Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 34D / a / Solution

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.