Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 20 / 3 / i / Solution

On a $2n$-dimensional symplectic manifold, a completely integrable Hamiltonian system has $n$ first integrals in involution $F_1=H,F_2,\dots ,F_n$ whose differentials are independent almost everywhere, hence on a dense open regular set:An equivalent formulation allows $H$ to be a function of the $n$ commuting independent integrals. The independence requirement is on the regular set, permitting exceptional critical points and singular fibres.

For example, on ${\mathbb{R}}^{2n}$ take $H={\sum }_{j=1}^n{E}_j$ with $E_j=(p_j^2+{\alpha }_j^2{q}_j^2)/2$ and ${\alpha }_j>0$. The disjoint coordinate pairs make all Poisson brackets $\{E_i,E_j\}$ zero, and the differentials are independent whenever no pair $(q_j,p_j)$ vanishes. Replacing $E_1$ by $H$ is an invertible linear map of this list. Thus $F_1=H,F_j=E_j$ for $j\ge 2$ give the required example. Irrational frequency ratios affect orbit density, but not complete integrability.