# How to find canonical transformation for given $P_i$ which are constants

We have a 2-D harmonic oscillator with Hamiltonian $H(r,\theta,p_r,p_\theta)=\frac{p_r^2}{2m}+\frac{p_\theta^2}{2mr^2}+\frac{1}{2}kr^2$. I need a canonical transformation to $(Q_1,Q_2,P_1,P_2)$ with $P_1=H, P_2=p_\theta$ which are conserved quantities.

I've tried to find a generating function for the transformation but I don't really know how to, besides guesswork. I'm not even sure this is the right method. Since there is no explicit time dependence I feel like the new Hamiltonian should be $K=H=P_1$ so that $\dot{Q_1}=\frac{\partial K}{\partial P_1}=1$ and $\dot{Q_2}=\frac{\partial K}{\partial P_2}=0$. But this doesn't feel right and it doesn't give me $Q_1,Q_2$ in terms of the old variables anyways.

• – Cosmas Zachos Feb 11 '19 at 15:35