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 1 2. Hamiltonian Physics Overview 2 3. From Manifolds to Flows — A Way of Thinking About Physical Systems 5 3.1. Manifolds 5 3.2. Vector Bundles 6 3.3. The Tangent Bundle 7 3.4. Tensors 13 3.5. Flows 15 4. Differentiation on Manifolds 17 5. Symplectic Geometry 22 5.1. A Review of Exterior Forms and Differential Forms 22 5.2. Symplectic Geometry 26 6. Noether’s Theorem 28 7. Conclusion 30 Acknowledgments 31 References 31

1. Introduction Noether’s Theorem underlies much of modern theoretical physics and furnishes a 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 space translational symmetry and time translational symmetry respectively. Yet, in spite of its auspicious statement and implications, Noether’s theorem is seldom proved for physics undergraduates and is instead touted as a frightfully interesting, 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 state and 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 curious mathematician who has studied undergraduate analysis (and thus been exposed to differential 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 using any outside resources. That said, all of the listed sources are highly recommended and the reader is encouraged to reference them to gain a deeper understanding. Readers who have already studied manifolds are encouraged to at least read the opening paragraph of each section, as this is where most of the motivation and physical intuition is presented.

2. Hamiltonian Physics Overview A fundamental assumption of the Hamiltonian formulation of physics is that the total energy of a physical system may be written as a function of the 2n coordinates q1, . . . , 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 coordinates taken together (qi, pi) are known as the canonical coordinates for the system. The space in which these 2n coordinates live is known as phase space. A point in phase space is known as a state, since the phase space coordinates of a point uniquely specify the state of the physical system. When the energy is written this way it is known as the Hamiltonian, denoted H. That is1,

(2.1) H = H(q1, . . . , qn, p1, . . . , pn) Given the Hamiltonian for a physical system, Hamilton’s equations determine the time-evolution of the system. They are: Hamilton’s Equations ∂H (2.2) p˙i = − ∂qi ∂H (2.3) q˙i = ∂pi 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 simple harmonic oscillator is given by: p2 1 H(q, p) = + mω2q2 2m 2 where p is the momentum of the particle in question, m is its mass, ω is the angular frequency of oscillation, and q is the displacement of the particle from its equilibrium position.

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 p q˙ = m Newton’s second law states F =p ˙ where F is the force on an object. Thus, the first of the two equations furnished by Hamilton’s equations tells us F = −kq is the 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 from the rate of change of position and vice versa. This is equivalent to the familiar relation p = mv, which holds universally in classical (read: nonrelativistic, non- quantum) mechanics. Differentiating the second equation with respect to time and substituting the first, 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 predictions of Newton’s second law. Indeed, the Hamiltonian formalism may be ”derived” from Newton’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 a certain class of systems. That is not to say, however, that the two formulations are on equal footing when it comes to theoretical niceties. There are two particular theoretical advantages of the Hamiltonian formalism that will be central to this paper. The first is that Hamilton’s equations lay bare the symplectic geometry of physical systems (that peculiar minus sign in Hamilton’s equations is not an accident). The second is that the Hamiltonian formulation lends itself nicely to studying conservation laws. This should not come as a surprise: some rudimentary conservation laws are baked right into Hamilton’s equations. We can see 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 first developing a good deal of theory. We may at least notice, however, that Hamilton’s equations may be more suggestively written in matrix form as:

   ∂H  q˙1 ∂q1 .  .   .   .   .         ∂H  q˙n 0 In ∂qn (2.6)   = ·  ∂H  p˙1  −In 0    ∂p1   .     .   .   .   .  p˙n ∂H ∂pn where In is the n × n identity matrix. As for the second advantage, we can already say much more (here we follow the development in [2, Lecture 15]). Suppose we have some function f : R2n × R → R of phase space (identified with R2n) and time (identified with R). Let’s compute 4 CHARLES HUDGINS df 3 dt . df ∂f dqi ∂f dpi ∂f ∂f ∂f ∂f = + + = q˙i + p˙i + dt ∂qi dt ∂pi dt ∂t ∂qi ∂pi ∂t Substituting Hamilton’s equations, we have: df ∂f ∂H ∂f ∂H ∂f = − + dt ∂qi ∂pi ∂pi ∂qi ∂t This motivates the definition: Definition 2.7. (Poisson Brackets in the Hamiltonian Formalism) If f, g are two functions of phase space and time which map into R, we define the Poisson bracket of f and g, denoted {f, g}, by: ∂f ∂g ∂f ∂g (2.8) {f, g} = − ∂qi ∂pi ∂pi ∂qi So, returning to our computation, we have: df ∂f (2.9) = {f, H} + dt ∂t This yields the following proposition: Proposition 2.10. (Our First Conservation Law) If f is a function of phase space which 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 Hamiltonian vanishes. 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 function corresponding to this observable is the ith-coordinate projection map. The propo- sition gives a necessary and sufficient condition for determining when an observable is constant as a function of time. We shall see that the Poisson bracket measures a purely geometrical property of the 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 phase space. Finally, one can go even further with the conservation properties that arise from the 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 naturally from the geometrical structure of the physical system, which will be far more general and 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 index in a term should be summed over. UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 5

3. From Manifolds to Flows — A Way of Thinking About Physical Systems 3.1. Manifolds. Why manifolds? In the Hamiltonian formalism above, it sufficed to think of physical systems with n position coordinates as existing in a phase space represented by R2n. What if, however, there are constraints on the position coordinates? A particle traveling along a 3-dimensional track may be parametrized by 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 of information, such as the magnitude of the constraining force felt by the particle at any given time. If, for some reason, we wish to parametrize the particle by its position in R3, we will find it difficult to use the Hamiltonian formalism. The manifolds perspective can systematically eliminate this difficulty, albeit at the cost of more complicated mathematical machinery. Quantum mechanics furnishes one of the most striking examples of the necessity of a manifold perspective in general physical systems. In finite dimensional quantum mechanical 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 multiplication by some complex scalar are physically identical. This is a long-winded way of saying that physically distinct quantum mechanical states are elements of CP n, where n is 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 local coordinates, but the object itself is independent of the choice of coordinates. This property is desirable since any physical system may be parametrized any number of 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 statement and proof of Noether’s theorem given in this paper. That said, [5] is a wonderful resource for physicists curious about the rigorous math underlying much of physical theory. 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 to a 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: S • (M1) i Ui = S. • (M2) If Ui ∩ Uj 6= ∅ for two charts (Ui, ϕi) and (Uj, ϕj), then ϕi(Ui ∩ Uj) = −1 Uji must be an open set, and the overlap map ψji = (ϕj ◦ ϕi )|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. This is just the statement that two atlases are equivalent precisely when their charts are compatible 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, which will 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 equivalence class 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 atlas A = {(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 . −1 Then the local representative of f with respect to ϕ and ψ is fϕψ = ψ ◦ fϕ : ϕ(U) 7→ ψ(f(U)). Suppose that for each x ∈ M and each local chart (V, ψ) with f(x) ∈ V there is a chart (U, ϕ) of M with x ∈ U and f(U) ⊂ V . Then we say that f is Cr precisely when 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, which are very well behaved and understood. The local representative of f lets us think of f, at least locally, as a map between open subsets of finite dimensional real vector spaces, which means, for example, that we may think about taking derivatives of f by taking derivatives of fϕψ. Going forward, a local representative of a map will always involve composing and precomposing by local charts such that the resulting map is a map between open 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 evolve in time. That is, they will give physical law to our manifold of physical states. The precise sense in which this is true will not be clear until our discussion of integral curves. For now, we will simply introduce the relevant definitions. Definition 3.7. Suppose E and F are finite-dimensional, real vector spaces and U ⊂ 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 0 × F 0 between local ∞ vector bundles which is C and of the form ϕ(u, f) = (ϕ1(u), ϕ2(u) · f), with 0 5 ∞ ϕ2(u) ∈ L(F,F ) for each u ∈ U. 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 a local vector bundle isomorphism . Definition 3.8. Given a set S and a subset U ⊂ S, a local bundle chart ϕ : U → 0 0 U × F is a bijection. A vector bundle atlas on S is a collection B = {(Ui, ϕi)} such that: S • (VBA 1) i Ui = S.

