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Home , Differentiable manifold, Hamiltonian vector field

U.U.D.M. Project Report 2021:36

The Hamiltonian formulation of

Victor Hildebrandsson

Examensarbete i matematik, 15 hp Handledare: Georgios Dimitroglou Rizell Examinator: Martin Herschend Juli 2021

Department of Mathematics Uppsala University

Abstract We explore the Hamiltonian formulation of the equation. We start with the definition of a differentiable . Then we continue with studying and differential forms. We define a (M, g) and geodesics, the shortest path between two points, on such . Lastly, we define the (T ∗M, d(pdq)), and study the between geodesics and the flow of Hamilton’s equations.

2 Contents

1 Introduction 4

2 Background 5 2.1 of variations ...... 5 2.2 Differentiable manifolds ...... 8 2.3 Tensors and vector bundles ...... 10

3 Differential forms 13 3.1 Exterior and differential forms ...... 13 3.2 and exterior of differential forms ...... 14

4 Riemannian manifolds 17

5 Symplectic manifolds 20

References 25

3 1 Introduction

In this thesis, we will study Riemannian manifolds and symplectic manifolds. More specifically, we will study geodesics, and their connection to the Hamilto- nian flow on the symplectic cotangent . A manifold is a topological which locally looks like . The study of Riemannian manifolds is called Riemannian , the study of symplectic manifolds is called . was first studied by in his habil- itation address ”Uber¨ die Hypothesen, welche der Geometrie zugrunde liegen” (”On the Hypotheses on which Geometry is Based”). In particular, it was the first time a mathematician discussed the concept of a differentiable manifold [1]. Symplectic geometry has its roots in Hamiltonian formulation of , where phase spaces take the form of symplectic manifolds. We will begin the thesis with discussing the calculus of variations, where we will get an understanding of concepts used in classical mechanics. Then we continue to define differentiable manifolds. We also define the and of a differentiable manifold. Moving on, we will discuss tensors, vector bundles, and differential forms. Tensors and differential forms are important tools when studying differential geometry, and the can be considered as a family of vector spaces parametrized by a manifold. We then define a Riemannian manifold and derive the geodesic equation on such manifolds. Lastly, we define a symplectic manifold and derive a natural symplectic structure for the . On the cotangent bundle we can define a vector field, the Hamiltonian vector field, and also define the cogeodesic flow: Theorem. Let (M, g) be a Riemannian manifold. Let (q, p) be local coordi- ∗ 1 ij nates of T M and H(q, p) = 2 g (q)pipj be our Hamiltonian. Then the geodesic equation is equivalent to the cogeodesic flow on T ∗M, given by:

i ∂H ij q˙ = = g pj, ∂pi ∂H 1 p˙ = − = − gjkp p . i ∂qi 2 ,i j k

Example. On the flat Tn = (S1)n we have the cotangent bundle

∗ n 1 n n T T = (S ) × R .

∗ n The local coordinates on T T are (θ1, . . . , θn, p1, . . . , pn), where (θ1, . . . , θn) are 1 n standard angular coordinates of (S ) , and (p1, . . . , pn) the standard coordinates of Rn. The cogeodesic flow is then given by ˙i θ = pi,

p˙i = 0.

4 2 Background

In this chapter, we will explain some concepts that will be crucial for our un- derstanding of the content in the later chapters. Each concept will have a separate subsection. However, before we can do that, we need to refresh (and in some cases, introduce) definitions, notations, and terminology that will be used throughout this thesis.

With h·, ·i we will, if not otherwise stated, mean the Euclidean inner product. If we say that a function f is smooth, we mean that the function is continuous and all its partial exists and are continuous. Definition 2.1 (Einstein summation convention). If a free index appears twice in a single term, once elevated, and once lowered, we sum over the values of that index. The easiest way to understand Einstein summation is with an example:

Example 2.1. Let a = (a1, . . . , an) and b = (b1, . . . , bn). Let n X y = aibi. i=1 With Einstein summation, we instead notate the vector b = (b1, . . . , bn), and then we write i y = aib . Remark. In some instances below, we will use Einstein summation over partial ∂ derivatives ∂xi . Then, an elevated index in the ”demoninator” is understood as a lowered index.

Definition 2.2 (Disjoint union). Let {Ai : i ∈ I} be a family of sets indexed by I. The disjoint union of this family is the G [ Ai = {(i, x): x ∈ Ai}. i∈I i∈I F If Ai = A for all i ∈ I, then i∈I Ai = I × A.

2.1 Calculus of variations In the calculus of variations we consider the extremals of functionals, contin- uous functions from the space C1(U) to the real numbers. Here U ⊂ Rn is compact, and C1 has

0 |f|C1 = sup |f(x)| + sup |f (x)|. x∈U x∈U

2 In euclidean plane R , let γ = {(t, x): x = x(t), t0 ≤ t ≤ t1} be a . 1 Consider the approximation γh = {(t, x): x = x(t) + h(t), h ∈ C (U)} of γ. Let Φ be a functional, consider the increment Φ(γh) − Φ(γ).

5 Definition 2.3 (Differentiable). A functional Φ is differentiable if

Φ(γh) − Φ(γ) = F (h) + R(γ, h),

1 where F : C (U) −→ R is linear and continuous, and R(γ, h) = O(|h|C1 ). F (h) is then called the differential.

