Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 8A (Classical Dynamics)

Hamilton's equations give $\dot{H}=H_q\cdot H_p-H_p\cdot H_q=0$. Also $\mathrm{div}{V}_H={\sum }_a({\partial }_{q_a}{\partial }_{p_a}H-{\partial }_{p_a}{\partial }_{q_a}H)=0$.

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.

The energy sublevel $H\le 1/2$ is compact and has finite phase volume because both $p$ and $q$ are bounded. If all ${\Phi }_{nT}(U)$ were disjoint, volume preservation would put infinitely many equal positive volumes inside it, impossible. Thus two iterates intersect. Pulling the intersection back shows a point of $U$ returns to $U$; applying the argument to arbitrarily small neighbourhoods gives Poincare recurrence theorem.