= Bertrand's theorem
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<Joseph Bertrand>'s theorem states that, among smooth attractive <central potentials> admitting stable <circular orbits>, requiring every <bound orbit> to be closed singles out the inverse-distance <potential energy> and the isotropic quadratic <potential energy>, up to additive constants. Rational radial-to-angular <frequency> ratios near a <circular orbit> provide only a necessary local closure test; finite-amplitude closure is the additional content. In $U=Br^n$, the radial-to-angular <frequency> ratio near a stable circle is $\sqrt{n+2}$, and stability requires $nB>0$ and $n>-2$. The all-bound-orbit cases are $n=-1$ and $n=2$.
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