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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.

Describe many-particle systems through distributions on phase space. Transport moves particles along trajectories, while collision or diffusion terms modify their velocity distributions. The Boltzmann equation, Hamiltonian Liouville equation and Fokker-Planck equation are central models.

Write $z=(x,v)$ and let $\Phi (t,s,z)$ denote the Hamiltonian flow from time $s$ to time $t$. The characteristic curves satisfy Hamilton's equations:The Hamiltonian Liouville equation then reduces along each curve toThe signs and derivative variables here are those in the PDF.

The global characteristic flow for a Hamiltonian with bounded Hessian follows, for example, from $H\in C^2(\mathbb{R}\times {\mathbb{R}}^{2d})$ and, for every finite $T$,Thus the Hamiltonian vector field $b=({\nabla }_{v}H,-{\nabla }_{x}H)$ is globally Lipschitz continuous in $z$ on each finite time interval and satisfies $|b(t,z)|\le A_T+L_T|z|$. The Picard-Lindelof theorem gives local existence and uniqueness, while the Gronwall inequality gives, for example,This excludes finite-time escape. There is a unique Hamiltonian flow for all finite forward and backward times, and $\Phi (s,t)$ is the inverse of $\Phi (t,s)$. These sufficient conditions are deliberately stronger than necessary.

For homogeneous Hamiltonian Liouville equation transport with autonomous $H$, compact initial support remains in the fixed energy sublevel determined by its maximum initial energy. If these sublevels are compact, this proves uniform compact support without solving the flow. The harmonic energy $H_{\omega }=(|v{|}^2+{\omega }^2|x{|}^2)/2$ has this property exactly when $\omega \ne 0$; at zero frequency spatial free streaming can escape every compact set.