Theorem 2.1. If the function L : R × R × R −→ R is differentiable, then the functional Φ(γ) = R t1 L(x(t), x˙(t), t)dt is differentiable, with differential t0

Z t1     t1 ∂L d ∂L ∂L F (h) = − hdt + h ∂x dt ∂x˙ ∂x˙ t0 t0 at any x ∈ C2(U) ⊆ C1(U). Proof. See p. 56-57 in [2]. Definition 2.4 (Extremal). An extremal of a differentiable functional Φ(γ) is a curve γ such that F (h) = 0 for all h that vanish along the boundary. Remark. Compare the definition of extremals to the definition of extreme points of a function f(x).

2 Theorem 2.2. The curve γ = {(t, x): x = x(t) ∈ C (R), t0 ≤ t ≤ t1} is an extremal of the functional Φ(γ) = R t1 L(x(t), x˙(t), t)dt if and only if t0 ∂L d ∂L − = 0 (1) ∂x dt ∂x˙ along the curve x. Proof. See p. 57-58 in [2]. Equation (1) is called the Euler-Lagrange equation, or Lagrange’s equa- tion. It can be generalized to higher . In euclidean space Rn, let ∂ ∂ ∂  x = (x1, . . . , xn) and = ,..., . ∂x ∂x1 ∂xn

Theorem 2.3. The curve γ = {(t, x): x = x(t), t0 ≤ t ≤ t1} is an extremal of the functional Φ(γ) = R t1 L(x(t), x˙(t), t)dt if and only if t0 ∂L d ∂L − = 0, i = 1, . . . , n ∂xi dt ∂x˙ i along the curve x. The condition of γ being an extremal is independent of n on R . Therefore we will use generalized coordinates q = (q1, . . . , qn). From the Euler-Lagrange equations, we get n second-order differential equa- tions. We would like to reduce them to first-order equations instead. To do this, we first need to define the .

6 Definition 2.5 (Legendre transformation). Let f : Rn −→ R be a convex T function, i.e. x Hf x is positive definite, where Hf is the hessian matrix of f. The Legendre transformation is the function

g(p) = hp, x(p)i − f(x(p)) = maxx[hp, xi − f(x)] ∂f where p = ∂x . Remark. The Legendre transformation is involutive, i.e. if the Legendre trans- form takes f to g, then the Legendre transform takes g to f.

n 2 2 Example 2.2. Let f : R −→ R,(x1, . . . , xn) 7−→ x1 + . . . xn be our function. T Then Hf = 2I, so x Hf x is positive definite, and f is convex. Then ∂f p p  p(x) = = (2x ,..., 2x ) ⇐⇒ x(p) = 1 ,..., n ∂x 1 n 2 2 and g(p) = hp, x(p)i − f(x(p)) p2 p2 p2 p2  = 1 + ··· + n − 1 + ··· + n 2 2 4 4 p2 p2 = 1 + ··· + n 4 4 is the Legendre transformation of f.

Given a Lagrangian L : Rn × Rn × R −→ R, assumed to be convex with ∂L ∂L respect toq ˙, consider the equationsp ˙ = ∂q , where p = ∂q˙ . Theorem 2.4. The system of n second-order Euler-Lagrange equations is equiv- alent to the system of 2n first-order equations ∂H p˙ = − , i ∂q i (2) ∂H q˙i = ∂pi where H(p, q, t) = hp, q˙(p)i − L(q, q˙(p), t) is the Legendre transformation of the Lagrangian viewed as a function of q˙. Proof. By definition, the Legendre transform of L(q, q,˙ t), with respect toq ˙, is ∂L the function H(p, q, t) = hp, q˙i − L(q, q,˙ t), whereq ˙ =q ˙(p) and p = ∂q˙ . The total differential of H = H(p, q, t) ∂H ∂H ∂H dH = dp + dq + ∂p ∂q ∂t is equal to the total differential of H = hp, q˙i − L(q, q,˙ t) ∂L ∂L dH =qdp ˙ − dq − . ∂q ∂t

7 The expressions of dH are equal if and only if ∂H =q, ˙ ∂p ∂H ∂L = − , ∂q ∂q ∂H ∂L = − ∂t ∂t

∂L Plugging inp ˙ = ∂q , we get the equations (2). The equations (2) are called Hamilton’s equations. Later in the thesis, we will come back to these equations as they are strongly linked to the study of symplectic manifolds.

2.2 Differentiable manifolds A manifold is a which locally looks like Euclidean space. To study these manifolds, we often need to use calculus. Then we need our mani- folds to be smooth, or differentiable [3]. Definition 2.6 (). Let M be a second-countable Hausdorff space such that for every point p ∈ M there is a neighbourhood U of p that is homeomorphic to an open set V of Rn. Then M is a n-dimensional mani- fold. Such , φ : U −→ V , is called a (coordinate) chart.