5By L(F,F 0) we mean the set of linear maps ϕ : F → F 0. UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 7

• (VBA 2) Given two local bundle charts (Ui, ϕi) and (Uj, ϕj) such that Ui ∩ Uj 6= ∅, Uji is a local vector bundle and the overlap map ψji is a local vector 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: 0 E0 = {e0 ∈ E | ∃(U, ϕ) ∈ B ∈ [B] such that e0 ∈ U and ϕ(e0) = (u , 0)} 0 0 0 Given e0 ∈ E0, and (U, ϕ) with ϕ(U) = U × F , where ϕ(e0) = (u , 0), we define −1 0 0 −1 Ee0,ϕ = ϕ ({u } × F ) with the obvious vector space structure induced by ϕ . A map f : E → E0 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) ⊂ V so 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 bundle atlas is also an atlas. The following proposition shows that E0 is a manifold and that there is a well defined projection operator that maps from E into E0. The proof is omitted, but for a complete proof, see [5, p. 38].

Proposition 3.10. (1) Ee0,ϕ1 = Ee0,ϕ2 , that is, Ee0,ϕ does not depend on the choice of local chart.

(2) If e ∈ E, there exists a unique e0 ∈ E0, such that e ∈ Ee0,ϕ ((1) shows that the 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 a

finite-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. That is, instead of saying, ”let E be a vector bundle,” one might say, ”let π : E → B be a 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 the concept 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 map ι : B → E of class Cr such that, for each b ∈ B, π(ι(b)) = b. We denote the set of all 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 zero section 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-in manifold in the form of its zero section. Is the converse true? Is every manifold naturally 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 space with 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 that squares with our notion of ”tangent” and then to check that the object is in fact a vector bundle. In particular, we want the tangent bundle of a manifold to be an object which has the set of all vectors tangent to the manifold at a particular point attached 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 of the particle. This is why we are interested in the tangent bundle, and it is this intuitive notion of tangency which we will abstract to the case of manifolds in order to define the tangent bundle.

Definition 3.12. Given a manifold M and a point on the manifold m ∈ M, a curve at m is a C1 map c : I → M, where I is an open interval containing 0, with c(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 to ϕ ◦ 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 dependent on the choice of local chart. The next proposition assures us that this is not the case.

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 at m ∈ 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 the exact 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. suppose D(ϕ1 ◦ c1)(0) = D(ϕ1 ◦ c2)(0). Then, making prominent use of the chain rule:

−1 D(ϕ2 ◦ c1)(0) = D(ϕ2 ◦ ϕ1 ◦ ϕ1 ◦ c1)(0) −1 = D(ϕ2 ◦ ϕ1 )((ϕ1 ◦ c1)(0)) · D(ϕ1 ◦ c1)(0) −1 = D(ϕ2 ◦ ϕ1 )((ϕ1 ◦ c1)(0)) · D(ϕ1 ◦ c2)(0) −1 = D(ϕ2 ◦ ϕ1 ◦ ϕ1 ◦ c2)(0)

= D(ϕ2 ◦ c2)(0) which shows c1 and c2 are tangent at m ∈ M with respect to ϕ2, completing the proof. 

Thus, we may speak of c1 and c2 being tangent at m ∈ M without reference to any 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 curves at m ∈ M. An equivalence class of curves at m defined by the tangency relation will be denoted by [c]m, where c is a representative of the equivalence class. Thus, we may make the following definition: UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 9

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

(3.15) TmM = {[c]m | c is a curve at m} S 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 bundle in a natural way. This, however, cannot be accomplished before thinking about how 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, T f : TM → TN, by:

(3.17) T f([c]m) = [f ◦ c]f(m) Note that, since f is C1, f ◦ c is a curve through N at f(m), which ensures T f([c]m) = [f ◦ c]f(m) ∈ TN. Thus, the map T f does indeed have the target-space claimed in the definition. Since equivalence classes are involved in the definition of T f, well-definedness must be checked, i.e. it must be shown that T f([c]m) does not depend on the choice of equivalence class representative.

Proposition 3.18. Suppose c1 : I1 → M and c2 : I2 → M are curves which are tangent at m ∈ M. Suppose, moreover, that f : M → N is a C1 map between manifolds. 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). Observe that f ◦ c1 and f ◦ c2 are curves at f(m) ∈ N. We compute: −1 D(ψ ◦ f ◦ c1)(0) = D(ψ ◦ f ◦ ϕ ◦ ϕ ◦ c1)(0) −1 = D(ψ ◦ f ◦ ϕ )((ϕ ◦ c1)(0)) · D(ϕ ◦ c1)(0) −1 = D(ψ ◦ f ◦ ϕ )((ϕ ◦ c1)(0)) · D(ϕ ◦ c2)(0) −1 = D(ψ ◦ f ◦ ϕ ◦ 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 between manifolds. Then: (1) f ◦ g : K → N is C1 and T (f ◦ g) = T f ◦ T g; (2) T (idM ) = idTM ; (3) If f : M → N is a diffeomorphism, then T f : TM → TN is a bijection with T f −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 clever form of the identity, a trick which, by now, should be familiar. As for the second part, one computes:

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

= [f ◦ (g ◦ c)]f(g(m))

= T f([g ◦ c]g(m))

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

= (T f ◦ T g)([c]m)

(2) T (idM )([c]m) = [id ◦ c]id(m) = [c]m = idTM ([c]m) −1 −1 −1 −1 (3) T f ◦ T (f ) = T (f ◦ f ) = T (idN ) = idTN , and T (f ) ◦ T f = T (f ◦ f) = T (idM ) = idTM .  With maps between tangent bundles understood, we now seek a more concrete way of thinking about TM, at least locally. This will also furnish a concrete way of thinking about T f.

