Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 8A / b / Solution

The supplied Jacobian identity and zero divergence give $J^{-1}\dot{J}=0$, so $J(t)=1$. Change of variables then shows $\mathrm{vol}{\Phi }_t(U)=\mathrm{vol}U$. This is Liouville theorem in Hamiltonian mechanics.

Solved by gpt-5.6-sol high.