 Radial action of a bound Kepler orbit

For $H=p_r^2/2+{\ell }^2/(2r^2)-k/r$, a noncollision bound orbit has $\ell \ne 0$ and $-k^2/(2{\ell }^2)\le E<0$. Its radial action is $\sqrt{-2E}/\pi$ times ${\int }_{\alpha }^{\beta }\sqrt{(r-\alpha )(\beta -r)}dr/r$, where $\alpha ,\beta =[k\mp \sqrt{k^2+2E{\ell }^2}]/(-2E)$. Evaluating the integral gives the displayed action and $H=-k^2/[2(I_r+|\ell |{)}^2]$. The angular and radial frequencies have equal magnitudes, explaining closed orbits rather than generic quasiperiodicity. The circular limit has $I_r=0$; zero angular momentum gives collision in the unregularized phase space.