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 there exists 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 is tangent to some c ∈ [c]u at u. In other words, fixing a c ∈ [c]u, we must show that there exists a unique e ∈ E such that D(cu,e)(0) = Dc(0). We have D(cu,e)(t) = e, so D(cu,e)(0) = e. Thus, if e = Dc(0), then D(cu,e)(0) = Dc(0), which shows existence. Uniqueness follows by the uniqueness of Dc(0) and D(cu,e)(0).  As a corollary to this proposition, we may define a useful map which will facilitate thinking about tangent bundles, as promised. Corollary 3.22. Suppose U ⊂ E is a local manifold. Define i : U × E → TU by −1 i(u, e) = [cu,e]u. Then i is a bijection. Moreover, E = i (TuU).

Proof. This is just a reformulation of the preceding proposition.  We now obtain a formula on local manifolds for the conjugation of T f by i, i−1 ◦ T f ◦ i, which is often taken as the definition of T f on local manifolds (for example, this definition is given in [5]). Proposition 3.23. Suppose U ⊂ E and U 0 ⊂ E0 are local manifolds, and suppose f : U → U 0 is a C1 mapping between manifolds. Then (3.24) (i−1 ◦ T f ◦ i)(u, e) = (f(u), Df(u) · e) UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 11

Proof. We compute: −1 −1 (i ◦ T f ◦ i)(u, e) = (i ◦ T f)([cu,e]u) −1 = i ([f ◦ cu,e]f(u))

−1 0 0 0 0 Now, by definition, i ([f ◦ cu,e]f(u)) is the unique point (u , e ) ∈ U × E such 0 that [cu0,e0 ]u0 = [f ◦ cu,e]f(u). We must have u = f(u), or else the two curves are not even at the same point. We must also require D(cu0,e0 )(0) = D(f ◦ cu,e)(0). Observe:

D(f ◦ cu,e)(t) = Df(cu,e(t)) · D(cu,e)(t) = Df(u + te) · e Thus, 0 e = D(cu0,e0 )(0)

= D(f ◦ cu,e)(0) = Df(u) · e −1 Therefore, i ([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 0 ⊂ E0 is a diffeomorphism of local manifolds. Then i−1 ◦T f ◦i : U ×E → U 0 ×E0 is a local vector bundle isomorphism. Proof. Proposition 3.23 shows that i−1 ◦ T f ◦ i is a local vector bundle mapping. Proposition 3.19 shows that T f is a bijection, which, since i is a bijection, ensures that i−1 ◦ T f ◦ i is a bijective local vector bundle mapping, in particular, it ensures −1 that (i ◦ T f ◦ i)2(u) = Df(u) is an isomorphism at each u ∈ U. Finally, we have −1 seen that (i ◦T f ◦i)1 = f, which is a diffeomorphism since f is a diffeomorphism. −1 Therefore, i ◦ T f ◦ 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 = −1 (TUi, i ◦ T ϕi). Proof. Since TU = S T M, (VBA 1) follows from (M1). We now check (VBA i m∈Ui m −1 −1 2). Suppose TUi ∩ TUj 6= ∅ for two charts (TUi, i ◦ T ϕi) and (TUj, i ◦ T ϕj). Because i sets up a bijection between tangent spaces of local manifolds and local −1 vector bundles, we are guaranteed that Uij;TM = i ◦ T ϕi(TUi ∩ TUj) is a local −1 vector bundle, and i ◦ T ϕi 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 atlases of manifolds), which is a diffeomorphism onto its image (which is a local manifold, −1 by (M2)). Thus, by Proposition 3.25, i ◦ T ψji;M ◦ i is a local vector bundle isomorphism. But, −1 −1 −1 i ◦ T ψji;M ◦ i = i ◦ T (ϕi ◦ ϕj ) ◦ i −1 −1 = i ◦ T ϕi ◦ (T ϕj) ◦ i −1 −1 −1 = (i ◦ T ϕi) ◦ (i ◦ T ϕj)

= ψji;TM 12 CHARLES HUDGINS

(the TM subscript denotes the fact that this is the overlap map as defined for vector 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 from the manifold structure on M. Finally, we check that M may be thought of as the zero section of (TM)0 in the sense that M and (TM)0 are diffeomorphic with respect to the projection map.

Proposition 3.27. The map τM |(TM)0 :(TM)0 → M is a diffeomorphism. Proof. We have: −1 (TM)0 = {[c]m ∈ TM | ∃(T U, i ◦ T ϕ) such that −1 0 [c]m ∈ TU and (i ◦ T ϕ)([c]m) = (u , 0)} −1 Take [c]m ∈ (TM)0 and a local bundle chart (T U, i ◦ T ϕ) such that [c]m ∈ TU. −1 We take the local representative of τM |(TM)0 with respect to ϕ and i ◦T ϕ, which is given by −1 −1 (τM |(TM)0)ϕ = ϕ ◦ (τM |(TM)0) ◦ (i ◦ T ϕ) −1 = ϕ ◦ (τM |(TM)0) ◦ T ϕ ◦ i

