an element of Γ∞(ωkM ),where ωkM ≡ ωk(τM ). It is a useful convention to let Ω0(M) = F (M) and Ω1(M) =F 0

1 (M).If α ∈ Ωk(M) and β ∈ Ωl(M), define α ∧ β ∈ Ωk+l(E) such that (α ∧ β)(m) =

α(m) ∧ β(m).

Definition 5.11. (Some Useful Notation) Let (U,ϕ) be a chart on a manifoldM with U ′ = ϕ(U) ⊂ Rn. Let ei denote the standard basis of Rn and letei(u) = Tϕ(u)ϕ

−1(ϕ(u), ei). Similarly, let αi denote the dual basis of ei and

αi(u) = (Tuϕ)∗(ϕ(u), αi). [Thus, for fixed u ∈ U , ei(u) and αi(u) are dual bases ofthe fiber TuM with respect to ϕ. Indeed, one may check:

αi(u)(ej(u)) = (Tuϕ)∗(ϕ(u), αi)(Tϕ(u)ϕ−1(ϕ(u), ej))

= (ϕ(u), αi)((Tuϕ) Tϕ(u)ϕ−1(ϕ(u), ej))

= (ϕ(u), αi)(Tϕ(u)(ϕ ϕ−1)(ϕ(u), ej))

= (ϕ(u), αi)(Tϕ(u)(id)(ϕ(u), ej))

= (ϕ(u), αi)(id(ϕ(u), ej))

= (ϕ(u), αi)(ϕ(u), ej)

= (ϕ(u), αi(ej))

= (ϕ(u), δij)

which verifies the assertion]. If ϕ(u) = (x1(u), . . . , xn(u)) ∈ Rn, we define:

(5.12)∂f

∂xi= Leif = df(ei)

24 CHARLES HUDGINS

Remark 5.13. Observe that:9

∂f

∂xi(u) = Leif(u)

= df(u)(ei(u))

= P2TufTϕ(u)ϕ−1(ϕ(u), ei)

= P2Tϕ(u)(f ϕ−1)(ϕ(u), ei)

= D(f ϕ−1) · (ϕ(u), ei)

= D(fϕ)(ϕ(u), ei)

= D(fϕ)(ϕ(u)) · ei

=∂fϕ∂xi

(ϕ(u))

=

(∂fϕ∂xi ϕ)

(u)

Therefore, more explicitly,

(5.14)∂f

∂xi=∂fϕ∂xi ϕ

at points u ∈ U .Defining:

(5.15) Df =

∂f∂x1

...∂f∂xn

We may write:

(5.16) Df = D(fϕ) ϕ

or, equivalently:

(5.17) (Df)ϕ = D(fϕ)

Remark 5.18. With xi defined as above, u′ = ϕ(u) so (u′)i = xi(u), and Pi(u′) =

(u′)i, the projection map onto the ith coordinate, we have:

xiϕ(u′) = (xi ϕ−1)(u′) = xi(ϕ−1(u′)) = xi(u) = (u′)i = Pi(u′)

Hence: xiϕ = Pi on U ′.

9Our convention will be that ∂∂xi

is the usual multivariable calculus operation when acting on

a function between finite-dimensional, real vector spaces.


Therefore

dxi(u)(ej(u)) =∂xi

∂xj(u)

=

(∂xiϕ∂xj ϕ

)(u)

=∂Pi∂xj

(ϕ(u))

=∂Pi∂xj

(u′)

= δij

= αi(u)(ej(u))

Which shows that

(5.19) dxi(u) = αi(u).

Remark 5.20. The previous remark showed that dxi(u) is a basis for T ∗u (M), sincewe already showed that αi(u) is a basis for T ∗u (M). We now exploit this fact, relyingon previous results about tensor bases.

To begin, df(u) ∈ T ∗u (M), so:

(5.21) df(u) = df(ei(u))dxi(u) =∂f

∂xi(u)dxi(u)

That is,

(5.22) df =∂f

∂xidxi

which should be a familiar formula from multivariable calculus.In just the same way, for each t ∈ T r

s (U) we have:

(5.23) t = ti1···irj1···jsei1 ⊗ · · · ⊗ eir ⊗ dxj1 ⊗ · · · ⊗ dxjs

and for each ω ∈ Ωk(U) we have:

(5.24) ω =∑

i1<···<ik

ωi1···ikdxi1 ∧ · · · ∧ dxik

where

(5.25) ti1···irj1···js = t(dxi1 , . . . , dxir , ej1 , . . . , ejs)

and

(5.26) ωi1···ik = ω(ei1 , . . . , eik)

We are now ready to extend d (as it was defined for F (M)) to Ωk(M).

