Duhamel formula for Hamiltonian transport ($f(t,z)=f_0(\Phi (0,t,z))+{\int }_0^{t}h(s,\Phi (s,t,z))ds$)

For a complete invertible Hamiltonian flow $\Phi (t,s)$, integrate the source along the backward characteristic ending at $z$ at time $t$. The displayed formula solves the Hamiltonian Liouville equation. Volume preservation makes each composition with $\Phi (s,t)$ an isometry of Lp spaces.