= Hamiltonian Liouville equation
{c}
{title2=$\partial_tf+\nabla_vH\cdot\nabla_xf-\nabla_xH\cdot\nabla_vf=h$}
= Liouville transport equation
{c}
{synonym}
Transport a phase-space density along <Hamilton's equations>. The <Hamiltonian vector field> is $b=(\nabla_vH,-\nabla_xH)$ and the equation is $\partial_tf+b\cdot\nabla_{x,v}f=h$. Its zero <divergence> gives volume-preserving transport. For $h=0$, the density is constant along characteristics; a source is integrated by the <Duhamel formula for Hamiltonian transport>. This is distinct from the elliptic Liouville equation.
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