 Solution

Hamilton's equations generate the phase-space vector field $(H_{p_1},\dots ,H_{p_n},-H_{q_1},\dots ,-H_{q_n})$. Its divergence vanishes:by equality of mixed derivatives. The Jacobian $J(t)$ of the flow obeys $\dot{J}=(\mathrm{div}V)J=0$ with $J(0)=1$. Consequently $d\tau =d{q}_1\cdots d{q}_{n}d{p}_1\cdots d{p}_n$ is preserved for every time on which the smooth flow exists. This proves Liouville theorem in Hamiltonian mechanics directly; explicit time dependence of the Hamiltonian would not change the divergence argument.