Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 32A / b / Solution

The Arnold-Liouville theorem states that if $F_1,\dots ,F_n$ are independent commuting first integrals and a connected component $L$ of a regular common levelis compact, then $L$ is diffeomorphic to the $n$-torus. A neighbourhood of $L$ admits action-angle variables $(I_1,\dots ,I_n,{\theta }_1,\dots ,{\theta }_n)$ in whichand the motion is linear on each invariant torus: