 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-3/14a/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Take

H (x, y) = y^{2} /2 + x^{2} /2 - x^{4} /4

\dot{H} = y (x^{3} - x) + (x - x^{3}) (y + a x - b x^{3}) = (x - x^{3}) (a x - b x^{3}) .

Integrating over a periodic orbit returns

H

to its starting value, and hence

\oint_{Γ} (x - x^{3}) (a x - b x^{3}) d t = 0.

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