Note that the domain of definition of (τM |(TM)0)ϕ, which must be some subset of U 0 × E0 = (i−1 ◦ T ϕ)(TU), is determined precisely by the requirement that −1 0 0 −1 (T ϕ ◦ i)(u , e ) ∈ (TM)0. That is, we require, for some chart (T V, i ◦ T ψ) with (T ϕ−1 ◦ i)(u0, e0) ∈ TV that there exists (v0, 0) ∈ V 0 × F 0 = (i−1 ◦ T ψ)(V ) such that: ((i−1 ◦ T ψ) ◦ (T ϕ−1 ◦ i))(u0, e0) = (v0, 0) The function on the right hand side above is just an overlap map on TM, and thus a local vector bundle isomorphism. Therefore, we must have e0 = 0 by the injectivity of the linear part of a local vector bundle isomorphism. We have: (i−1 ◦ T ψ) ◦ (T ϕ−1 ◦ i) = i−1 ◦ T (ψ ◦ ϕ−1) ◦ i = (ψ ◦ ϕ−1,D(ψ ◦ ϕ−1)) Since ψ ◦ ϕ−1 : U 0 → V 0 is guaranteed to be a diffeomorphism of local manifolds by (M2), we may require that, for all u0 ∈ U 0,(ψ ◦ ϕ−1)(u0) = v0 for some v0 ∈ V 0. Thus, we see that for all u0 ∈ U 0 ((i−1 ◦ T ψ) ◦ (T ϕ−1 ◦ i))(u0, 0) = (v0, 0) may be satisfied for some v0 ∈ V 0. 0 Therefore, the domain of (τM |(TM)0)ϕ is precisely U × {0}. Moreover: 0 −1 0 (τM |(TM)0)ϕ(u , 0) = (ϕ ◦ (τM |(TM)0) ◦ T ϕ ◦ i)(u , 0) −1 = (ϕ ◦ (τM |(TM)0) ◦ T ϕ )([cu0,e0 ]u0 ) −1
= (ϕ ◦ (τM |(TM)0)) [ϕ ◦ cu0,e0 ]ϕ−1(u0) = ϕ(ϕ−1(u0)) = u0 0 0 0 0 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 UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 13 coordinate, it is actually a diffeomorphism. Therefore τM |(TM)0 :(TM)0 → M is a diffeomorphism, since our choice of chart was arbitrary.  To summarize, we found that, for each manifold M, there is a vector bundle TM that is induced naturally by the structure of M. Moreover, we constructed −1 this vector bundle TM such that the fiber over a point m ∈ M (defined by τM (m) = TmM) was the set of ”ways” a curve could be tangent at the point m. Then, we studied the map i, which showed that the sets TmM have a natural vector space structure, and allowed us to give TM a vector bundle structure that was a natural consequence of the manifold structure of M. Finally, we saw that M is realized in TM as the zero section. This concludes our development of tangent bundles, but, before moving on, it will be important to establish some simplifying conventions. We defined TM as a set of equivalence classes of curves to be sure that it was indeed ”tangent” to M 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 maps between 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 will exclusively think of it this way. As such, the map i will be suppressed. If at any time this becomes confusing, refer back to this section to see where the i’s should be 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 we will gain generality and notational uniformity at the cost of almost no additional difficulty by considering tensors in general. Definition 3.28. Suppose E is a finite-dimensional, real vector space. E∗ will r denote the set of linear maps from E to , denoted L(E, ). A tensor of type R R s is a multi-linear map t from r copies of E∗ and s copies of E to R. The set of such r tensors with the natural real vector space structure is denoted Ts (E). The tensor product of two tensors t ∈ T r1 (E) and t ∈ T r2 (E), denoted t ⊗ t 1 s1 2 s2 1 2 is given by:

1 r1 1 r2 (t1 ⊗ t2)(α , . . . , α , β , . . . , β , e1, . . . , es1 , f1, . . . , fs2 )

1 r1 1 r2 = t1(α , . . . , α , e1, . . . , es1 ) · t2(β , . . . , β , f1, . . . , fs2 ) 1 ∗ 0 We will make the identifications E = T0 (E) and E = T1 (E). If these identifications are not obvious, it helps to choose a basis for E and 1 think of T0 (E) as the set of column matrices acting on row matrices by right 0 matrix multiplication. Think of T1 (E) as the set of row matrices acting on column matrices by left matrix multiplication. We have the following proposition which tells us how to construct a basis for r Ts (E) given a basis of E. We omit the proof. See [5] for details.

Proposition 3.29. Suppose dim E = n and let eˆ = (e1, . . . en) be an ordered basis of E. Let eˆ∗ = (α1, . . . , αn) be the dual basis, i.e. the basis of E∗ which satisfies j j α (ei) = δi . r r+s Then Ts (E) has dimension n and admits a basis:

j1 js (3.30) {ei1 ⊗ · · · ⊗ eir ⊗ α ⊗ · · · ⊗ α | ik, jk ∈ {1, . . . , n}} 14 CHARLES HUDGINS

j j where ei(α ) ≡ α (ei). r We define an action of L(E,F ) on Ts (E). r r Definition 3.31. Suppose ϕ ∈ L(E,F ) is an isomorphism. We define Ts ϕ = ϕs ∈ r Ts (F ) by: r 1 r ∗ 1 ∗ r −1 −1 (3.32) ϕst(β , . . . , β , f1, . . . , fs) = t(ϕ β , . . . ϕ β , ϕ (f1), . . . ϕ (fs)) ∗ k ∗ where ϕ β · e ≡ β(ϕ(e)), β ∈ F , and fk ∈ F . r 7 We find that Ts has the same nice properties as T from before . We omit the proof as it is a rather tedious, but nevertheless straightforward computation. Proposition 3.33. Take isomorphisms ϕ ∈ L(F,G) and ψ ∈ L(E,F ). r r r (1) (ϕ ◦ ψ)s = ϕs ◦ ψs r (2) (id ) = id r E s Ts (E) r r r (3) ϕs : Ts (E) → Ts (F ) is an isomorphism. We generalize to maps between local vector bundles: Definition 3.34. Suppose ϕ : U × F → U 0 × F 0 is a local vector bundle mapping r such that, for each u ∈ U, ϕ2(u) is an isomorphism. Then we define ϕs : U × r 0 r 0 Ts (F ) → U × Ts (F ) by: r r (3.35) ϕs(u, t) = (ϕ1(u), (ϕ2(u))st) 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 0 × F 0 is a local vector bundle map and r ϕ2(u) is an isomorphism for each u ∈ U, then ϕs is a local vector bundle map and r (ϕ2(u))s is an isomorphism for each u ∈ U. Moreover, if ϕ is a local vector bundle r isomorphism, then so is ϕs. With this, we may make the following definition:

−1 Definition 3.37. Suppose π : E → B is a vector bundle. Let Eb = π (b). We r S r r r define Ts (E) = b∈B Ts (Eb), and we define πs (e) = b ⇐⇒ e ∈ Ts (Eb). r r Remark 3.38. If (U, ϕ) is a local bundle chart of E, then (Ts (U), ϕs) is also a local bundle chart, which will, by construction, preserve the linear structure on each fiber. r r r The local charts (Ts (U), ϕs) give rise to a natural vector bundle atlas on Ts (E), r turning Ts (E) into a vector bundle. This remark, of course, must be checked, but the proof is analogous to what was done in the previous section with tangent bundles, and so is omitted. We now connect what we have done with tensors to the tangent bundle.

r Definition 3.39. Let M be a manifold with tangent bundle TM. We call Ts (M) ≡ r Ts (TM) the vector bundle of tensors of contravariant order r and covariant order r s, i.e. of type . s 0 We call T1 (M) the cotangent bundle of M.

