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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ii/paper-1/30b/iv/solution

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Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 30B / iv / Solution by

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

For the quadratic potential the characteristic curves are harmonic oscillations:

x = x_{0} cos t + v_{0} sin t, v = v_{0} cos t - x_{0} sin t .

(1)

The inverse is

x_{0} = x cos t - v sin t

v_{0} = v cos t + x sin t

. Therefore the Liouville equation has solution

f (x, v, t) = f_{I} (x cos t - v sin t, v cos t + x sin t) .

(2)

This formula works for smooth data or by transport for locally integrable data, and for measures by pushforward. Although the quadratic

V

itself is not globally Lipschitz as in the initial sufficient hypotheses, its gradient is globally Lipschitz, which is the property actually needed for this global flow.

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