Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 34D (Integrable Systems)

A Hamiltonian system on a $2n$-dimensional symplectic manifold is completely integrable when it has $n$ first integrals $F_1=H,F_2,\dots ,F_n$ whose differentials are independent on the regular set and which are in involution:

The Arnold--Liouville theorem states that every compact connected regular common level set of these integrals is an $n$-torus. In a neighbourhood of it there are action--angle coordinates $(I_1,\dots ,I_n,{\theta }_1,\dots ,{\theta }_n)$ in whichand Hamilton's equations becomeThus the motion on each invariant torus is linear in the angles.

SetThen ${\sum }_i{F}_i=H$. Each $F_i$ involves only $(p_i,q_i)$, sofor all $i,j$. Their differentials have disjoint coordinate supports and are independent wherever none of them vanishes, which is an open dense regular set.

Hamilton's equations areAfter the shiftthey become $n$ independent harmonic oscillators,Thus the $n$ independent commuting integrals establish complete integrability.

Completing the square givesHence the positive oscillator energy isUsing the conventionthe integral is the phase-plane ellipse area divided by $2\pi$. With oscillator frequency ${\omega }_i=|W_i|$, that area is $2\pi E_i/{\omega }_i$. ThereforeEquivalently,so the angle velocities are the constant frequencies $|W_i|$.