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:
- $U_M = U + \dfrac {M^2} {2 r^2}$
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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).
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Recall that if $\Phi$ is commensurable with $\pi$, then an orbit is closed.
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Sources
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- 1974: V.I. Arnold: Mathematical Methods of Classical Mechanics: $\S 2.\text D$, problems $1$. to $6$.