Past exam of the theoretical physics I course of the University of Cambridge / 2012 / Paper 1 / 2 / b / Solution

Treat all transformation coefficients as $O(\lambda )$, so products of them with $\lambda$ are second order. Substitution into the Hamiltonian gives its harmonic part plusSet these four coefficients to zero and impose the canonical constraints. The result isThus the first-order canonical removal of a cubic oscillator perturbation isup to $O({\lambda }^2)$. It yields $K=P^2/(2m)+m{\omega }^2{X}^2/2+O({\lambda }^2)$. The inverse map, if old initial data are specified, is $X=x+\lambda x^2/(m{\omega }^2)+2\lambda p^2/(m^3{\omega }^4)$ and $P=p-2\lambda xp/(m{\omega }^2)$ to the same order. There is no time-dependent generating term because the map is autonomous.