Say I have $M_1$ a 3-dimensional compact Riemannian manifold, and $M=M_1^{n}$ the product manifold representing n particles on $M_1$. I can identify $TM$ with $T^*M$ via the metric $g$ and then the Hamiltonian flow (on $T^*M$ as a symplectic manifold) given by $(1/2)g(v,v)$ is just the geodesic flow. The $6n$-1-form $\theta\wedge d\theta^{3n-1}$ (\theta being the tautological one-form) defines a volume form on the constant energy codimension-one leaves which, from the $TM$ perspective, have a local description as the Riemannian volume form times Euclidean volume on any tangent sphere (since spheres correspond to constant energy leaves). All this is in Part I, Chapter 5 of Katok-Hasselblatt. The following is not.
I now want my $n$ particles to interact in some way, so I add a potential $V(x)$ on $M$ (which might depend only on the $M_1$ distance between the particles). The function $(1/2)g(v,v)+V(x)$ on $TM$ yields a different Hamiltonian flow on $T^*M$, which no longer corresponds to the geodesic flow on $TM$ and whose invariant leaves no longer correspond to spherical tangent subbundles of $TM$. Still the codimension-one contact form defines an invariant measure on these leaves. My question is then: when I project/pushforward this invariant measure on a leaf (say the leaf with energy H=1) to the $3n$-manifold $M$, what do I get? I feel that I should not get the Riemannian volume measure, as it was the case for the geodesic flow, since configurations with high potential should have lower frequency/probability (at least that's how stat mech-oriented people working with probabilistic models tend to speak). But is this heuristic correct? Do I actually get something like $ke^{-\beta V(\xi)}d\xi$ ($k$ and $\beta$ constants, $d\xi$ the Riemannian measure)?