= Radial action of a bound Kepler orbit
{c}
{title2=$I_r=k/\sqrt{-2E}-|\ell|$}
For $H=p_r^2/2+\ell^2/(2r^2)-k/r$, a noncollision bound orbit has $\ell\ne0$ and $-k^2/(2\ell^2)\leq E<0$. 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.
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