 Solution

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

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

F_{1}, \dots, F_{n}

L

of a regular common level

F_{1} = c_{1}, \dots, F_{n} = c_{n}

is compact, then

L

is diffeomorphic to the

n

L

(I_{1}, \dots, I_{n}, θ_{1}, \dots, θ_{n})

in which

ω = \sum_{j} d I_{j} \land d θ_{j}, H = H (I),

and the motion is linear on each invariant torus:

\dot{I}_{j} = 0, \dot{θ}_{j} = \frac{\partial H}{\partial I _{j}} .

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