Theorem 5.27. Let M be a manifold. Then there is a unique family of mappingsdk(U) : Ωk(U) → Ωk+1(U) (k = 0, 1, 2, . . . , n, and U is open in M), which wedenote by d, called the exterior derivative on M , such that:

(1) d is a ∧ antiderivation. That is, d is R-linear and for α ∈ Ωk(U), β ∈Ωl(U),

(5.28) d(α ∧ β) = dα ∧ β + (−1)kα ∧ dβ(2) If f ∈ F (U), df = df , with the latter as already defined.

26 CHARLES HUDGINS

(3) d d = 0.(4) d is local.

The proof of this theorem is too lengthy to reproduce here, but the punchline isthat for ω ∈ Ωk(E) with

ω =∑

i1<···<ik

ωi1···ikdxi1 ∧ · · · ∧ dxik

the unique exterior derivative d is given by:

dω =∑

i1<···<ik

d(ωi1···ik) ∧ dxi1 ∧ · · · ∧ dxik

The exterior derivative promotes a k-form to a k+ 1-form. The inner product isan operator which does the opposite.

Definition 5.29. Let M be a manifold and X ∈X (M) and ω ∈ Ωk+1(M). Thendefine iXω ∈ Ωk(M) by:

(5.30) iXω(X1, . . . , Xk) = ω(X,X1, . . . , Xk)

By convention, if ω ∈ Ω0(M), we have iXω = 0. We call iX the inner product ofX and ω.

The interior product has many magical properties, but we shall only need a fewof them.

Proposition 5.31. For X ∈X (M), α ∈ Ωk(M), and β ∈ Ωl(M), we have:

(1) iX is an antiderivation.(2) iXdf = LXf

Proof. This is a simple matter of computation.

5.2. Symplectic Geometry. The theory of differential forms is worth study inits own right and essential for developing the ideas presented in this paper further.In truth, however, we only developed the theory so that we could look at 2-forms,which will be used to encode the natural geometry of phase space.

Definition 5.32. A symplectic form ω is a closed (dω = 0), non-degenerate differ-ential 2-form on M . By nondegenerate, we mean that, if, for all Y ∈X (M):

(5.33) ω(m)(X(m), Y (m)) = 0

then X(m) = 0.A symplectic manifold is a manifold M equipped with a symplectic form ω.

To see why a symplectic manifold is the natural home of physical systems, werecall some basic facts about exterior 2-forms.

Proposition 5.34. Suppose E is a finite-dimensional, real vector space and sup-pose that ω ∈ Ω2(E). Then, there exists a basis e = (e1, . . . , en) with dual basise∗ = (α1, . . . , αn) such that:

(5.35) ω =

k∑i=1

αi ∧ αi+k


Take αi ⊗ αj | i, j ∈ 1, . . . , n to be the basis for T 02 (E). Then, in this basis, ω

has the following matrix representation(ωij = ω(ei, ej)):

(5.36) ω =

0 Ik 0−Ik 0 0

0 0 0

If, in particular, ω is nondegenerate, then:

(5.37) ω =

(0 In/2

−In/2 0

)Equation 5.37 is precisely the matrix prefactor in Equation 2.6, which indicates

that symplectic manifolds will have the right geometric structure for classical me-chanics. On manifolds, the above proposition takes the form of a theorem due toDarboux.:

Theorem 5.38. (Darboux) If ω is a closed, nondegenerate 2-form on a 2n-manifoldM , then at each m ∈M there exists a chart (U,ϕ) with m ∈ U , ϕ(m) = 0, and

(5.39) ϕ(u) = (xi(u), . . . , xn(u), y1(u), . . . , yn(u))

such that:

(5.40) ω|U =

n∑i=1

dxi ∧ dyi

Definition 5.41. The charts guaranteed by Darboux’s theorem are called sym-plectic charts and the component functions xi, yi are called canonical coordinates.

Having reproduced the geometry implied by Equation 2.6, we must now findsome mathematically precise way of reproducing Hamilton’s equations of motion.This is where the concept of a Hamiltonian vector field comes in.

Definition 5.42. Suppose (M,ω) is a symplectic manifold and H : M → R is agiven Cr function. The vector field XH determined by the condition:

(5.43) iXHω = dH

is called the Hamiltonian vector field with energy function H. We call (M,ω,XH)a Hamiltonian system.