Definition 2.7 (Differentiable manifold). An is a family {Uα, φα} of S charts such that Uα is a covering of M. A chart is compatible with an atlas if their union is again an atlas. An atlas is called maximal if any chart compatible with it is already contained in it. An atlas is called differentiable if all chart transitions −1 φβ ◦ φα : φα(Uα ∩ Uβ) −→ φβ(Uα ∩ Uβ) are smooth. A maximal differentiable atlas is called a differentiable struc- ture, and a differentiable manifold of n is a n-dimensional manifold with a differantiable structure. At a point x in a manifold M we have a tangent space, which is a real . Intuitively, the tangent space is the span of the directional derivatives at the point x. We can ”glue together” all tangent spaces of a manifold to get its tangent bundle. The tangent bundle is itself a manifold of twice the dimension of the original manifold.

We start by defining the tangent space and tangent bundle of an open set of Rn. Let x1, . . . , xn be euclidean coordinates of Rn,U ⊆ Rn open, p ∈ U. The  ∂ ∂ tangent space of U at p is TpU := {p} × E, where E = span ,..., . ∂x1 ∂xn Here, ∂ is the i-th at p. ∂xi

8 Let U ⊆ Rn,U 0 ⊆ Rm be open, f : U −→ U 0 be differentiable, and p ∈ U. Define the derivative dfp as

0 dfp : TpU −→ Tf(p)U , j i ∂ i ∂f ∂ v i 7−→ v i (p) . ∂x ∂x ∂fj

S ∼ n n n Let TU := p∈U TpU = U × E = U × R . Hence, TU ⊆ R × R is open, and in particular a differentiable manifold. Let π : TU −→ U be the projection onto the first factor. The triple (T U, π, U) is the tangent bundle of U. Similar to the tangent space, we define

df : TU −→ TU 0    j  i ∂ i ∂f ∂ x, v i 7−→ f(x), v i (x) ∂x ∂x ∂fj Now we can define the tangent space and the tangent bundle of a general differentiable manifold M. Definition 2.8 (Tangent space). Let p ∈ M. On

{(φ, v): φ : U −→ V chart with p ∈ U, v ∈ Tφ(p)V } define the equivalence relation (φ, v) ∼ (ψ, w) ⇐⇒ w = d(ψ ◦φ−1)v. The space of equivalence classes is called the tangent space to M at p, denoted TpM.

Given a map f : M −→ N between manifolds M, N , let (UM, φ), (UN , ψ) be charts for p ∈ M and f(p) ∈ N , respectively. We then get the map ˜ f : VM −→ VN f˜ = ψ ◦ f ◦ φ−1

m n between open sets VM = φ(UM) ⊆ R , VN = ψ(UN ) ⊆ R . Represented in local coordinates by the map df˜, df is the unique map such that the following diagram commutes:

df T M T N dφ dψ df˜ TVM TVN

Definition 2.9 (Tangent bundle). T M is the disjoint union of the tangent spaces TpM, p ∈ M. Define the projection π : T M −→ M by π(p, w) = p for w ∈ TpM. (T M, π, M) is called the tangent bundle of M, T M is called the total space of the tangent bundle. Usually, we use the total space T M to denote the tangent bundle.

9 If (Uα, φ) is a chart for an n-dimensional differentiable manifold M we let TU = F T M, and define p∈Uα p

dφ : TUα −→ TV,

F n n where TV = p∈V TpR = V × R , as the charts for T M. Then [ [ G G TUα = TpM = TpM = T M

α α p∈Uα p∈M

−1 −1 and dφα ◦ dφβ = d(φα ◦ φβ ) are smooth. So T M is an 2n-dimensional differ- entiable manifold, in particular, it can be checked that T M is Haussdorf and second countable. Now we can find extremals of functionals on a differentiable manifold M. Consider a smooth function L : T M −→ R, and a map γ : R −→ M.

Theorem 2.5. In local coordinates (q1, . . . , qn) of a point γ(t), γ is an extremal of the functional Φ(γ) = R t1 L(q, q˙)dt, if it satisfies the Euler-Lagrange equation t0 with Lagrangian L(q, q˙), where (q, q˙) are local coordinates of T M.

Proof. It suffices to consider variations supported in some local chart, then use the results from 2.1.

2.3 Tensors and vector bundles A generalization of the tangent bundle is that of a vector bundle. Given an abstract vector bundle, there are induced vector bundles of tensors, so called bundles. Sections of such bundles are called tensor fields. Both differen- tial forms and Riemannian metrics are examples of tensor fields. [4].

Definition 2.10 (Vector bundle). A smooth k-dimensional vector bundle is a triple (E, π, M) of smooth manifolds E (the total space) and M (the base space), and a surjective map π : E −→ M (the projection), satisfying

−1 (a) Each set Ep := π (p) is a vector space. (b) For each p ∈ M, there is a neighbourhood U of p and a diffeomorphism φ : π−1(U) −→ U × Rk such that

π = π1 ◦ φ

where π1 is the projection onto the first factor.

k (c) The restriction of φ to φ : Ep −→ {p} × R is a linear .

Ep is called the fiber of E over p. The diffeomorphism φ is called a local trivialization of E.

10 Proposition 2.6. Let M be a smooth manifold, E a set, and π : E −→ M a surjective map. Suppose we are given an open covering {Uα} of M together with bijective maps −1 k φα : π (Uα) −→ Uα×R satisfying π1◦φ = π, such that whenever Uα∩Uβ 6= ∅, the composite map

−1 k k φα ◦ φβ : Uα ∩ Uβ × R −→ Uα ∩ Uβ × R is on the form −1 φα ◦ φβ (p, V ) = (p, τ(p)V ) for some smooth map τ : Uα ∩Uβ −→ GL(k, R). Then E has a unique structure as a smooth k-dimensional vector bundle over M for which the maps φα are local trivializations.

