Solution

Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 30A / ii / Solution

{c}
{title2=$f_t+H_v\cdot f_x-H_x\cdot f_v=0$}

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.

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 Incoming links: Hamiltonian transport equation