Remark 5.44. The nondegeneracy of ω guarantees the unique existence of suchan XH . This is because nondegeneracy guarantees that the map f : X (M) →X ∗(M);X 7→ iXω is a linear isomorphism.

We will now see that integral curves on a Hamiltonian system obey Hamilton’sequations.

Proposition 5.45. Let (q1, . . . , qn, p1, . . . , pn) be canonical coordinates for ω (thecoordinate names here are meant to be suggestive), so ω =

∑i dq

i ∧ dpi. Then, inthese coordinates (and neglecting base points):

(5.46) XH =

(∂H

∂pi,−∂H

∂qi

)= J · dH

28 CHARLES HUDGINS

where J =

(0 I−I 0

). Thus (q(t), p(t)) is an integral curve of XH iff Hamilton’s

equations hold:

(5.47) qi =∂H

∂pi, pi = −∂H

∂qi, i = 1, . . . , n

Proof. If we can show that XH =(∂H∂pi

,−∂H∂qi)

is a Hamiltonian vector field, then,

by uniqueness, we will have shown that it is the Hamiltonian vector field givenenergy function H.

We have, by the definition of dpi and dqi,

iXHdqi =

∂H

∂pi

iXHdpi =∂H

∂qi

Hence:

iXHω =∑

iXH (dqi ∧ dpi) =∑

(iXHdqi) ∧ dpi −

∑dqi ∧ iXHdpi

=∑ ∂H

∂pidpi +

∂H

∂qidqi = dH

Remark 5.48. The canonical coordinates furnished by Darboux’s theorem are theprecise analog of the canonical coordinates from Hamiltonian mechanics. Frompurely geometric considerations, we furnished a set of coordinates which have ex-actly the analytical properties of the position and momentum in Hamiltonian me-chanics.

This concludes our reformulation of Hamiltonian mechanics in the context ofmanifolds. We have seen that the faint glimmer of geometric structure in Equa-tion 2.6 was no accident. On the contrary, we have found that, for a Hamiltonianvector field over a symplectic manifold, one inevitably arrives at Hamilton’s equa-tions.

6. Noether’s Theorem

The stage is at last set to prove a mathematically precise statement of Noether’stheorem. The star of our particular formulation of the theorem10 will be an objectknown as the momentum map. Much of the work in this section will be devoted tounderstanding the terms that appear in the definition of the momentum map.

Definition 6.1. A Lie Group G is a manifold which is equipped with smooth groupoperations G × G → G; (g, h) 7→ gh and G → G; g 7→ g−1 and which contains anidentity element e ∈ G.

The Lie Algebra g corresponding to the Lie group G is defined as the fiber overe in TG. That is, g = TeG. The Lie bracket on this Lie algebra is:

(6.2) [ξ, η] ≡ [Xξ, Xη](e)

10There are many formulations of Noether’s theorem. Some are more general than others andsome neither imply nor are implied by others.


There is a map between g and G known as the exponential map which will becrucial in our proof of Noether’s theorem.

Definition 6.3. For every ξ ∈ g = TeG, let ϕξ : R → G; t 7→ exptξ denote theintegral curve of Xξ passing through e at t = 0.

We now have all we need to define the momentum mapping.

Definition 6.4. (The Momentum Map) Suppose (M,ω) be a connected symplecticmanifold and Φ : G ×M → M is a symplectic action of the Lie group G on M .That is, suppose Φ defines a group action of G on M , and that for each g ∈ G,Φg : M → M ;m 7→ g · m is a symplectomorphism. A symplectomorphism is adiffeomorphism where Φ∗gω = ω, where, by definition:

(6.5) (Φ∗gω)m(X1, X2) = ωΦg(m)(dΦg(X1), dΦg(X2))

We say that a mapping:

(6.6) J : M → g∗ = T ∗eG

is a momentum mapping for the action provided that for every ξ ∈ g = TeG:

(6.7) dJ(ξ) = iξMω

or, equivalently:

(6.8) XJ(ξ) = ξM ≡d

dt(ϕξ(t), x)|t=0

where J(ξ) : M → R is defined by J(ξ)(x) = J(x) · ξ, and ξM is called theinfinitesimal generator of the action corresponding to ξ.

We will also need to generalize the Poisson bracket to our current context inorder to prove Noether’s theorem. This should come as no surprise since it wasconstructed to sniff out conservation laws in our first formulation of Hamiltonianmechanics.

