 Hamiltonian transport equation

Transport along a Hamiltonian flow. The solution is $f(\cdot ,t)=f_I\circ {\Phi }_{-t}$ wherever the inverse characteristic exists. Equality of mixed derivatives makes the transport vector field divergence-free, so its flow preserves phase-space volume. Smoothness of the Hamiltonian alone gives local characteristics, not a complete flow on all of phase space.