 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-1/32a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

\overset{q}{˙}_{j} = \frac{\partial H}{\partial p _{j}}, \overset{p}{˙}_{j} = - \frac{\partial H}{\partial q _{j}} .

F : M \to R

is a first integral when it is constant along every Hamiltonian trajectory. Since

\frac{d F}{d t} = {F, H},

this is equivalent to

{F, H} = 0

The

2 n

-dimensional Hamiltonian system is integrable when it possesses

n

F_{1} = H, F_{2}, \dots, F_{n}

that are functionally independent on a dense open set and in mutual involution:

{F_{j}, F_{k}} = 0 (1 \leq j, k \leq n) .

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