The smooth, GL(k, R) valued maps τ in the proposition are called transi- tion functions for E. In the following example we find the transition functions for T M. Example 2.3. Given an open set U of M with local coordinates x = (x1, . . . , xn), any V ∈ TpM at a point p ∈ U can be written in terms of the i ∂ coordinate basis as V = v ∂xi . Define a bijection

−1 n φ : π (U) −→ U × R V 7−→ (p, v).

Where two sets of coordinates, x andx ˜, overlap, the coordinate bases are related by ∂ ∂x˜j ∂ = , ∂xi ∂xi ∂x˜j and then the same vector V is represented by

∂ ∂ ∂x˜j ∂ V =v ˜j = vi = vi . ∂x˜j ∂xi ∂xi ∂x˜j

j i ∂x˜j Hencev ˜ = v ∂xi and thus

φ˜ ◦ φ−1(p, v) = φ˜(V ) = (p, v˜) = (p, τ(p)v)

∂x˜j  where τ(p) = ∂xi ∈ GL(n, R). Remark. These are explicit coordinates (xi, vi) on π−1(U), called the standard coordinates of the tangent bundle. Definition 2.11 (Section). If (E, π, M) is a vector bundle over M, a section of E is a map F : M −→ E such that π ◦ F = idM . Equivalently we can define the section of E as F (p) ∈ Ep for all p.

11 In order to define the tensor bundle, we first need to present the construction in a vector space. Let V be a finite-dimensional real vector space, and V ∗ its . Definition 2.12 (Tensors). A covariant k-tensor on V is a multilinear map

T : V × · · · × V −→ R. | {z } k times A contravariant l-tensor is a multilinear map ∗ ∗ T : V × · · · × V −→ R. | {z } l times k A tensor of type l is a multilinear map ∗ ∗ T : V × · · · × V × V × · · · × V −→ R. | {z } | {z } l times k times k However, the order of the Cartesian product in a l -tensor can be mixed. The space of covariant tensors on V is denoted by T k(V ), of contravariant tensors k k by Tl(V ), and the space of mixed l -tensors by Tl (V ). k p Definition 2.13 (). If T ∈ Tl (V ) and S ∈ Tq (V ), their tensor k+p product T ⊗ S ∈ Tl+q is defined as

T ⊗ S(ω1, . . . , ωl+q,X1,...,Xk+p)

= T (ω1, . . . , ωl,X1,...,Xk)S(ωl+1, . . . , ωl+q,Xk+1,...,Xk+p) k Definition 2.14 (). A tensor T ∈ Tl (V ) is antisymmet- ric if v T (ωi1 , . . . , ωik ,X1,...,Xl) = (−1) T (ω1, . . . , ωk,X1,...,Xl), or v T (ω1, . . . , ωk,Xi1 ,...,Xil ) = (−1) T (ω1, . . . , ωk,X1,...,Xl), where  0, if the permutation i , . . . , i (or i , . . . , i ) is even v = 1 k 1 l 1, if the permutation i1, . . . , ik (or i1, . . . , il) is odd k The space of antisymmetric tensors is denoted Ωl (V ). Definition 2.15 (Tensor bundle and tensor field). Let M be a differentiable k manifold. We define the bundle of l -tensors on M as k G k Tl M = Tl (TpM). p∈M

k A tensor field on M is a smooth section of some tensor bundle Tl M. The k k space of l -tensor fields on M is denoted Tl (M). The tensor bundle is, as mentioned, a smooth vector bundle, with fibres k Tl (TpM).

12 3 Differential forms

When studying geometry, one useful tool is the exterior form, which in its most basic shape is a . The differential form, an exterior form which varies smoothly on the tangent spaces of a manifold, is especially important in symplectic geometry, as we will see in Chapter 6.

3.1 Exterior and differential forms Definition 3.1 (Bundle of antisymmetric tensors). The bundle of antisymmet- ric k-tensors is defined as

k G k Ω (M) := Ω (TpM), p∈M and is a subbundle of T kM. Definition 3.2 ((Exterior) k-form). A k-form ωk is an antisymmetric k-covariant tensor on the vector space Rn, i.e. ωk ∈ Ωk(Rn). Remark. The set of all k-forms becomes a real vector space if we define by k k k k (ω1 + ω2 )(x1, . . . , xk) = ω1 (x1, . . . , xk) + ω2 (x1, . . . , xk) and multiplication with scalars by

k k (λω )(x1, . . . , xk) = λω (x1, . . . , xk).

Example 3.1. The most familiar k-form is the 1-form, which is a linear function ω : Rn −→ R. The vector space of 1-forms is (Rn)∗, the dual space of Rn.

Definition 3.3 (Exterior product). Suppose we are given k 1-forms ω1, . . . , ωk. We define their exterior product as

ω1(x1) ··· ωk(x1)

. .. . (ω1 ∧ · · · ∧ ωk)(x1, . . . , xk) = . . . .

