Isochrony in 3D radial potentials. From Michel H\'enon ideas to isochrone relativity: classification, interpretation and applications

TL;DR

This paper provides a comprehensive geometric and algebraic classification of isochrone potentials, revealing their symmetries, relativity, and connections to Keplerian orbits, with implications for potential theory and classical mechanics.

Contribution

It introduces a novel characterization of all isochrone potentials using affine transformations and parabola representations, extending Michel Hénon's ideas and linking them to relativity and classical symmetries.

Findings

Complete classification of isochrone potentials via affine group actions.

Mapping of isochrone orbits to Keplerian orbits through a generalized Bohlin transformation.

Insight into symmetries like Kepler's Third Law and Bertrand's theorem through isochrone relativity.

Abstract

Revisiting and extending an old idea of Michel H\'enon, we geometrically and algebraically characterize the whole set of isochrone potentials. Such potentials are fundamental in potential theory. They appear in spherically symmetrical systems formed by a large amount of charges (electrical or gravitational) of the same type considered in mean-field theory. Such potentials are defined by the fact that the radial period of a test charge in such potentials, provided that it exists, depends only on its energy and not on its angular momentum. Our characterization of the isochrone set is based on the action of a real affine subgroup on isochrone potentials related to parabolas in the $\mathbb{R}^2$ plane. Furthermore, any isochrone orbits are mapped onto associated Keplerian elliptic ones by a generalization of the Bohlin transformation. This mapping allows us to understand the isochrony property of a given potential as relative to the reference frame in which its parabola is represented. We detail this isochrone relativity in the special relativity formalism. We eventually exploit the completeness of our characterization and the relativity of isochrony to propose a deeper understanding of general symmetries such as Kepler's Third Law and Bertrand's theorem.