Let $H: \mathbb{R}^{2n} \rightarrow \mathbb{R}$ be smooth and let $\phi^t$ be the Hamiltonian flow of $H$. Consider the restriction of the flow to a regular energy hypersurface, $\Gamma$, with the associated Liouville measure $\mu_\Gamma$.
Question: is every orbit of the flow measurable with measure zero (with respect to $\mu_\Gamma$)?
Thoughts: In the general case of jointly measurable ergodic flows, it is possible to have both 1) non-measurable orbits and 2) measurable orbits with positive measure (e.g. translations on the torus). My guess is that differentiability of $\phi^t$ with respect to time should ensure measurability (through Lipschitz continuity?).