A new derivation of the flenon's isochrone potentials

Author(s)

Saa, Alberto ; Venegeroles, Roberto

• Document type: Journal article
• Source: CANADIAN JOURNAL OF PHYSICS; v. N/A, p. 7-pg., 2022-12-21.

Abstract

We revisit in this note the Henon's isochrone problem. By using the standard Abel inversion technique for one-dimensional motion, we recover in a simple way the Henon'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., T2 proportional to |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. (AU)

• FAPESP's process: 21/09293-7 - Black hole spectroscopy: physical and mathematical aspects
• Grantee: Alberto Vazquez Saa
• Support Opportunities: Scholarships abroad - Research