Geodesic flows: notes 1

Geodesic flows, notes 1: symplectic geometry

Our first topic is to understand geodesics, first defined as curves \(\gamma(t)\), in terms of a Hamilton flow in the cotangent bundle. Hamilton flows belong to the realm of symplectic geometry. This is an area of mathematics which comes up in a variety of contexts, such as:

1. Historically, symplectic geometry arose from the study of classical mechanics. Consider the \(N\)-body problem, where \(N\) celestial objects (planets, stars etc) interact with each other gravitationally. If the objects have mass \(m_i\), position \(q_i(t) \in \mathbb{R}^3\), and momentum (or velocity) \(p_i(t) = \dot{q}_i(t) \in \mathbb{R}^3\), by Newton's laws one has the equations \( m_i \ddot{q}_i(t) = \sum_{j \neq i} \frac{G m_i m_j}{|q_i - q_j|^2}\) where \(G\) is the gravitational constant. These can be rewritten as the Hamilton equations \begin{align*} \dot{q}(t) &= \nabla_p H(q(t),p(t)), \\ \dot{p}(t) &= -\nabla_q H(q(t),p(t)) \end{align*} where \(H\) is the Hamilton function \(H = T + U\), with \(T\) the kinetic energy and \(U\) the self-potential energy (source: Wikipedia). The underlying symplectic structure makes it possible to consider these equations in many different (symplectic) coordinate systems, while still retaining the basic properties.

2. The geodesic flow on a Riemannian manifold \((M,g)\) can be understood as a Hamilton flow on the cotangent space \(T^* M\). This explains the facts that geodesics have constant speed, that geodesic flow preserves volume, and that the geodesic vector field on \(SM\) is divergence free and formally skew-adjoint.

3. If \(A(x,D)\) is a linear differential operator in an open set \(\Omega \subset \mathbb{R}^n\), so \(A(x,D) u = \sum_{|\alpha| \leq m} a_{\alpha}(x) D^{\alpha} u\) where \(a_{\alpha} \in C^{\infty}(\Omega)\), the principal symbol of \(A(x,D)\) is

\begin{align*} a(x,\xi) = \sum_{|\alpha| = m} a_{\alpha}(x) \xi^{\alpha}, \qquad (x,\xi) \in \Omega \times \mathbb{R}^n. \end{align*}

It turns out that under a change of coordinates, the principal symbol of \(A\) transforms like an invariant function on the cotangent bundle \(T^* \Omega\). Moreover, if \(A\) and \(B\) are two differential operators, their commutator \([A,B] = AB - BA\) has principal symbol \(\{a,b\}\) where \(\{\,\cdot\,, \,\cdot\,\}\) is the Poisson bracket on \(T^* \Omega\): \begin{align*} \{ a, b \}(x,\xi) = \nabla_{\xi} a \cdot \nabla_x b - \nabla_x a \cdot \nabla_{\xi} b. \end{align*} These basic facts indicate that symplectic geometry has a fundamental role in the study of linear partial differential equations.

4. The billiard map in the context of the broken geodesic ray transform preserves the related symplectic structure and hence volumes.

Symplectic manifolds

We proceed to explain some basic ideas in the setting of a general \(2n\)-dimensional symplectic manifold.

Definition. A symplectic manifold is a pair \((N, \sigma)\) where \(N\) is an \(2n\)-dimensional smooth manifold, and \(\sigma\) is symplectic form, that is, a closed \(2\)-form on \(N\) which is nondegenerate in the sense that for any \(\rho \in N\), the map \(I_{\rho}: T_{\rho} N \to T^*_{\rho} N, I_{\rho}(s) = \sigma(s, \,\cdot\,)\) is bijective.

Example 1. The space \(\mathbb{R}^{2n}\) has a standard symplectic structure given by the \(2\)-form \(\sigma = dx_1 \wedge \,dx_{n+1} + \ldots + dx_n \wedge \,dx_{2n}\).