7 r Again, this means that Ts is a functor UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 15

r Remark 3.40. Given a local chart (U, ϕ) on M, the natural chart on Ts M is r r r (Ts U, (T ϕ)s). This follows easily from the functorial properties of T and (·)s after Remark 3.38 has been verified. We are now ready to introduce the idea of a tensor field on a manifold. Before we do, we should review our strategy so far. We developed the idea of the tangent bundle TM of a manifold M to give us a vector bundle with only tangent vectors attached at each point of M. Then, given a vector bundle π : E → B, we developed r the vector bundle Ts (E) which gives us a meaningful way of attaching tensors of r type at each point in B, where previously only vectors were attached. s Definition 3.39 brings these two ideas together; it tells us what it means for a tensor to be tangent at a point on a manifold. r Definition 3.41. A tensor field of type is a C∞ section of T r(M). We denote s s r ∞ r Ts (M) = Γ (Ts (M)). Let F (M) denote the set of C∞ maps from M into R. 1 A vector field on M is an element of X (M) = T0 (M). ∗ 0 A covector field on M is an element of X (M) = T1 (M) Remark 3.42. The definitions for vector fields and covector fields are motivated by the identification discussed at the beginning of the section. In words, a vector field is some smooth way of assigning to each point of a 1 manifold, m ∈ M, a vector in the fiber over m, TmM ≡ T0 (TmM). The same goes for 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 fields yield the laws of our physical system. The physical laws of a system govern how the system evolves from one state to another as a function of time. That said, there are some evolutions which may be out-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 expect that 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 small increment of time later must be nearby and the ”vector” connecting the initial and final states must be tangent at the initial state. To give a crude example, consider a particle constrained to move on the surface of a sphere. Certainly no valid evolution of the system has the particle move away from the sphere. Rather, the particle must move 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 that point. We shall see that a vector field on our state space, which is, again, taken to be a manifold, furnishes time evolutions of our system which are physical in the ways described above. First, however, we give a definition.

Definition 3.43. Given a manifold M and a vector field X ∈ X (M), an integral curve 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)) = c0(t) ≡ T c(t, 1) To make sense of this equation, we must look at local representatives. Take a local chart of M,(U, ϕ), such that c(0) ∈ M. We have:

cϕ(t) = (ϕ ◦ c)(t)(3.45) 0 0 (c )ϕ(t, 1) = (T ϕ ◦ T c)(t, 1) = T (ϕ ◦ c)(t, 1) = (cϕ) (t)(3.46) −1 Xϕ(m) = (T ϕ ◦ X ◦ ϕ )(m)(3.47) −1 (3.48) (X ◦ c)ϕ(t) = (T ϕ ◦ X ◦ c)(t) = (T ϕ ◦ X ◦ ϕ ◦ ϕ ◦ c)(t) = (Xϕ ◦ cϕ)(t) Therefore, the defining equation in Definition 3.43 is equivalent to: 0 (3.49) Xϕ(cϕ(t)) = (cϕ) (t) in a local chart (U, ϕ). Ignoring base points (i.e. the first coordinate), this is just a system of first order ordinary differential equations. In physical terms, Equation 3.44, says that the ”velocity” of the system in state space 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 described above) evolution of states as a function of time. We also want to have a notion of flow on our physical systems. Given a state of the system, a flow outputs the state of that system after the system has evolved forward by time t. A flow box is a structure which defines both integral curves and flows on an open subset of a manifold. Definition 3.50. (Flow Box)8 Let M be a manifold and X ∈ X (M). A flow box of 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 Fλ ◦Fµ = Fλ+µ. That is, evolving the system forward by time µ and then time λ yields the same result as simply evolving the system forward by time λ + µ. This is another way of saying that the flow of states on the state space manifold induced by the vector field X is deterministic. We have the following existence and uniqueness theorem, whose proof is well beyond the scope of this paper. Theorem 3.52. Let M be a manifold, X ∈ X (M), and m ∈ M. Then there 0 0 0 exists a unique flow box of X at m ∈ M. That is, (U0, a, F ) and (U0, a ,F ) are 0 0 flow boxes of X at m, then F = F on U0 × Ia ∩ U0 ∩ Ia0 . In summary, given a state space manifold M, a vector field X ∈ X (M) induces a unique and ”physical” flow of states on M.

8This is the definition as stated in [5]. UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 17

4. Differentiation on Manifolds Having seen how we may precisely formulate the time evolution of a physical system as a flow, we must now turn to the question of conservation, i.e. how quantities change with time along these flows. The usual tool for this sort of thing in Rn is the standard multivariable derivative combined with the chain rule. In this section, we will develop the analogous tool for manifolds. By ”quantities”, as in our introduction to Hamiltonian mechanics, we mean functions which take on real values when evaluated at states in our state space. In our new manifolds setting, then, the set of ”quantities” which may be measured on our state space is precisely the set F (M). Again, as in the Hamiltonian mechanics section, we will call elements of F (M) observables. Definition 4.1. Let f ∈ F (M), so that: (4.2) T f : TM → T R = R × R and

(4.3) Tmf = T f|TmM ∈ L(TmM, {f(m)} × R) ∗ We define df : M → T M by df(m) = P2 ◦ Tmf, where P2 denotes projection onto the second coordinate. That is, df is the part of T f corresponding to a linear transformation between vector spaces, ignoring base points. Given X ∈ X (M), the Lie derivative of f with respect to X is LX f : M → R given by LX f(m) = df(m)(X(m)), where we forget the base point associated with X(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 ◦ T f ◦ X; that is, df(X)(m) = P2 ◦ Tmf(X(m)). Finally, we have LX f ∈ F (M). ∗ 0 Proof. If df is smooth, then df ∈ X (M), since df is already a section of T1 (M), essentially by definition. Let (U, ϕ) be an admissible chart on M so that the local 0 −1 0 0 ∗ representative of df in the natural charts is (df)ϕ = (T ϕ)1 ◦df ◦ϕ : U → U ×E , where ϕ : U ⊂ M → U 0 ⊂ E. Then (taking arbitrary u0 = ϕ(u), e ∈ E): 0 0 −1 0 (df)ϕ(u ) · e = ((T ϕ)1 ◦ df ◦ ϕ )(u ) · e 0 = (Tuϕ)1 · df(u) · e −1 = df(u)(Tu0 ϕ) · e −1 = (df(u) ◦ Tu0 ϕ ) · e −1 = (P2 ◦ Tuf ◦ Tu0 ϕ ) · e −1 = (P2 ◦ Tu0 (f ◦ ϕ )) · e = D(f ◦ ϕ−1)(u0) · e 0 = D(fϕ)(u ) · e at which point, the composite mapping theorem establishes that (df)ϕ and hence df is of class C∞. Therefore df ∈ X ∗(M). One checks that (P2 ◦ T f ◦ X)(m) = P2(T f(X(m))) = P2(Tmf(X(m))) = df(m)(X(m)) = df(X)(m) = df(m)(X(m)). This shows that LX f = df(X) = P2 ◦ T f ◦ X. 18 CHARLES HUDGINS

With the same chart as above and taking arbitrary u0 = ϕ(u), we compute: 0 0 (LX f)ϕ(u ) = (P2 ◦ T f ◦ X)ϕ(u ) −1 0 = ((P2 ◦ T f ◦ X) ◦ ϕ )(u ) −1 0 = ((P2 ◦ T f ◦ idTM ◦ X) ◦ ϕ )(u ) −1 0 = ((P2 ◦ T f ◦ T idM ◦ X) ◦ ϕ )(u ) −1 −1 0 = ((P2 ◦ T f ◦ T (ϕ ◦ ϕ) ◦ X) ◦ ϕ )(u ) −1 −1 0 = (P2 ◦ T f ◦ T ϕ ◦ T ϕ ◦ X ◦ ϕ )(u ) −1 −1 0 = (P2 ◦ T (f ◦ ϕ ) ◦ (T ϕ ◦ X ◦ ϕ ))(u ) 0 = ((P2 ◦ Tu0 (fϕ)) · Xϕ)(u ) 0 0 = D(fϕ)(u ) · Xϕ(u ) at which point, the composite mapping theorem establishes that (LX f)ϕ and hence ∞ LX f : M → R is of class C . Therefore, LX f ∈ F (M). 

