Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 8 / 1 / ii

For the Arnold-Liouville theorem, work on a symplectic manifold of dimension $2d$. Let $F_1=H,F_2,\dots ,F_d$ be smooth functions with pairwise vanishing Poisson brackets, and supposeon a connected component $N$ of a common level. If $N$ is compact, then $N$ is a $d$-dimensional torus. Its commuting Hamiltonian vector fields generate translations, and the Hamiltonian motion on it is linear in angular coordinates. Compactness ensures completeness of these restricted vector fields.

Moreover a neighbourhood of $N$ admits action-angle variables $(I_1,\dots ,I_d,{\theta }_1,\dots ,{\theta }_d)$, with each angle defined modulo $2\pi$, for whichOne compatible symplectic form convention is $\omega ={\sum }_{i}d{\theta }_i\wedge d{I}_i$. The equations becomeThe resulting motion is periodic when the frequencies are commensurable and otherwise quasiperiodic on a subtorus. This is Liouville integrability with $d$ independent first integrals in involution. Compactness and regularity cannot be discarded from the torus conclusion. On a general Poisson manifold, apply this theorem on a regular symplectic leaf of dimension $2d$; the ambient dimension alone does not specify the required number of integrals.