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Communications in Mathematical Physics

, Volume 363, Issue 2, pp 605–653 | Cite as

Isochrony in 3D Radial Potentials

From Michel Hénon’s Ideas to Isochrone Relativity: Classification, Interpretation and Applications
  • Alicia Simon-Petit
  • Jérôme Perez
  • Guillaume Duval
Article

Abstract

Revisiting and extending an old idea of Michel Hénon, 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.

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Notes

Acknowledgements

This work is supported by the “IDI 2015” Project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02. JP especially thanks Jean-Baptiste Fouvry for helpful discussions about Bertrand’s theorem. ASP especially thanks Alain Albouy for his great remarks on Bertrand’s theorem and for sharing his deep historical knowledge. The authors are grateful to Faisal Amlani for his detailed copy-editing of the paper and thank the referees of the article for their helpful comments and fruitful suggestions.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Applied Mathematics LaboratoryEnsta ParisTech, Paris Saclay UniversityPalaiseauFrance
  2. 2.Mathematics and Informatics LaboratoryINSA RouenSaint-Étienne-du-RouvrayFrance

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