 Solution

The Hamiltonian flow of $H$ solvesCompactness and the absence of a boundary make this smooth vector field complete, so the flow exists for all real $t$. The defining equation and Cartan's magic formula giveTherefore the pullback of a differential form differentiation rule yieldsso $({\varphi }_H^t{)}^{\ast }\omega =\omega$. Pullback commutes with the wedge product of differential forms, and consequentlyThus the flow preserves the symplectic volume, in fact the entire symplectic form.