A simple derivation of the Henon's isochrone potentials: https://arxiv.org/abs/2110.01953
Alberto Saa, Roberto Venegeroles
By using well-known techniques of one-dimensional motion, we get all isochrone central potentials as exact solutions of simple quadratic equations. Moreover, we show that all potentials with Keplerian radial period T2∝|E|−3 are necessarily isochrone. Bertrand's theorem, which states that only Newtonian and harmonic central potentials have closed orbits for arbitrary initial conditions, is a direct consequence of our main result.
Comments: 5 pages