ω1(xk) ··· ωk(xk)

Definition 3.4 (Exterior multiplication). The exterior multiplication ωk ∧ ωl of a k-form ωk on Rn and a l-form ωl on Rn is the k + l-form on Rn, whose value on k + l Rn-vectors is equal to

k l X v k l (ω ∧ ω )(x1, . . . , xk+l) = (−1) ω (xi1 , . . . , xik )ω (xj1 , . . . , xjl ) where i1 < ··· < ik and j1 < ··· < jl, (i1, . . . , ik, j1, . . . , jl)) is a permutation of (1, . . . , k + l)), and v is the same as in definition 2.14.

13 Remark. Exterior multiplication is

• Skew-commutative: ωk ∧ ωl = (−1)klωl ∧ ωk,

k k l k l k l • Distributive: (λ1ω1 + λ2ω2 ) ∧ ω = λ1ω1 ∧ ω + λ2ω2 ∧ ω , • Associative: (ωk ∧ ωl) ∧ ωm = ωk ∧ (ωl ∧ ωm). Definition 3.5 (Differential k-form). Let M be a differentiable manifold. At a k point x ∈ M a differential k-form ω |x is an exterior k-form on the tangent k space TxM. A differential k-form on the manifold M is a section of Ω (M).

For a point x ∈ M, let x1, . . . , xn be local coordinates. The differential 1-forms dx1, . . . , dxn form a basis of the space of 1-forms on TxM. Also, the n k k-forms

dxi1 ∧ · · · ∧ dxik , i1 < ··· < ik form a basis of the space of k-forms on TxM. Therefore, every differential k-form on M can be written as

k X ω = ai1,...,ik (x)dxi1 ∧ · · · ∧ dxik

i1<···

where ai1,...,ik are smooth functions.

3.2 Integral and of differential forms We want to define an integral of a differential form. We start with the integral of a differential 1-from.

Let M be a differentiable manifold and let γ : [0, 1] −→ M be a smooth map. We define the tangent vectors as follows: Divide 0 ≤ t ≤ 1 into parts ∆i : ti ≤ t ≤ ti+1, by points ti (∆i can be seen as a tangent vector to the t-axis at ti). Then

ξi = dγ|ti (∆i) ∈ Tγ(ti)M is the tangent vector of ∆i at γ(ti). Definition 3.6 (Integral of a differential 1-form). The integral of the 1-form ω on γ is the limit of Riemann sums

n Z X ω = lim ω(ξi). ∆→0 γ i=1 For the integral of a k-form on an arbitrary differentiable manifold, we first need to look at the integral of a k-form on oriented space Rk. k Definition 3.7 (Integral of a differential k-form on R ). Let x1, . . . , xk be an oriented coordinate system on Rk. Then every k-form has the form

k ω = φ(x)dx1 ∧ · · · ∧ dxk

14 where φ(x) is a smooth function on Rk. Let D be a bounded convec polyhedron in Rk. The integral of ωk on D is Z Z k ω = φ(x)dx1 . . . dxk D D where the right integral is the usual Riemann integral. Definition 3.8 (). Let f : M −→ N be a differentiable map on smooth manifolds, and ωk a differential k-form on N . Then

∗ k k (f ω )(ξ1, . . . , ξk) = ω (dfξ1, . . . , dfξk) is a k-form on M, for any tangent vectors ξ1, . . . , ξk ∈ TxM. This k-form is called the pullback of f by ω. Now we can define the integral of a k-form on an arbitrary differentiable manifold. Let ωk be a differentiable k-form on a n-dimensional differentiable manifold M. Let D be a bounded convex k-dimensional polyhedron in Rk.A k-dimensional cell σ of M is a triple σ = (D, f, Or) consisting of

• a convex polyhedron D ⊆ Rk, • a differentiable map f : D −→ M,

• an orientation on Rk, denoted Or. Definition 3.9 (Integral of a differential k-form on an n-dimensional manifold). The integral of the k-form ωk over the k-dimensional cell σ = (D, f, Or) is the integral Z Z ωk = f ∗ωk σ D where f ∗ is the pullback of f. Definition 3.10 (Chain). A k-dimensional chain on M consists of a finite number of cells σ1, . . . , σr and integers m1, . . . , mr ∈ Z called muliplicities.A chain is denoted ck = m1σ1 + ··· + mrσr. Definition 3.11 (Integral of a form over a chain). Let ωk be a k-form on M, P k and ck = miσi a chain on M. The integral of ω over ck is Z Z k X k ω = mi ω . ck σi

We want to define the value of the differential form dωk on k + 1 vectors ξ1, . . . , ξk+1 ∈ TxM. To do this, we choose a coordinate system in a neighbour- hood of x ∈ M, i.e. a differentiable map f of a neighbourhood of 0 ∈ Rn to

15 a neighbourhood of x ∈ M. The preimage of the vectors ξ1, . . . , ξk+1 ∈ TxM n ∼ n under dfx lie in T0R = R , so we consider the preimage to be vectors

∗ ∗ n ξ1 , . . . , ξk+1 ∈ R .

Consider the parallelepiped Π∗ in Rn spanned by these vectors. The map f takes Π∗ to a (k + 1)-cell Π in M. The boundary of Π, ∂Π, is a k-chain. Definition 3.12 (Exterior derivative of a k-form). Let ωk be a differential k- form, and ∂Π the k-chain discussed above. The exterior derivative of ωk at x is the integral Z k k dω (ξ1, . . . , ξk+1) = ω . ∂Π

Proposition 3.1. If, in the local coordinate system x1, . . . , xn on M, the form ωk is written as k X ω = ai1,...,ik (x)dxi1 ∧ · · · ∧ dxik then dωk is written as

k X dω = dai1,...,ik (x) ∧ dxi1 ∧ · · · ∧ dxik .

