Liouville theorem for Hamiltonian volume

Statement

On a $2n$ -dimensional symplectic manifold $(M,\omega)$ , the Liouville volume form is $\omega^n/n!$ , using the symplectic orientation. A Hamiltonian flow preserves this form wherever the flow is defined:

(\Phi^t)^*(\omega^n/n!)=\omega^n/n!.

If the flow is complete and the total volume is finite and nonzero, normalization gives an invariant probability measure.

Reason

Hamiltonian flow preserves $\omega$ . Pullback respects wedge products, so it preserves its top exterior power. In canonical coordinates the form is the usual phase-space volume $dq_1\,dp_1\cdots dq_n\,dp_n$ .

Energy levels

Restricting dynamics to an energy surface requires the appropriate invariant surface measure; simply restricting the ambient volume as a set measure usually gives zero. Volume preservation by itself does not imply ergodicity.