 Kepler-harmonic epicyclic precession

For $\Phi =-GM/r+\alpha {\omega }^2{r}^2/2$, nearly circular motion has ${\Omega }^2=GM/r^3+\alpha {\omega }^2$ and ${\kappa }^2=GM/r^3+4\alpha {\omega }^2$. Their generally nonrational ratio makes small-eccentricity orbits precess even if $\alpha$ was added slowly. This illustrates adiabatic actions do not preserve orbital closure.