Bertrand's Theorem/Lemma

Preliminary Lemma
For simplicity we set $m = 1$, so that the effective potential becomes:
 * $U_M = U + \dfrac {M^2} {2 r^2}$

Consider the apsidial angle:
 * $\ds \Phi = \sqrt 2 \int_{r_\min}^{r_\max} \frac {M \rd r} {r^2 \sqrt {E - U_M} }$

where:
 * $E$ is the energy
 * $r_\min, r_\max$ are solutions to $\map {U_M} r = E$.

By definition, this is the angle between adjacent apocenters (pericenters).

Recall that if $\Phi$ is commensurable with $\pi$, then an orbit is closed.