A new derivation of the Henon's isochrone potentials

Abstract

We revisit in this note the Hénon's isochrone problem. By using the standard Abel inversion technique for one-dimensional motion, we recover in a simple way the Hénon's parabolae and get all isochrone central potentials under mild smoothness assumptions on the potential function. Our approach also allows us to conclude that isochronous radial periods with explicit energy dependence are necessarily Keplerian, i.e., $T^2\propto |E{|}^{-3}$, and that their corresponding orbits can be easily integrated by mapping them into the usual Kepler problem. It can also be employed to study some other inverse central-force problems and, in particular, it provides a proof of Bertrand's theorem.

7 pages. Text revised, new figure, conclusions unchanged. Final version to appear in Can. J. Phys

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