Action variable of a shifted harmonic oscillator

For $F=(p^2+W^2{q}^2+aq)/2$, shifting $Q=q+a/(2W^2)$ gives oscillator energy $E=F+a^2/(8W^2)$ and action $I=E/|W|$.

The action $(2\pi {)}^{-1}\oint pdq$ is the area enclosed by a periodic phase-plane orbit divided by $2\pi$.