Definition 6.9. Define the Poisson bracket of two maps f, g : M → R by:

(6.10) f, g = −iXf iXgω

Remark 6.11. To see why this is the natural generalization of the Poisson bracket,observe:

(6.12) f, g = −iXf iXgω = −iXf dg = −dg ·Xf = −LXf gThat is, f, g measures (up to sign) how g changes along flows of Xf and how

f changes along flows of Xg. If we let g = H, then LXHf = 0 ⇐⇒ f,H = 0.Thus f is constant along flows governed by Hamilton’s equations if and only if thePoisson bracket vanishes. This is made precise in the next lemma.

Lemma 6.13. With f, g as above, f is constant on flows of Xg iff g is constanton flows of Xf iff f, g = 0.

Proof. Suppose Ft is the flow of Xf , then:

(6.14)d

dt(g Ft) =

d

dt(F ∗t g) = F ∗t LXf g = −F ∗t f, g

The result follows from the fact that F ∗t f, g = 0 for all t iff f, g = 0.

At last, we may formulate and prove Noether’s theorem.

30 CHARLES HUDGINS

Theorem 6.15. (Noether’s Theorem) Let Φ be a symplectic action of G on (M,ω)with a momentum mapping J . Suppose H : M → R is invariant under the action,i.e.:

(6.16) H(x) = H(Φg(x))

Then J is an integral for XH , i.e. if Ft is the flow of XH , then:

(6.17) J(Ft(x)) = J(x)

Proof. Let Ft be the flow of XH . For ξ ∈ g, we have H(Φexp tξ(x)) = H(x), sinceH is invariant. We differentiate this expression with respect to time at t = 0 toobtain:

(6.18) dH(x) · ξM = 0

So, by the definition of L:

(6.19) LξMH = 0

which, by Equation 6.8 is equivalent to :

(6.20) LXJ(ξ)H = 0

Therefore:

(6.21) H, J(ξ) = 0

So, by the lemma:

(6.22)d

dt(J(ξ) Ft) = 0

Since Ft is a flow, F0 = idM . Hence, by the above statement:

(6.23) J(ξ) Ft = J(ξ) F0 = J(ξ)

Hence:

(6.24) J(Ft(x)) · ξ = J(ξ)(Ft(x)) = (J(ξ) Ft)(x) = J(ξ)(x) = J(x) · ξTherefore:

(6.25) J(Ft(x)) = J(x)

In words, we have shown that J is constant along flows of XH when H is pre-served under a group action. We call this constant value of J the momentum.

7. Conclusion

We saw that Hamiltonian mechanics hints at a connection between the geometryof the system and and conservation laws by way of the Poisson bracket. We thenstaged our physical system as a symplectic manifold equipped with a Hamiltonianvector field, and found that this connection is an inevitable consequence of thinkingof points on a manifold as states and vector fields as laws governing how those statesmay evolve.

A clever choice of map, which we called the momentum map, produced for usthe conserved quantity of Noether’s theorem. Although, as of now, it is merely aclever map. But, one may develop the theory further and see examples of how thismap arises in situ and even write it down quite explicitly if the state space is apseudo-Riemannian manifold. In this context, the momentum map is no longer an


abstruse contrivance, but rather something quite natural and, perhaps, even morefundamental than Noether’s theorem itself [7].

Acknowledgments. I would like to thank my mentor Catherine Ray for her pa-tient and helpful guidance. She provided me with references and notes which wereindispensable in learning the mathematical formalism at work in this paper. I wouldalso like to thank Peter May for providing me with a wonderful REU experiencewhich exposed me to interesting math I might never have seen otherwise.

References

[1] H. Goldstein et al. Classical Mechanics Third Edition. Addison Wesley. 2000.

[2] M. Oreglia. Lecture Notes on PHYS 18500. University of Chicago. 2017.[3] John Baez Noether’s Theorem in a Nutshell. University of California Riverside. 2002.

http://math.ucr.edu/home/baez/noether.html

[4] B. Jia and X. Lee Quantum States and Complex Projective Space. Chinese Academy of Sci-ences. https://arxiv.org/pdf/math-ph/0701011.pdf

[5] R. Abraham and J. Marsden. Foundations of Mechanics. Addison-Wesley. 1978.

[6] M. Spivak. Calculus on Manifolds. Addison-Wesley. 1995.[7] P. Woit. Not Even Wrong. Use the Moment Map, not Noether’s Theorem.

https://www.math.columbia.edu/ woit/wordpress/?p=7146

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