Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 32A (Integrable Systems)

Hamilton's equations areA time-independent function $F:M\to \mathbb{R}$ is a first integral when it is constant along every Hamiltonian trajectory. Sincethis is equivalent to $\{F,H\}=0$.

The $2n$-dimensional Hamiltonian system is integrable when it possesses $n$ first integrals $F_1=H,F_2,\dots ,F_n$ that are functionally independent on a dense open set and in mutual involution:

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:

From $z_j=q_j+i{p}_j$ and ${\bar{z}}_j=q_j-i{p}_j$, the chain rule givesSubstituting these identities into the canonical Poisson bracket and cancelling the $f_{z_j}{g}_{z_j}$ and $f_{{\bar{z}}_j}{g}_{{\bar{z}}_j}$ terms givesThis is the Poisson bracket in complex canonical coordinates.

Anti-Hermiticity makes ${\bar{z}}_j{A}_{jk}{z}_k$ purely imaginary, so $I_A$ is real. Its complex derivatives areThe complex-coordinate bracket from part (c) therefore givesHence the anti-Hermitian quadratic Hamiltonian identity is

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.