Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 32B / b / i / Solution

At $\epsilon =0$,is conserved, and its orbit of amplitude $a>0$ may be writtenThe energy change over one period, to first order in $\epsilon$, isHere the stated trigonometric integrals, together with
${\int }_0^{2\pi }{\cos }^2\theta {\sin }^4\theta d\theta =\pi /8$, were used. The two candidate leading-order amplitudes are therefore