Duhamel formula for Hamiltonian transport
= Duhamel formula for Hamiltonian transport
{c}
{title2=$f(t,z)=f_0(\Phi(0,t,z))+\int_0^th(s,\Phi(s,t,z))\,ds$}
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>.