Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 8B / b / Solution

Hamilton's equations givesoUseThenand $P_2=p_2-(n_2/n_1)p_1$ is the conserved combination. The transformed Hamiltonian depends on $Q_1$ but not $Q_2$. Thus $H$ and $P_2$ are two independent first integrals in involution for two degrees of freedom, proving Liouville integrability.