= Solution
<Hamilton's equations> generate the phase-space vector field $(H_{p_1},\ldots,H_{p_n},-H_{q_1},\ldots,-H_{q_n})$. Its divergence vanishes:
$$
\sum_k\left(\partial_{q_k}H_{p_k}-\partial_{p_k}H_{q_k}\right)=0,
$$
by equality of mixed derivatives. The Jacobian $J(t)$ of the flow obeys $\dot J=(\operatorname{div}V)J=0$ with $J(0)=1$. Consequently \b[$d\tau=dq_1\cdots dq_n\,dp_1\cdots dp_n$ 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.
Back to article page