Proof. See p. 190-191 in [2].

Theorem 3.2 (Stoke’s theorem). Let c be any (k +1)-chain on M, and ωk any k-form on M. Then Z Z ωk = dωk. ∂c c Proof. See p. 192-193 in [2]. Definition 3.13 (Closed form). A differential form ω on a manifold M is closed if dω = 0. Corollary 3.2.1. The integral of a closed form ωk over the boundary of any (k + 1)-chain ck+1 is zero: Z ωk = 0. ∂ck+1

16 4 Riemannian manifolds

Sometimes when studying manifolds, we want to be able to measure lengths and angles between vectors. The Riemannian allows us to do exactly that. Together, a differentiable manifold and a Riemannian metric form a Riemannian manifold. When we can measure distances, we want a curve which optimizes the distance between two points. Such a curve is called a geodesic [1]. Definition 4.1 (Riemannian manifold). A Riemannian metric on a smooth manifold M is a 2-tensor field g ∈ T 2(M) that is symmetric and positive definite: g(X,Y ) = g(Y,X) (symmetric) g(X,X) > 0 if X 6= 0 (positive definite) The pair (M, g) is called a Riemannian manifold. Let x = (x1, . . . , xn) be local coordinates. Then, a metric is represented by a positive definite, symmetric matrix

(gij(x))i,j=1,...,n where the coefficients depend smoothly on x. The product of two vectors v, w ∈ i ∂ i ∂ TpM, where v = v ∂xi , w = w ∂xi , is

i j hv, wi := gij(x(p))v w .

Also, the norm of v is given by ||v|| = phv, vi.

n Example 4.1. Consider R with standard coordinates x1, . . . , xn. We have the metric

n n gp : TpR × TpR −→ R  X ∂ X ∂  X a , b 7−→ a b . i ∂x j ∂x i i i i j j i

(R, g) is clearly a Riemannian metric. Infact, it is usually referred to as Eu- clidean space, and g is the Euclidean metric. Its matrix is the identity matrix In. Theorem 4.1. Each differentiable manifold can be equipped with a Riemannian metric. Proof. The proof uses , defined on p.7 in [1]. For the proof itself, see p. 19 in [1].

Now, let [a, b] ⊂ R be a closed interval, (M, g) a Riemannian manifold, and γ :[a, b] −→ M a smooth curve. The length of γ is defined as

Z b dγ

L(γ) = (t) dt, a dt

17 and the energy of γ as

1 Z b dγ 2

E(γ) = (t) dt. 2 a dt With the coordinates (x1(γ(t)), . . . , xn(γ(t))), we write d x˙ i(t) := (xi(γ(t))). dt Then Z b q i j L(γ) = gij(x(γ(t)))x ˙ (t)x ˙ (t)dt, a and Z b 1 i j E(γ) = gij(x(γ(t)))x ˙ (t)x ˙ (t)dt. 2 a Proposition 4.2. The Euler-Lagrange equations for the energy E are

i i j k x¨ (t) + Γjk(x(t))x ˙ (t)x ˙ (t) = 0, i = 1, . . . , n with 1 Γi = gil(g + g − g ), (3) jk 2 jl,k kl,j jk,l and where ∂ gij = (g )−1, g = g . ij ij,k ∂xk ij Proof. Recall that for a functional

Z b I(x) = f(t, x(t), x˙(t))dt a the Euler-Lagrange equations are given by d ∂f ∂f − = 0, i = 1, . . . , n. dt ∂x˙ i ∂xi In our case, with Z b 1 j k E(γ) = gjk(x(t))x ˙ (t)x ˙ (t)dt 2 a as our functional, we get 1 d 1 g (x(t))x ˙ k(t) + g (x(t))x ˙ j(t) − g (x(t))x ˙ j(t)x ˙ k(t) = 0 2 dt ik ji 2 jk,i for i = 1, . . . , n. Then 1 1 1 1 1 g x¨k + g x¨j + g x˙ lx˙ k(t) + g x˙ lx˙ j − g x˙ jx˙ k = 0. 2 ik 2 ji 2 ik,l 2 ji,l 2 jk,i

18 After renaming i to l, and using that g is symmetric, so gij = gji, we get 1 g x¨m + (g + g − g )x ˙ jx˙ k = 0, l = 1, . . . , n lm 2 lk,j jl,k jk,l and then 1 gilg x¨m + gil(g + g − g )x ˙ jx˙ k = 0, i = 1, . . . , n. lm 2 lk,j jl,k jk,l

il il m i We have that g glm = δim so g glmx¨ =x ¨ . Then we get the desired formula.

i Remark. The expressions Γjk from (3) are called the Christoffel symbols. Definition 4.2 (Geodesic). A smooth curve γ :[a, b] −→ M, which satisfies i d i (with x˙ = dt x (γ(t)) etc.)

i i j k x¨ (t) + Γjk(x(t))x ˙ (t)x ˙ (t) = 0, i = 1, . . . , n (4) is called a geodesic. The equation (4.2) is called the geodesic equation. The intuition behind this definition is that, from classical mechanics, we want a geodesic to follow the principle of least action. This means that the curve should use the least amount of energy to go from a point a to a point b, which is exactly what it means to be an extremal of E.

