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

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: