Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 8E / a

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- Solution a

Solution

Let

ω = \frac{k}{m}, H (x, p) = \frac{p ^{2}}{2 m} + \frac{1}{2} m ω^{2} x^{2} .

An orbit of energy

E > 0

is an ellipse in phase space, with intercepts

x = \pm 2 E / (m ω^{2})

and

p = \pm 2 m E

. Its enclosed area is

2 π E / ω

, so the action variable is

I = \frac{1}{2 π} \oint p d x = \frac{E}{ω} .

The transformation

x = \frac{2 I}{mω} sin θ, p = 2 mω I cos θ

is canonical and gives the action-angle variables. Substitution yields

H (I) = ω I, \dot{I} = 0, \dot{θ} = ω .

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