Isochrony in 3D radial potentials

Revisiting and extending an old idea of Michel Hénon [1], we geometrically and algebraically characterize the whole set of isochrone potentials [2]. Such potentials are fundamental in the potential theory. They appear in spherically symmetrical systems formed by a large amount of charges (electric or gravitational) of the same type considered in a mean-field theory. Such potentials are defined by the fact that the radial period of any test charges 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 R^2 plane. Furthermore, any isochrone orbits are mapped to 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. Finally, we propose to show that isochrony is also the after the collapse initial condition for singular stellar like globular systems and Low Surface Brightness galaxies [3].
This seminar will be introductive and dedicated to a large public of physicist.

References
[1] Hénon M., L’amas isochrone, Annales d’Astrophysique, Vol. 22, p.126, 1959
[2] Simon-Petit A.,Perez J. and Duval G., Isochrony in 3D Radial Potentials. From Michel Hénon’s Ideas to Isochrone Relativity: Classification, Interpretation and Applications, Communications in Mathematical Physics, Vol. 363, pp 605-653, 2018
[3] Simon-Petit A., Perez J. and Plum G., The status of isochrony in the formation and evolution of self-gravitating systems, Monthly Notices of the Royal Astronomical Society, Vol. 484, pp 4963-4971, 2019