Bertrand's Theorem/Lemma

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Theorem

Let $U: \R_{>0} \to \R$ be analytic for $r > 0$.

Let $M > 0$ be a nonvanishing angular momentum such that a stable circular orbit exists.

Suppose that every orbit sufficiently close to the circular orbit is closed.


Then $U$ is either $k r^2$ or $-\dfrac k r$ (for $k > 0$) up to an additive constant.


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.




Sources