We now make precise the statement that LX f is the rate of change of the ob- servable f along the flow of a vector field X. Recall that the governing equation for an integral curve c of a vector field X is: X(c(t)) = c0(t) which, in a local chart, may be expressed as (ignoring base points): 0 Xϕ(cϕ(t)) = (cϕ) (t) = T (cϕ)(t, 1) = D(cϕ)(t) · 1. Now, observe that (ignoring base points): (f ◦ c)0(t) = (f ◦ ϕ−1 ◦ ϕ ◦ c)0(t) 0 = (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

= (LX f)ϕ(cϕ(t)) · 1 −1 = (LX f ◦ ϕ )((ϕ ◦ c)(t)) · 1 −1 = (LX f ◦ ϕ ◦ ϕ ◦ c)(t) · 1

= (LX f ◦ c)(t) · 1

= (LX f)(c(t)) · 1 Therefore, suppressing the multiplication by 1, we have: 0 (4.5) (LX f)(c(t)) = (f ◦ c) (t) or, equivalently: 0 (4.6) LX f ◦ c = (f ◦ c)

That is, (LX f)(c(t)) is precisely the rate of change, i.e. time derivative, of the observable f along the integral curve c of X. UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 19

The explicit expression for (LX f)ϕ we derived in Proposition 4.4 makes it clear that LX : F (M) → F (M) will be R-linear and obey the product rule on F (M) as a consequence of the R-linearity of D and · and the fact that D obeys the product rule. 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) = (LX f)g + f(LX g)

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

LX c1 = LX (c1c1) = (LX c1)c1 + c1(LX c1) = 2c1(LX c1) = 2(LX c1) Therefore:

LX c1 = 0

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

LX (cλ) = LX (λc1) = λ(LX c1) = λ · 0 = 0



Remark 4.10. This corollary admits a partial converse. We found that (LX f)(c(t)) = 0 0 (f ◦ c) (t), where c is an integral curve of X. Thus, if LX f = 0, then (f ◦ c) (t) = 0, so f ◦ c = constant . Therefore, if LX f = 0, then the value of the observable f is constant along integral curves of X. Finally, we note the following corollary, which has little to add in the way of physical 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 state the key insight which makes the proof work.

Theorem 4.12. If θ is an R-linear derivation on F (M), then there exists a unique vector 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) of M: 0 0 1 0 n 0 0 1 0 n 0 Xϕ(u ) = (u ,Xϕ(u ),...,Xϕ (u )) = (u , (LX ϕ )ϕ(u ),..., (LX ϕ )ϕ(u )) n X ∂fϕ (L f) (u0) = (u0)(L ϕi) (u0)(4.13) X ϕ ∂xi X ϕ i=1 where ϕi is the i-th component of ϕ. This is the computation underlying the perhaps 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 for LX and LY .  In light of this proposition and the above theorem, we may make the following definition:

Definition 4.15. [X,Y ] = LX Y ∈ X (M) is the unique vector field such that L[X,Y ] = [LX ,LY ]. Remark 4.16. The brackets [·, ·] defined above turn X (M) into a Lie algebra. That is, [·, ·] is bilinear, alternating ([X,X] = 0 ∀X ∈ X (M), and satisfies the Jacobi identity, which, in this case, may be written:

(4.17) LX [Y,Z] = [LX Y,Z] + [Y,LX Z] 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). i 0 i 0 [X,Y ]ϕ(u ) = (L[X,Y ]ϕ )(u ) i 0 = ([LX ,LY ]ϕ )ϕ(u ) i 0 = ((LX ◦ LY − LY ◦ LX )ϕ )ϕ(u ) i i 0 = (LX (LY ϕ ) − LY (LX ϕ ))ϕ(u ) i 0 i 0 = (LX (LY ϕ ))ϕ(u ) − (LY (LX ϕ ))ϕ(u ) i 0 i 0 = (LX Yϕ)ϕ(u ) − (LY Xϕ)ϕ(u ) i 0 0 i 0 0 = (D(Yϕ)(u ) · Xϕ(u ) − D(Xϕ)(u ) · Yϕ(u )) ∂Y i ∂Xi = ϕ (u0) · Xj (u0) − ϕ (u) · Y j(u0) ∂xj ϕ ∂xj ϕ That is: ∂Y i ∂Xi (4.19) [X,Y ]i = ϕ · Xj − ϕ · Y j ϕ ∂xj ϕ ∂xj ϕ

Finally, we see that LX obeys the Leibniz rule on F (M) ⊗ X (M). UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 21

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

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

Proof. For g ∈ F (M), we have:

L[X,fY ]g = LX (Lfyg) − LfY (LX g)

= LX (fLY g) − f(LY (LX g))

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

= (LX f)(LY g) + fL[X,Y ]g Hence

(4.22) L[X,fY ] = (LX f)LY + fL[X,Y ] Thus, by Theorem 4.12:

(4.23) [X, fY ] = (LX f)Y + f[X,Y ] Therefore:

(4.24) LX fY = (LX f)Y + fLX Y which shows LX is a derivation on (F (M), X (M)) (we already know LX is R-linear on both sets). 

The next theorem, which we shall not prove, as its proof is beyond the scope of this 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. r ∗ (DO 3) For t ∈ Ts (M), α1, . . . , αr ∈ X (M), X1,...,Xs ∈ X (M), we have:

D(t(α1, . . . , αr,X1,...,Xs)) = (Dt)(α1,...,Xs) r X = t(α1, . . . , Dαj, . . . , αr,X1,...,Xs) j=1 r X = t(α1, . . . , αr,X1,...,DXk,...,Xs) j=1

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 to restriction, i.e. are local, and are R-linear derivations. That is:

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

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 . It satisfies the following property, which is sometimes taken as the definition of LX ∗ r [5]. If Fλ t ≡ (TF−λ)s ◦ t ◦ Fλ, then: d (4.27) F ∗t = F ∗L t dλ λ λ X ∗ As an obvious corollary, since F0 = id: ∗ (4.28) LX t = 0 ⇐⇒ t = Fλ t

5. Symplectic Geometry Of course, not just any flow of a system is physical: masses don’t fall up, helium filled balloons don’t fall down, etc. By adding additional geometric structure to our physical systems, however, we will obtain flows that are as physical as we could want them 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 exterior forms 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, 0 0 the alternation mapping A : Tk (E) → Tk (E) is defined by: 1 X (5.2) At(e , . . . e ) = (sign σ)t(e , . . . , e ) 1 k k! σ(1) σ(k) σ∈Sk where Sk is the set of permutations on {1, . . . , k} and (sign σ) is 1 if σ is an even permutation and −1 otherwise. 0 k We define A(Tk (E)) = Ω (E), which we call the set of exterior k-forms on E. This is the set of k-multilinear, alternating maps on E. 0 0 (k+l)! k+l Given α ∈ Tk (E) and β ∈ Tl (E), define α ∧ β = k!l! A(α ⊗ β) ∈ Ω (E). The ∧ operator is known as the wedge product. 0 ∗ 0 0 ∗ For ϕ ∈ L(E,F ) and α ∈ Tk (E), define ϕ : Tk (E) → Tk (F ) by ϕ α(e1, . . . , ek) = α(ϕ(e1), . . . , ϕ(ek)).

