Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 33B / a / Solution

Hamilton's equations are ${\dot{x}}_i=p_i$ and ${\dot{p}}_i=-{\partial }_{i}V$. A first integral $F$ satisfies $\{F,H\}=0$. See integrable Hamiltonian system: in three degrees of freedom, integrability requires three independent commuting first integrals.