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The Hamiltonian Formulation of Geodesics U.U.D.M. Project Report 2021:36 The Hamiltonian formulation of geodesics 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 geodesic equation. We start with the definition of a differentiable manifold. Then we continue with studying tensors and differential forms. We define a Riemannian manifold (M; g) and geodesics, the shortest path between two points, on such manifolds. Lastly, we define the symplectic manifold (T ∗M; d(pdq)), and study the connection between geodesics and the flow of Hamilton's equations. 2 Contents 1 Introduction 4 2 Background 5 2.1 Calculus 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 Integral and exterior derivative 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 bundle. A manifold is a topological space which locally looks like Euclidean space. The study of Riemannian manifolds is called Riemannian geometry, the study of symplectic manifolds is called symplectic geometry. Riemannian geometry was first studied by Bernhard Riemann 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 classical mechanics, 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 tangent space and tangent bundle 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 vector bundle 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 cotangent bundle. 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 torus 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 derivatives 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 fAi : i 2 Ig be a family of sets indexed by I. The disjoint union of this family is the set G [ Ai = f(i; x): x 2 Aig: i2I i2I F If Ai = A for all i 2 I, then i2I 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 norm 0 jfjC1 = sup jf(x)j + sup jf (x)j: x2U x2U 2 In euclidean plane R , let γ = f(t; x): x = x(t); t0 ≤ t ≤ t1g be a curve. 1 Consider the approximation γh = f(t; x): x = x(t) + h(t); h 2 C (U)g 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(jhjC1 ). 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 2 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 = f(t; x): x = x(t) 2 C (R); t0 ≤ t ≤ t1g 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 dimensions. In euclidean space Rn, let @ @ @ x = (x1; : : : ; xn) and = ;:::; . @x @x1 @xn Theorem 2.3. The curve γ = f(t; x): x = x(t); t0 ≤ t ≤ t1g 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 coordinate system 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 Legendre transformation. 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).
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