0 Remark 5.3. It is a fact that A ◦ A = A, i.e. A is a projection from Tk (E) onto Ωk(E). The wedge product has the following properties:

0 0 0 Proposition 5.4. For α ∈ Tk (E), β ∈ Tl (E), and γ ∈ Tm(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. UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 23

j Proposition 5.5. Let n = dim E. If ei is a basis for E and α is the dual basis, then

i1 ik (5.6) {α ∧ · · · ∧ α | 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:

k [ k (5.8) ω (E)|A = Ω (Eb) b∈A If A = E, denote ωk(E)|E = ωk(E). Define ωk(π): ωk(E) → B by ωk(π)(t) = k b ⇐⇒ t ∈ Ω (Eb).

k Theorem 5.9. If {(Ui, ϕi)} is a vector bundle atlas of π, then (ω (E)|Ui, ϕi∗) is k r a vector bundle atlas of ω (π), where ϕi∗ is defined in the same manner as ϕs is defined 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 of manifolds.

∞ k Definition 5.10. A differential k-form on a manifold M is an element of Γ (ωM ), k k 0 1 where ωM ≡ ω (τM ). It is a useful convention to let Ω (M) = F (M) and Ω (M) = 0 F1 (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 manifold 0 n n M with U = ϕ(U) ⊂ R . Let ei denote the standard basis of R and let −1 i ei(u) = Tϕ(u)ϕ (ϕ(u), ei). Similarly, let α denote the dual basis of ei and ∗ i αi(u) = (Tuϕ) (ϕ(u), α ). [Thus, for fixed u ∈ U, ei(u) and αi(u) are dual bases of the fiber TuM with respect to ϕ. Indeed, one may check: i ∗ i −1 α (u)(ej(u)) = (Tuϕ) (ϕ(u), α )(Tϕ(u)ϕ (ϕ(u), ej)) i −1 = (ϕ(u), α )((Tuϕ) ◦ Tϕ(u)ϕ (ϕ(u), ej)) i −1 = (ϕ(u), α )(Tϕ(u)(ϕ ◦ ϕ )(ϕ(u), ej)) i = (ϕ(u), α )(Tϕ(u)(id)(ϕ(u), ej)) i = (ϕ(u), α )(id(ϕ(u), ej)) i = (ϕ(u), α )(ϕ(u), ej) i = (ϕ(u), α (ej)) i = (ϕ(u), δj) which verifies the assertion]. If ϕ(u) = (x1(u), . . . , xn(u)) ∈ Rn, we define: ∂f (5.12) = L f = df(e ) ∂xi ei i 24 CHARLES HUDGINS

Remark 5.13. Observe that:9

∂f (u) = L f(u) ∂xi ei = df(u)(ei(u)) −1 = P2TufTϕ(u)ϕ (ϕ(u), ei) −1 = P2Tϕ(u)(f ◦ ϕ )(ϕ(u), ei) −1 = D(f ◦ ϕ ) · (ϕ(u), ei)

= D(fϕ)(ϕ(u), ei)

= D(fϕ)(ϕ(u)) · ei ∂f = ϕ (ϕ(u)) ∂xi ∂f  = ϕ ◦ ϕ (u) ∂xi

Therefore, more explicitly,

∂f ∂f (5.14) = ϕ ◦ ϕ ∂xi ∂xi at points u ∈ U. Defining:

 ∂f  ∂x1  .  (5.15) Df =  .  ∂f ∂xn

We may write:

(5.16) Df = D(fϕ) ◦ ϕ or, equivalently:

(5.17) (Df)ϕ = D(fϕ)

i 0 0 i i 0 Remark 5.18. With x defined as above, u = ϕ(u) so (u ) = x (u), and Pi(u ) = (u0)i, the projection map onto the ith coordinate, we have:

i 0 i −1 0 i −1 0 i 0 i 0 xϕ(u ) = (x ◦ ϕ )(u ) = x (ϕ (u )) = x (u) = (u ) = Pi(u )

i 0 Hence: xϕ = Pi on U .

9Our convention will be that ∂ is the usual multivariable calculus operation when acting on ∂xi a function between finite-dimensional, real vector spaces. UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 25

Therefore ∂xi dxi(u)(e (u)) = (u) j ∂xj ! ∂xi = ϕ ◦ ϕ (u) ∂xj ∂P = i (ϕ(u)) ∂xj ∂P = i (u0) ∂xj i = δj i = α (u)(ej(u)) Which shows that (5.19) dxi(u) = αi(u). i ∗ Remark 5.20. The previous remark showed that dx (u) is a basis for Tu (M), since i ∗ we already showed that α (u) is a basis for Tu (M). We now exploit this fact, relying on previous results about tensor bases. ∗ To begin, df(u) ∈ Tu (M), so: ∂f (5.21) df(u) = df(e (u))dxi(u) = (u)dxi(u) i ∂xi That is, ∂f (5.22) df = dxi ∂xi which should be a familiar formula from multivariable calculus. r In just the same way, for each t ∈ Ts (U) we have: (5.23) t = ti1···ir e ⊗ · · · ⊗ e ⊗ dxj1 ⊗ · · · ⊗ dxjs j1···js i1 ir and for each ω ∈ Ωk(U) we have:

X i1 ik (5.24) ω = ωi1···ik dx ∧ · · · ∧ dx

i1<···

(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 mappings dk(U):Ωk(U) → Ωk+1(U)(k = 0, 1, 2, . . . , n, and U is open in M), which we denote 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 is that for ω ∈ Ωk(E) with

X i1 ik ω = ωi1···ik dx ∧ · · · ∧ dx

i1<···

X i1 ik dω = d(ωi1···ik ) ∧ dx ∧ · · · ∧ dx

i1<···

(5.30) iX ω(X1,...,Xk) = ω(X,X1,...,Xk) 0 By convention, if ω ∈ Ω (M), we have iX ω = 0. We call iX the inner product of X and ω. The interior product has many magical properties, but we shall only need a few of them. Proposition 5.31. For X ∈ X (M), α ∈ Ωk(M), and β ∈ Ωl(M), we have:

