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

The Hamiltonian in the question isDefineEach $I_j$ is a first integral, the functions involve disjoint canonical coordinate pairs and hence satisfy $\{I_j,I_k\}=0$, and their differentials are independent wherever every $z_j\ne 0$. Since $H={\sum }_j{I}_j$, these $n$ integrals prove Liouville integrability of the isotropic harmonic oscillator.

For constants $c_j>0$, the common level isso the regular invariant set isHamilton's equations give ${\dot{z}}_j=-i{z}_j$, so every angle evolves as ${\theta }_j(t)={\theta }_j(0)-t$. If some $c_j=0$, the corresponding circle collapses and the level is a lower-dimensional singular invariant torus.