Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 1 / 15C / b / Solution

The invariant is phase-space volume, or area $dqdp$ for one degree of freedom. It does not require an autonomous Hamiltonian. The flow vector $v=(H_p,-H_q)$ has divergence ${\partial }_q{H}_p-{\partial }_p{H}_q=0$ whenever the mixed derivatives agree. If $M(t)$ is the Jacobian of the flow with respect to initial coordinates, then $\dot{M}=DvM$. Differentiating its determinant givesThus the change-of-variables formula shows that any region carried by the flow retains its area. This proves Liouville theorem in Hamiltonian mechanics even when $H=H(q,p,t)$ depends on time.