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.
Best Answer
My reading of the question is this: we're given $H\in C^\infty(M)$ with $M$ symplectic, and we want to know whether there's a submanifold $L\subset M$, a Riemannian metric $g$ on $L$, and a symplectomorphism $T^\ast L \cong M$ under which $H$ pulls back to the norm-square function. And we want to know if $(L,g)$ is unique.
Uniqueness is easy: we recover $L$ as $H^{-1}(0)$, and $g$ as the Hessian form of $H$ on the vertical tangent bundle (determined by the symplectomorphism) along $L$.
Basic necessary conditions:
(1) $L:=H^{-1}(0)$ is a Lagrangian submanifold of $M$.
(2) $L$ is a non-degenerate critical manifold of $H$ of normal Morse index 0.
These conditions imply that a neighbourhood of $L$ embeds symplectically in $T^\ast L$, and also (by the Morse-Bott lemma) that $H$ is quadratic in suitable coordinates near $L$. These two sets of coordinates needn't be compatible, so let's replace (2) by something much stronger (but still intrinsic):
(3) There's a complete, conformally symplectic vector field $X$ (i.e., $\mathcal{L}_X\omega=\omega$), whose zero-set is exactly $L$, along which $H$ increases quadratically (i.e., $dH(X)=2H$).
I claim that (1) and (3) are sufficient. With these data, you can locate a point $x\in M$ in $T^\ast L$. Flow $X$ backwards in time starting at $x$ to obtain the projection to $L$; pay attention to the direction of approach to $L$ to get a tangent ray, and use the metric (i.e., the Hessian of $H$ on the fibres of projection to $L$) to convert it to a cotangent ray. Pick out a cotangent vector in this ray by examining $H(x)$. If I'm not mistaken, this will single out a symplectomorphism with the desired properties.