19 5 Symplectic manifolds

A symplectic structure on a manifold is a closed, nondegenerate differential 2- form. If we equip a manifold with a symplectic structure we get a symplectic manifold. On a symplectic manifold, we can find a vector field which corresponds to the differential of a function. This vector field is called a Hamiltonian vector field [2]. Definition 5.1 (Symplectic structure and symplectic manifold). Let M be an even dimensional, differentiable manifold. A symplectic structure on M is a closed, nondegenerate differential 2-form ω2 on M: dω2 = 0 (closed), 2 ω (ξ, η) = 0 ∀η ∈ TxM =⇒ ξ = 0 (nondegenerate).

The pair (M, ω2) is called a symplectic manifold.

2n 1 n Example 5.1. Let M = R have coordinates (q , . . . , q , p1, . . . , pn), and let 2 i 2n 2 ω = dpi ∧ dq . Then (R , ω ) is a symplectic manifold. Definition 5.2 (). Let M be a differentiable manifold, and x ∈ M. The vector space dual to the tangent space TxM is called the cotangent ∗ space and is denoted by Tx M. Definition 5.3 (Cotangent bundle). The vector bundle over M whose fibers are the cotangent spaces of M is called the cotangent bundle, and is denoted by T ∗M. Remark. Elements of T ∗M are differential 1-forms at some x ∈ M and are called covectors, sections of T ∗M are differential 1-forms on M (recall definition 3.5.) The cotangent bundle of a n-dimensional manifold M is itself a 2n-dimensional manifold. We can define a natural symplectic structure on T ∗M: ∗ Let V ∈ T(q,p)(T M) be a vector tangent to the cotangent bundle at a point (q, p) ∈ T ∗M. Since T ∗M is a vector bundle, it has a projection π : T ∗M −→ M. The derivative dπ : T (T ∗M) −→ T M takes V to a vector dπV tangent to ∗ M at q. Since p ∈ Tq M and dπV ∈ TqM, we can define the 1-form θ1(V ) = (π∗p)(V )

∗ on Tq M. We then define our symplectic form as ω2 = dθ1.

1 n 1 n If we let (q , . . . , q ) be local coordinates of M, and (q , . . . , q , p1, . . . , pn) the induced coordinates T ∗M, we can write the symplectic structure using these local coordinates. From the projection π : T ∗M −→ M we get a pullback π∗ sending 1-forms on M to 1-forms on T ∗M, which in local coordinates is the mapping

i i αidq 7−→ (αi ◦ dπ)dq .

20 We get a linear map on the fibers

∗ ∗ ∗ ∗ π(q,p) : Tq M −→ T(q,p)(T M) ξ 7−→ (ξ, 0). in the splitting of T ∗M induced by the above local coordinates (q, p). We get that 1 ∗ i i θ = π(q,p)p = (p, 0) = pidq + 0dpi = pidq and then it follows that 2 i ω = dpi ∧ dq . This shows Proposition 5.1. The cotangent bundle T ∗M of a differentiable manifold M carries a symplectic structure given by

2 1 i ω = dθ = d(pidq ) which is globally well-defined.

On a symplectic manifold M, we can define an isomorphism between TxM ∗ ∗ and Tx M by mapping a vector V ∈ TxM to a 1-form ηV ∈ Tx M, where

2 ηV (W ) = ω (W, V ).

If we let H : M −→ R be a smooth function, then dH is a differential 1-form on M. For every x ∈ M, dHx is then associated with a vector in TxM. We get the following definition.

Definition 5.4 (Hamiltonian vector field). Let H : M −→ R be a smooth func- tion on the symplectic manifold M. We then have a vector field XH satisfying

2 ω (·,XH (x)) = dHx.

XH is called the Hamiltonian vector field associated to the Hamiltonian function H.

Proposition 5.2. The flow generated by the vector filed XH is defined by the ordinary differential equation x˙ = XH (x), given in local coordinates (q, p) by

i ∂H ∂H q˙ = , p˙i = − i . ∂pi ∂q Proof. Let (q, p) be our local coordinates of T ∗M. Let H = H(q, p) be a ∗ 2 i ∗ Hamiltonian on T M, ω = dpi ∧ dq the symplectic form on T M, and XH = i ∂ ∂ A i + Bi the Hamiltonian vector field. Then ∂q ∂pi   2 i i ∂ ∂ i i ω (·,XH ) = dpi ∧ dq ·,A i + Bi = A dpi − Bidq . ∂q ∂pi

21 Moreover,

∂H i ∂H dH = i dq + dpi ∂q ∂pi

∂H ∂ ∂H ∂ which gives us that XH = i − i . Hence ∂pi ∂q ∂q ∂pi

i ∂H ∂H x˙ = XH (x) ⇐⇒ q˙ = , p˙i = − i . ∂pi ∂q

∗ Similarly to the definition of η : TxM −→ Tx M, if we let M be a Rieman- nian manifold, we can define another isomorphism

∗ αg : T M −→ T M (p, V ) 7−→ (p, g(V, −)).

