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-PetitEmail author
  • Jérôme Perez
  • Guillaume Duval


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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.


  1. 1.
    Albouy, A.: Lectures on the Two-Body Problem, Classical and Celestial Mechanics (Recife, 1993/1999), pp. 63–116 (2002)Google Scholar
  2. 2.
    Arnold V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)CrossRefGoogle Scholar
  3. 3.
    Arnold V.I.: Huygens & Barrow & Newton & Hooke, pp. 95. Birkhäuser Verlag, Basel (1990)CrossRefGoogle Scholar
  4. 4.
    Bertrand J.: Théorème relatif au mouvement d’un point attiré vers un centre fixe. C. R. Acad. Sci. Paris 77, 849–853 (1873)zbMATHGoogle Scholar
  5. 5.
    Binney, J.: Hénon’s isochrone model. In: Alimi, J.-M., Mohayaee, R., Perez, J. (Eds.) Une vie dédiée aux systèmes dynamiques, pp. 99–109. Hermann, Paris (2016). arXiv:1411.4937 [astro-ph]
  6. 6.
    Binney J., Tremaine S.: Galactic Dynamics, 2nd edn. Princeton University Press, Princeton (2008)zbMATHGoogle Scholar
  7. 7.
    Bohlin K.: Note sur le problème des deux corps et sur une intégration nouvelle dans le problème des trois corps. Bull. Astron. Ser. I 28, 113 (1911)zbMATHGoogle Scholar
  8. 8.
    Borel, E.: Introduction géométrique à à quelques théories physiques. Gauthier-Villars, (1914)Google Scholar
  9. 9.
    Brown L.S.: Forces giving no orbit precession. Am. J. Phys. 46, 930–931 (1978)ADSCrossRefGoogle Scholar
  10. 10.
    Castro-Quilantan J.L., Del Rio-Correa J.L., Medina M.A.R.: Alternative proof of Bertrand’s theorem using a phase space approach. Rev. Mex. Fis. 42, 867–877 (1996)Google Scholar
  11. 11.
    Chin, S.A.: A truly elementary proof of Bertrand’s theorem. Am. J. Phys. 83, 320 (2015)Google Scholar
  12. 12.
    Vaucouleurs G.: Recherches sur les Nebuleuses Extragalactiques. Ann. Astrophys. 11, 247 (1948)ADSGoogle Scholar
  13. 13.
    Dehnen W.: A family of potential-density pairs for spherical galaxies and bulges. Mon. Not. R. Astron. Soc. 265, 250 (1993)ADSCrossRefGoogle Scholar
  14. 14.
    Fasano A., Marni S.: Analytic Mechanics—An Introduction. Oxford University Press, Oxford (2006)Google Scholar
  15. 15.
    Féjoz, J., Kaczmarek, L.: Sur le théorème de Bertrand (d’après Michael Herman), Michael Herman Memorial Issue. Ergod. Theory Dyn. Syst. 24(5), 1583–1589 (2004)Google Scholar
  16. 16.
    Gidas B., Ni W.-M., Nirenberg L.: Symmetry of positive solutions of nonlinear elliptic equations in \({\mathbb{R}^{n}}\). Math. Anal. Appl. 79, 369–402 (1981)zbMATHGoogle Scholar
  17. 17.
    Goldstein H.: Classical Mechanics, pp. 601–605. Addison Wesley, New York (1981)Google Scholar
  18. 18.
    Gourgoulhon, E.: Special Relativity in General Frames, Graduate Texts in Physics. Springer, Berlin (2013)Google Scholar
  19. 19.
    Goursat E.: Les transformations isogonales en Mécanique. C. R. Hebd. Acad. Sci. 108, 446448 (1889)zbMATHGoogle Scholar
  20. 20.
    Grandati Y., Bérard A., Ménas F.: Inverse problem and Bertrand’s theorem. Am. J. Phys. 76, 782–787 (2008)ADSCrossRefGoogle Scholar
  21. 30.
    Lynden-Bell D.: Statistical mechanics of violent relaxation in stellar systems. Mon. Not. R. Astron. Soc. 136, 101 (1967)ADSCrossRefGoogle Scholar
  22. 31.
    Lynden-Bell D.: Bound central orbits. Mon. Not. R. Astron. Soc. 447(2), 1962–1972 (2015)ADSCrossRefGoogle Scholar
  23. 32.
    Lynden-Bell, D.: Variations on the theme of Michel Hénon’s isochrone. In: Alimi, J.-M., Mohayaee, R., Perez, J. (eds.) Une vie dédiée aux systèmes dynamiques. Hermann, pp. 81–86 (2016). arXiv:1411.4926 [astro-ph]
  24. 33.
    Lynden-Bell D., Jin S.: Analytic central orbits and their transformation group. Mon. Not. R. Astron. Soc. 386, 245–260 (2008)ADSCrossRefGoogle Scholar
  25. 34.
    MacLaurin C.: A Treatise on Fluxions. W. Baynes, London (1801)Google Scholar
  26. 35.
    Newton, I.: Philosophiae Naturalis Principia Mathematica, London (1756)Google Scholar
  27. 36.
    Navarro J., Frenk C., White S.: The Structure of Cold Dark Matter Halos. Astrophys. J. 463, 563 (1996)ADSCrossRefGoogle Scholar
  28. 37.
    Perez J., Aly J.-J.: Stability of spherical self-gravitating systems I: analytical results. Mon. Not. R. Astron. Soc. 280, 689 (1996)ADSCrossRefGoogle Scholar
  29. 38.
    Rartinez-y-Romero R.P., Nunez-Yepez H.N., Salas-Brito A.L.: Closed orbits and constants of motion in classical mechanics. Eur. J. Phys. 13, 26–31 (1992)MathSciNetCrossRefGoogle Scholar
  30. 39.
    Santos F., Soares V., Tort A.: Determination of the Apsidal Angle and Bertrand’s theorem. Phys. Rev. E 79, 036605-1–036605-6 (2009)ADSMathSciNetCrossRefGoogle Scholar
  31. 40.
    Tikochinsky Y.: A simplified proof of Bertrand’s theorem. Am. J. Phys. 56, 1073–1075 (1988)ADSMathSciNetCrossRefGoogle Scholar
  32. 41.
    Zarmi Y.: The Bertrand theorem revisited. Am. J. Phys. 70, 446–449 (2002)ADSCrossRefGoogle Scholar

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

Personalised recommendations