# Definition of Liouville measure on energy surface of Hamiltonian system

This is a reference request, as I can't for the life of me find anything that answers my question in the literature.

If $(M,\omega,H)$ is a Hamiltonian system, we know from Liouvile's theorem that its level sets $H^{-1}(c)$ for $c$ some regular value are tori. Several of the texts I am reading mention that there is a flow-invariant measure on these level sets called the Liouville measure.

As I don't think this is trivial, but don't know how to construct it, how is the Liouville measure defined for arbitrary symplectic manifolds, and why is it given by $$\frac{dS}{||\nabla H||},$$ where $S$ is "surface area," for $M=\mathbb{R}^{2n}$ with the canonical symplectic form?

If you don't care that the answer is not rigorous, here is the idea. Think of Liouville measure $dS$ on $H^{-1}([c,c+\delta c])$ for $\delta c$ small. Then the "thickness" of this "shell" will be proportional to $1/\|\nabla H\|$.
• Thanks, this does make sense. I think this means the Liouville measure is a sort of volume form on the level sets. If so, is it just something like $\omega^n$ for an arbitrary symplectic manifold $(M,\omega)$? – user117824 Dec 30 '13 at 5:53