nLab Bertrand's theorem

Bertrand’s theorem in classical mechanics: the only central forces for which all orbits are closed are $f(r) = a r$ and $f(r) = - b r^{-2}$ where $a,b\gt 0$. In both cases the orbits are ellipses, centered at origin in the first case and with the focus in the center in the second case.

• wikipedia