 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ii/paper-3/7d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

H (q, p) = p^{2} / (2 m) + V (q)

(θ, I)

in which

θ

is periodic modulo

2 π

and

H = H (I)

\dot{I} = 0

and

\dot{θ} = d H / d I

I = \frac{1}{2 π} \oint p d q = \frac{1}{π} \int_{q_{-}}^{q_{+}} 2 m (E - V (q)) d q .

The orientation is that of the motion, so the integral is the positive enclosed phase-space area divided by

2 π

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