(1) iX is an antiderivation. (2) iX df = LX f

Proof. This is a simple matter of computation.  5.2. Symplectic Geometry. The theory of differential forms is worth study in its 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, we recall some basic facts about exterior 2-forms. Proposition 5.34. Suppose E is a finite-dimensional, real vector space and sup- 2 pose that ω ∈ Ω (E). Then, there exists a basis eˆ = (e1, . . . , en) with dual basis eˆ∗ = (α1, . . . , αn) such that:

k X (5.35) ω = αi ∧ αi+k i=1 UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 27

i j 0 Take {α ⊗ α | i, j ∈ {1, . . . , n} to be the basis for T2 (E). Then, in this basis, ω has the following matrix representation(ωij = ω(ei, ej)):   0 Ik 0 (5.36) ω = −Ik 0 0 0 0 0 If, in particular, ω is nondegenerate, then:  0 I  (5.37) ω = n/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 to Darboux.: Theorem 5.38. (Darboux) If ω is a closed, nondegenerate 2-form on a 2n-manifold M, 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: n X (5.40) ω|U = dxi ∧ dyi i=1 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 find some 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 a r given C function. The vector field XH determined by the condition:

(5.43) iXH ω = 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 such an XH . This is because nondegeneracy guarantees that the map f : X (M) → ∗ X (M); X 7→ iX ω is a linear isomorphism. We will now see that integral curves on a Hamiltonian system obey Hamilton’s equations.

1 n Proposition 5.45. Let (q , . . . , q , p1, . . . , pn) be canonical coordinates for ω (the P i coordinate names here are meant to be suggestive), so ω = i dq ∧ dpi. Then, in these coordinates (and neglecting base points): ∂H ∂H  (5.46) XH = , − i = J · dH ∂pi ∂q 28 CHARLES HUDGINS

 0 I where J = . Thus (q(t), p(t)) is an integral curve of X iff Hamilton’s −I 0 H equations hold:

i ∂H i ∂H (5.47)q ˙ = , p˙ = − i , i = 1, . . . , n ∂pi ∂q

 ∂H ∂H  Proof. If we can show that XH = , − i is a Hamiltonian vector field, then, ∂pi ∂q by uniqueness, we will have shown that it is the Hamiltonian vector field given energy function H. We have, by the definition of dpi and dqi,

i ∂H iXH dq = ∂pi ∂H i dp = XH i ∂qi Hence: X i X i X i iXH ω = iXH (dq ∧ dpi) = (iXH dq ) ∧ dpi − dq ∧ iXH dpi

X ∂H ∂H i = dpi + i dq = dH ∂pi ∂q  Remark 5.48. The canonical coordinates furnished by Darboux’s theorem are the precise analog of the canonical coordinates from Hamiltonian mechanics. From purely 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 of manifolds. 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 Hamiltonian vector 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’s theorem. The star of our particular formulation of the theorem10 will be an object known as the momentum map. Much of the work in this section will be devoted to understanding 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 group operations G × G → G;(g, h) 7→ gh and G → G; g 7→ g−1 and which contains an identity element e ∈ G. The Lie Algebra g corresponding to the Lie group G is defined as the fiber over e 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 and some neither imply nor are implied by others. UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 29

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

Definition 6.3. For every ξ ∈ g = TeG, let ϕξ : R → G; t 7→ exptξ denote the integral 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 symplectic manifold 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 a ∗ diffeomorphism where Φgω = ω, where, by definition: ∗ (6.5) (Φgω)m(X1,X2) = ωΦg (m)(dΦg(X1), dΦg(X2)) We say that a mapping: ∗ ∗ (6.6) J : M → g = Te G is a momentum mapping for the action provided that for every ξ ∈ g = TeG: ˆ (6.7) dJ(ξ) = iξM ω or, equivalently: d (6.8) X = ξ ≡ (ϕ (t), x)| Jˆ(ξ) M dt ξ t=0 ˆ ˆ where J(ξ): M → R is defined by J(ξ)(x) = J(x) · ξ, and ξM is called the infinitesimal generator of the action corresponding to ξ. We will also need to generalize the Poisson bracket to our current context in order to prove Noether’s theorem. This should come as no surprise since it was constructed to sniff out conservation laws in our first formulation of Hamiltonian mechanics.

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

(6.10) {f, g} = −iXf iXg ω Remark 6.11. To see why this is the natural generalization of the Poisson bracket, observe:

(6.12) {f, g} = −iXf iXg ω = −iXf dg = −dg · Xf = −LXf g

That 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 LXH f = 0 ⇐⇒ {f, H} = 0. Thus f is constant along flows governed by Hamilton’s equations if and only if the Poisson 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 constant on flows of Xf iff {f, g} = 0.

Proof. Suppose Ft is the flow of Xf , then: d d (6.14) (g ◦ F ) = (F ∗g) = F ∗L g = −F ∗{f, g} dt t dt t t Xf t ∗ The result follows from the fact that Ft {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), since H is invariant. We differentiate this expression with respect to time at t = 0 to obtain:

(6.18) dH(x) · ξM = 0 So, by the definition of L:

(6.19) LξM H = 0 which, by Equation 6.8 is equivalent to : (6.20) L H = 0 XJˆ(ξ) Therefore: (6.21) {H, Jˆ(ξ)} = 0 So, by the lemma: d (6.22) (Jˆ(ξ) ◦ F ) = 0 dt t Since Ft is a flow, F0 = idM . Hence, by the above statement:

(6.23) Jˆ(ξ) ◦ Ft = Jˆ(ξ) ◦ F0 = Jˆ(ξ) Hence:

(6.24) J(Ft(x)) · ξ = Jˆ(ξ)(Ft(x)) = (Jˆ(ξ) ◦ Ft)(x) = Jˆ(ξ)(x) = J(x) · ξ Therefore:

(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 geometry of the system and and conservation laws by way of the Poisson bracket. We then staged our physical system as a symplectic manifold equipped with a Hamiltonian vector field, and found that this connection is an inevitable consequence of thinking of points on a manifold as states and vector fields as laws governing how those states may evolve. A clever choice of map, which we called the momentum map, produced for us the conserved quantity of Noether’s theorem. Although, as of now, it is merely a clever map. But, one may develop the theory further and see examples of how this map arises in situ and even write it down quite explicitly if the state space is a pseudo-Riemannian manifold. In this context, the momentum map is no longer an UNDERSTANDING NOETHER’S THEOREM WITH SYMPLECTIC GEOMETRY 31 abstruse contrivance, but rather something quite natural and, perhaps, even more fundamental 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 were indispensable in learning the mathematical formalism at work in this paper. I would also like to thank Peter May for providing me with a wonderful REU experience which 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