Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 32B / Solution

Hamilton's equations are ${\dot{q}}_j=\partial H/\partial p_j$ and ${\dot{p}}_j=-\partial H/\partial q_j$. The Arnold-Liouville theorem states that $n$ functionally independent first integrals in involution on a $2n$-dimensional symplectic manifold give local integrability; a compact connected regular common level is an $n$-torus. Near such a torus there are action-angle variables $(I,\vartheta )$ with $H=H(I)$, $\dot{I}=0$ and $\dot{\vartheta }=\partial H/\partial I$.

For the displayed diagonal oscillator Hamiltonian set $E_k=(p_k^2+{\omega }_k^2{q}_k^2)/2$. Each is constant, since ${\dot{q}}_k=p_k$ and ${\dot{p}}_k=-{\omega }_k^2{q}_k$. Different $E_k$ depend on disjoint canonical pairs, so their Poisson brackets vanish. Their differentials are independent wherever every $d{E}_k$ is nonzero, an open dense set; this proves complete integrability. For nonzero frequencies put ${\Omega }_k=|{\omega }_k|$. A positive-energy level in each pair is an ellipse with semiaxes $\sqrt{2E_k}$ and $\sqrt{2E_k}/{\Omega }_k$. Its enclosed area is $2\pi E_k/{\Omega }_k$, so the action variables areFor the usual positive oscillator frequencies this is $I_k=E_k/{\omega }_k$. If a printed “constant” frequency is allowed to be zero, that pair is a free particle: $p_k$ is a commuting integral, but its generic level is noncompact and has no closed-cycle oscillator action. Complete integrability persists, while the torus/action formula requires nonzero frequencies.