Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 32C / a / Solution

In canonical coordinates, Hamilton's equations areA function $F(q,p)$ is an integral of motion if it is constant along every Hamiltonian flow, equivalentlyfor a time-independent $F$. The system is an integrable Hamiltonian system when it has $n$ first integrals $F_1,\dots ,F_n$ whose differentials are linearly independent almost everywhere and which are in mutual Poisson involution: $\{F_i,F_j\}=0$ for all $i,j$.