−1 ij In local coordinates, αg = gij and its inverse αg = g . Definition 5.5 (Cogeodesic orbit). A geodesic γ in M parametrised by constant speed has a unique lift γ˙ to a curve in T M. The corresponding curve in T ∗M

∗ αg ◦ γ˙ : I −→ T M is called the cogeodesic orbit.

In theorem 2.4 we found that, in Rn, Hamilton’s equations were first order differential equations equivalent to the second order Euler-Lagrange equations. In definition 4.2 we said that geodesics are extremals of the energy of a curve. We now want to find Hamilton’s equations equivalent to the geodesic equation on a Riemannian manifold M. This leads us to the main theorem of this thesis. Theorem 5.3. Let (M, g) be a Riemannian manifold. Let (q, p) be local coor- ∗ 1 ij dinates of T M and H(q, p) = 2 g (q)pipj be our Hamiltonian. Then the co- geodesic orbits are equal to the Hamiltonian orbits of H. Recall that the Hamil- tonian orbits are given by the solutions of the ordinary differential equation γ˙ = XH (γ), which in local coordinates are given by ∂H q˙i = = gijp , ∂p j i (5) ∂H 1 p˙ = − = − gjkp p . i ∂qi 2 ,i j k Proof. From the first equation

i ij ij k q¨ = g p˙j + g,kq˙ pj ij ij k l = g p˙j + g,kq˙ gjlq˙

22 and with the second equation, then 1 q¨i = − gijglkp p + gijg q˙kq˙l 2 ,j l k ,k il 1 = gijglmg gnkg q˙rg q˙s + gimg gnjg q˙kq˙l 2 mn,j lr ks mn,k jl

ij im nj using g,l = −g gmn,lg 1 = gijg q˙mq˙n − gimg q˙kq˙n 2 mn,j mn,k 1 = gij(g − g − g )q ˙mq˙n 2 mn,j jn,m jm,n

k n 1 k n 1 k n since gmn,kq˙ q˙ = 2 gmn,kq˙ q˙ + 2 gmk,nq˙ q˙ , and after renumbering some in- dices,

i m n = −Γmnq˙ q˙ . We have seen that γ(t) = q(t) is a geodesic in M parametrized with constant speed. Since p(t) = αg ◦γ˙ , it follows that the cogeodesic orbit is the Hamiltonian orbit as claimed.

Example 5.2. Let the flat torus Tn = (S1)n be our manifold. Its cotangent ∗ n 1 n n bundle is T T = (S ) × R . We get local coordinates (θ1, . . . , θn, p1, . . . , pn), 1 n where (θ1, . . . , θn) are standard angular coordinates of (S ) , and (p1, . . . , pn) the standard coordinates of Rn. We will have a flat metric, which in a point θ ∈ Tn is given by gij(θ) = δij. We get the Hamiltonian 1 X H(θ, p) = p2. 2 i i The coegodesic orbits of Tn are then given by ˙i θ = pi,

p˙i = 0, which gives us that the geodesics are straight lines. Proposition 5.4. The Hamiltonian flow preserves the symplectic structure. Proof. See section 18.1 in [5].

The equation 2 dH(XH ) = ω (XH ,XH ) = 0 implies that along orbits of the cogeodesic flow, the Hamiltonian is constant. ∗ Thus, the cogeodesic flow maps the set Eλ = {(q, p) ∈ T M : H(q, p) = λ}

23 onto itself, for all λ ≥ 0. Eλ is called a level set of constant energy λ. We can define a element on a symplectic manifold (M, ω2) by exterior multiplication on the symplectic structure n times:

(ω2)n := ω2 ∧ · · · ∧ ω2 . | {z } n times We get a nonvanishing 2n-form on M, where dim M = 2n, i.e. a . Corollary 5.4.1. Hamiltonian flow preserves the volume defined by the sym- plectic structure. Theorem 5.5 (Poincar´erecurrence theorem). Let D be a bounded set. Let g be a volume preserving continuous one-to-one mapping such that gD = D. Then, in any neighbourhood U of any point of D, there is a point x ∈ U which returns to U, i.e. gnx ∈ U for some n > 0. From corollary 5.4.1, we have that the Hamiltonian flow preserves volume, and we know that it takes the level set Eλ onto itself. Moreover, if M is compact, then so is Eλ. Hence, for compact manifolds, the theorem applies to Eλ. This means that, while we cannot be sure that a point in Eλ will be mapped exactly to itself after some iteration of the Hamiltonian flow, we know that it may be mapped arbitrarily close to itself.

24 References

[1] J. Jost. Riemannian Geometry and Geometric Analysis (7th Edition). Springer International Publishing AG, Gewerbestrasse 11, 6330 Cham, Switzerland, 2017. [2] V.I. Arnold. Mathematical Methods of Classical Mechanics (2nd Edition). Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA, 1989. [3] J. Lee. Introduction to Smooth Manifolds (Version 3.0). 2000. [4] J. Lee. Riemannian Manifolds: An Introduction to Curvature. Springer- Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA, 1997.

[5] A. Cannas da Silva. Lectures on symplectic geometry. Springer, Berlin New York, 2001. [6] A. Cannas da Silva. Introduction to Symplectic and Hamiltonian Geometry. IMPA, Brazil, 2011.

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