Liouville theorem

Liouville's Theorem: The volume of any closed surface in phase space remains constant as that surface moves through phase space. This is a consequence of trajectories never crossing.

\Rightarrow \nabla \cdot v = \nabla \cdot \dot{z} = 0

Apply divergence theorem to Hamilton's equations:

\dot{z} = (\frac{\partial H}{\partial p}, - \frac{\partial H}{\partial q})

\nabla \cdot \dot{z} = \frac{\partial}{\partial q} \frac{\partial H}{\partial p} + \frac{\partial}{\partial p} (- \frac{\partial H}{\partial q}) = 0