Harmonic oscillator action-angle variables ($I=H/\omega ,\dot{\theta }=\omega$)

For a harmonic oscillator with Hamiltonian $H=p^2/(2m)+m{\omega }^2{q}^2/2$, where $m,\omega >0$, defineThese action-angle variables are defined for $I>0$, with a local angle $\theta$ modulo $2\pi$. A proof of canonicity is ${\theta }_q=p/(2I)$ and ${\theta }_p=-q/(2I)$, which give $\{\theta ,I\}=1$ using the Poisson bracket. Substitution gives $H=\omega I$. Hamilton's equations then prove $I(t)=I(0)$ and $\theta (t)=\theta (0)+\omega t$. The action integral around the corresponding phase space ellipse is $\oint pdq=2\pi I$, so $I$ is the usual normalized action variable. The angle is undefined at the zero-energy origin.