In potential theory, isochrony was introduced by Michel Hénon in 1959 to characterize the astrophysical observations of certain globular clusters. Today, Michel Hénon’s isochrone potential is mainly used for its integrable property in numerical simulations, but is generally not really known. This talk will present new results on isochrony which are of particular importance in self-gravitating dynamics.

After introducing Michel Hénon’s isochrone definition, based on a brilliant remark on gravitational dynamics, we will complete the set of isochrone potentials. This will allow us to highlight a particular relation between the isochrones, by generalizing the Bohlin transformation. In fact, we will determine the keplerian nature of isochrones, which is at the heart of the new isochrone relativity. Finally, the consequences of this relativity in celestial mechanics (a generalization of Kepler’s Third law, Bertrand’s theorem) will be applied to analyze the result of a gravitational collapse.

References :

Hénon M., L’amas isochrone, Annales d’Astrophysique,

Vol. 22, p.126, 1959

Simon-Petit A., Perez J., Duval G., Isochrony in 3D radial potentials, Comm. in Math. Phys., Accepted. Preprint : https://arxiv.org/abs/1804.11282.