TITLE: When is the time evolution of a Hamiltonian system described by the geodesic flow on a Riemannian manifold?
QUESTION [21 upvotes]: Here is my precise question.  Let $M, \omega$ be a symplectic manifold and let $H: M \to \mathbb{R}$ be any smooth function.  The symplectic form gives rise to an isomorphism between the tangent bundle and cotangent bundle of $M$, and in this way we can associate to the 1-form $dH$ a vector field $X_H$ which is characterized by the property that $\omega(X_H, Y) = Y(H)$ for any vector field $Y$.  The one parameter group of diffeomorphisms associated to $X_H$ is the "Hamiltonian flow" associated to $H$.
An interesting special case of this construction is furnished by Riemannian geometry.  For any manifold $M$, there is a canonical symplectic structure on $T^*M$ (regarded as a manifold in its own right) defined as follows.  Given a tangent vector $X \in T(T^*M)$ sitting over a covector $p \in T^*M$, define $\eta_p(X) = p(d\pi_p(X))$ where $\pi: T^*M \to M$ is the natural bundle projection.  Then $\eta$ is a 1-form on $T^*M$, and one checks that $d\eta$ is a symplectic form.  If $M$ is equipped with a Riemannian metric $g$ then the metric yields an isomorphism between $TM$ and $T^*M$, and the construction of the previous paragraph produces a Hamiltonian flow associated to any smooth function on $TM$.  If we consider the smooth function $H: TM \to \mathbb{R}$ given by $H(V) = g(V, V)$, then it is a fact that the resulting Hamiltonial flow $F_t$ is precisely the geodesic flow for $M$.  In other words, given a tangent vector $W \in T_p M$, $F_t(W)$ is the velocity vector at time $t$ of the unique geodesic $\gamma$ with $\gamma(0) = p$, $\gamma'(0) = W$.
So I am wondering if there are interesting invariants - dynamical, geometric, topological, or otherwise - which help to determine whether or not a given Hamiltonian system is secretly the geodesic flow on some Riemannian manifold.  This is kind of a screwy question from a geometric point of view, because it essentially asks if given a smooth function $H: M \to \mathbb{R}$ on a symplectic manifold there is a submanifold $N$ of $M$ such that there is a diffeomorphism $M \to TN$ which carries $H$ to a positive definite quadratic form on each fiber.  But dynamically it boils down to a fairly natural question: how can one characterize geodesic flows among all Hamiltonian dynamical systems?
If this question has any sort of reasonable answer, I can think of half a dozen follow-up questions.  Is there a natural notion of equivalence up to which $N$ is unique? To what extent does $H$ constrain the geometry and topology of $N$?  If a Lie group acts on the pair $M, H$ can we choose $N$ which is invariant under the group action?  For example, one idea along these lines that comes to mind immediately is the assertion that if the Hamiltonian flow for $H$ is not ergodic relative to a prescribed smooth invariant measure then $N$ cannot have nonpositive curvature.  If you have an answer to this question and you can elaborate on the relationship between $H$ and the geometry of $N$, please do so.

REPLY [2 votes]: There is the paper ARE HAMILTONIAN FLOWS GEODESIC FLOWS?
by CHRISTOPHER MCCORD, KENNETH R. MEYER, AND DANIEL OFFIN which treats this problem and gives examples of Hamiltonian flows (most cases of the planar $N$-body problem, for instance) that cannot be seen as geodesic flows.