Bertrand's Theorem/Asymptotic Proof

Preliminary Lemma
The substitution $x = M / r$ gives:
 * $\ds \Phi = \sqrt 2 \int_{x_\min}^{x_\max} \frac {\d x} {\sqrt {E - \map W x}}$

where $\map W x \equiv \map U {\dfrac M x} + \dfrac 1 2 x^2$.

In general, the frequency of oscillations around a stable equilibrium at $x = x_0$ for a particle of mass $m$ in a potential $V$ is given by:
 * $\omega = \sqrt {\dfrac {\map {V''} {x_0} } m}$

which means that:
 * $\Phi \approx 2 \pi \dfrac M {x_0^2} \sqrt {\map {W} {x_0} } = 2 \pi \sqrt {\dfrac {U'} {3 U' + x_0 U} }$

Suppose $\Phi$ were constant.

Then the solutions are:
 * $\map U r = k r^\alpha, \alpha \ge - 2, \alpha \ne 0$

and:
 * $\map U r = b \log r$

Therefore:
 * $\Phi = \dfrac {2 \pi} {\sqrt{\alpha + 2}}$

with $\alpha = 0$ corresponding to the logarithmic solution.

This is discarded, since in that case $\Phi$ is not commensurable with $\pi$.

Now consider $\alpha > 0$, so that $\map U r \to \infty$ as $r \to \infty$.

The substitution $x = y x_\max$ reduces $\Phi$ to:
 * $\ds \Phi = \sqrt 2 \int_{y_\min}^1 \frac {\d y} {\sqrt {\map W 1 - \map W y} }$

where:
 * $\map W y \equiv \dfrac {y^2} 2 + \dfrac 1 { {x_\max}^2} \map U {\dfrac M {y x_\max} }$

Therefore, when $E \to \infty$:
 * $x_\max \to \infty$ and $y_\min \to 0$

This means that:
 * $\ds \lim_{E \mathop \to \infty} \Phi = \pi$

Equating this with the previous result gives:
 * $\dfrac {2 \pi} {\sqrt {\alpha + 2}} = \pi$

Therefore:
 * $\alpha=2$

Now let $\map U r = -k r^{-\beta}, 0 < \beta < 2$.

A similar approach gives:
 * $\ds \lim_{E \mathop \to -0} \Phi = \frac {2 \pi} {2 - \beta}$

so that:
 * $\dfrac {2 \pi} {2 + \alpha} = \dfrac {2 \pi} {\sqrt {2 + \alpha} }$

It follows that:
 * $\alpha = -1$

and the result follows.