Example 2. More generally, if \(M\) is an \(n\)-dimensional \(C^{\infty}\) manifold, then \(N = T^* M\) becomes a symplectic manifold as follows: if \(\pi: T^* M \to M\) is the natural projection, there is a \(1\)-form \(\lambda\) on \(N\) (called the Liouville form) defined by \(\lambda_{\rho} = \pi^* \rho, \rho \in T^* M\). Then \(\sigma = d\lambda\) is a closed \(2\)-form. If \(x\) are local coordinates on \(M\), and if \((x,\xi)\) are associated local coordinates (called canonical coordinates) on \(T^* M\), then in these local coordinates \begin{align*} \lambda &= \xi_j \,dx^j, \\ \sigma &= d\xi_j \wedge \,dx^j. \end{align*} It follows that \(\sigma\) is nondegenerate and hence a symplectic form.

It is a theorem of Darboux that if \((N,\sigma)\) is a symplectic manifold, then near any point of \(N\) there are local coordinates \((x,\xi)\) so that \(\sigma = d\xi_j \wedge \,dx^j\) in these coordinates. Hence every symplectic manifold is locally symplectically equivalent to \(\mathbb{R}^{2n}\), and all obstructions to having a symplectic structure are global in nature: for instance if \(N\) is a closed manifold having a symplectic form \(\sigma\), then the de Rham cohomology groups \(H^2(N)\) and \(H^{2n}(N)\) are nontrivial (this follows since the \(n\)-fold wedge product of \(\sigma\) is non vanishing, more about this later).

Hamilton flows

Definition. Let \((N,\sigma)\) be a symplectic manifold. Given any function \(f \in C^{\infty}(N)\), the Hamilton vector field of \(f\) is the vector field \(H_f\) on \(N\) defined by \[ H_f = I^{-1}(df) \] where \(df\) is the exterior derivative of \(f\) (a \(1\)-form on \(N\)), and \(I\) is the isomorphism \(TN \to T^* N\) given by the nondegenerate \(2\)-form \(\sigma\).

Example 1. Let \(M\) be an \(n\)-manifold, and let \(\sigma\) be the standard symplectic form on \(T^* M\). If \((x,\xi)\) are canonical local coordinates, then (using that \(\alpha \wedge \beta(v, w) = \alpha(v) \beta(w) - \alpha(w) \beta(v)\) for \(1\)-forms \(\alpha, \beta\)) \[ I_{x,\xi}(s^j \partial_{x_j} + t^j \partial_{\xi_j})(\tilde{s}^j \partial_{x_j} + \tilde{t}^j \partial_{\xi_j}) = (d\xi_i \wedge \,dx^i)(s^j \partial_{x_j} + t^j \partial_{\xi_j}, \tilde{s}^k \partial_{x_k} + \tilde{t}^k \partial_{\xi_k}) = \sum_{i=1}^n (t^i \tilde{s}^i - s^i \tilde{t}^i) \] which gives that \begin{align*} I(s^j \partial_{x_j} + t^j \partial_{\xi_j}) &= t^j \,dx_j - s^j \,d\xi_j \\ H_f &= \partial_{\xi_j} f \partial_{x_j} - \partial_{x_j} f \partial_{\xi_j}. \end{align*}

Example 2. In particular, in \( \mathbb{R}^{2n} \) one has \(I(s,t)=(t,-s)\), \(s, t \in \mathbb{R}^n\), and \[ H_f = \nabla_{\xi} f \cdot \nabla_x - \nabla_x f \cdot \nabla_{\xi}. \]

Definition. Let \((N,\sigma)\) be a symplectic manifold, and let \(f \in C^{\infty}(N)\). Denote by \(\varphi_t\) the flow on \(N\) induced by \(H_f\), that is, \[ \varphi_t: \rho(0) \mapsto \rho(t) \text{ where } \dot{\rho}(t) = H_f(\rho(t)). \]

We will later see that any Hamilton flow map is symplectic (\((\varphi_t)^* \sigma = \sigma\)) and consequently volume-preserving.