When is the time evolution of a Hamiltonian system described by the geodesic flow on a Riemannian manifold?

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.

I wish to add an $\epsilon$ to the previous answer. Assume that your symplectic manifold is $T^*M$, where $M$ is a closed manifold, and consider a so called ``Tonelli Hamiltonian´´ $H:T^*M\rightarrow\mathbb R$, which is simply a function which is fiberwise (differentiably) convex and superlinear. Then consider the value $$ c(H)=\min_{u\in C^\infty(M;\mathbb R)} \max_{q\in M} H(q,du_q). $$

This special number in the literature is called ``Mañé critical value´´. Now, if you fix an energy value $h > C(H)$, you can easily build a new Hamiltonian $G$ such that

1) $H^{-1}(h)=G^{-1}(h')$ for some $h'\in\mathbb R$

2) $G(q,\lambda p)=\lambda^2 G(q,p)$ for each $\lambda>0$

3) $G$ is fiberwise convex

Now, the Legendre dual of the Hamiltonian $G$ will be a Finsler metric (not Riemannian in general).

The interesting thing here is that the critical value $c(H)$ can be defined in many other equivalent ways (in terms of Lagrangian action of closed loops, in terms of minimizing invariant measures, etc.). You can find more on this in the book by Contreras and Iturriaga: http://www.cimat.mx/~gonzalo/libro/lagrangians.pdf

This is a kinda stupid answer but it is the simplest most useful test I know. Does your Hamiltonian vector field have a zero? If `yes': sorry, not a geodesic flow!

(Could be the reduction of a geodesic flow though.)