Periodic Orbits of the Planar Anisotropic Manev Problem and of the Perturbed Hydrogen Atom Problem

  • Jaume LlibreEmail author
  • Pengfei Yuan
Part of the following topical collections:
  1. In Memoriam Florin Diacu


In this paper we use the averaging theory for studying the periodic solutions of the planar anisotropic Manev problem and of two perturbations of the hydrogen atom problem. When a convenient parameter is sufficiently small we prove that for every value \(e\in (0,1)\) a unique elliptic periodic solution with eccentricity e of the Kepler problem can be continued to the mentioned three problems.


Periodic solutions Averaging theory Anisotropic Manev problem Hydrogen atom problem 



We thank to the reviewer his/her many comments and suggestions which help us to improve the presentation of our results. The first author is partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca Grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The second author is partially supported by Fundamental Research Funds for the Central Universities (NO.XDJK2015C139), China Scholarship Council(N0.201708505030).


  1. 1.
    Abouelmagd, E.I., Llibre, J., Garcia Guirao, J.L.: Periodic orbits of the planar anisotropic kepler probelm. Int. J. Bifurc. Chaos Appl. Sci. Eng. 27(1750039–1), 1–6 (2017)zbMATHGoogle Scholar
  2. 2.
    Aparicio, I., Floría, L.: Canonical focal method treatment of a Gylden–Maneff problem. Posters IV Catalan Days Appl. Math. 1–16 (1998)Google Scholar
  3. 3.
    Barrabés, E., Ollé, M., Borondo, F., Farrelly, D., Mondelo, J.M.: Phase space structure of the hydrogen atom in a circularly polarized microwave field. Phys. D 241(4), 333–349 (2012)CrossRefGoogle Scholar
  4. 4.
    Brunello, A.F., Uzer, T., Farrelly, D.: Hydrogen atom in circularly polarized microwaves, chaotic ionization via core scattering. Phys. Rev. 55(5), 3730–3745 (1997)CrossRefGoogle Scholar
  5. 5.
    Buica, A., Francoise, J.P., Llibre, J.: Periodic solutions of nonlinear periodic differential systems with a small parameter. Commun. Pure Appl. Anal. 6, 103–111 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Craig, S., Diacu, F.N., Lacomba, E.A., Perez, E.: On the anisotropic Manev problem. J. Math. Phys. 40, 1359–1375 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cushman R.H., van der Meer J.C.: Orbiting dust under radiation pressure, In: Berger HB, Hennig JD, (eds.) Proceedings of the 15th International Conference on Differential Geometric Methods in Theoretical Physics, World Scientific, Germany, pp. 403–414. (1986)Google Scholar
  8. 8.
    Delgado, J., Diacu, F.N., Lacomba, E.A., Mingarelli, A., Mioc, V., Perez, E., Stoica, C.: The global flow of the Manev problem. J. Math. Phys. 37, 2748–2761 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Deprit, A.: Dynamics of orbiting dust under radiation pressure. In: Berger, A. (ed.) The Big Bang and Georges Lemaitre, pp. 151–180. D. Reidel, Dordrecht (1984)CrossRefGoogle Scholar
  10. 10.
    Devaney, R.L.: Blowing up singularities in classical mechanical systems. Am. Math. Monthly 89(8), 535–552 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Devaney, R.L.: Singularities in classical mechanical systems. In: Ergodic theory and dynamical systems, I (College Park, Md., 1979 C80), Progr. Math., vol. 10, pp. 211–333. Birkhauser, Boston (1981)Google Scholar
  12. 12.
    Diacu, F.N.: The planar isosceles problem fro Maneff’s gravitational law. J. Math. Phys. 34, 5671–5690 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Farrelly, D., Uzer, T.: Ionization mechanism of Rydberg atoms in a circularly polarized microwave field. Phys. Rev. Lett. 74(10), 1720–1723 (1995)CrossRefGoogle Scholar
  14. 14.
    Fu, P., Scholz, T.J., Hettema, J.M., Gallagher, T.F.: Ionization of Rydberg atoms by a circularly polarized microwave field. Phys. Rev. Lett. 64(5), 511–514 (1995)CrossRefGoogle Scholar
  15. 15.
    Gebarowski, R., Zakrzewski, J.: Ionization of hydrogen atoms by circularly polarized microwaves. Phys. Rev. A 51, 1508–1519 (1995)CrossRefGoogle Scholar
  16. 16.
    Griffiths, J.A., Farrelly, D.: Ionization of Rydberg atoms by circularly and elliptically polarized microwave fields. Phys. Rev. A 45(5), 2678–2681 (1992)CrossRefGoogle Scholar
  17. 17.
    Lacomba, E.A., Llibre, J., Nunes, A.: Invariant tori and cylinders for a class of perturbed Hamiltonian systems. In: The Geometry of Hamiltonian Systems (Berkeley, CA, 1989) (Math. Sci. Res. Inst. Publ. vol. 22), pp. 373–385. Springer, New YorkGoogle Scholar
  18. 18.
    Lanchares, V., Iãrrea, M., Salas, J.P.: Bifurcations in the hydrogen atom in the presence of a circularly polarized microwave field and a static magnetic field. Phys. Rev. A 56(3), 1839–1843 (1997)CrossRefGoogle Scholar
  19. 19.
    Nauenberg, M.: Comment on “Ionization of Rydberg atoms by a circularly polarized microwave field”. Phys. Rev. Lett. 64, 27–31 (1990)CrossRefGoogle Scholar
  20. 20.
    Maneff, G.: Die gravitation und das prinzip von wirkung und gegenwirkung. Z. Phys. 31, 786–802 (1925)CrossRefzbMATHGoogle Scholar
  21. 21.
    Maneff, G.: La gravitation et le principe de l’égalité de l’action et de la réaction. C. R. 178, 2159–2161 (1924)zbMATHGoogle Scholar
  22. 22.
    Maneff, G.: La gravitation et l’énergie au zéro. C. R. 190, 1374–1377 (1930)zbMATHGoogle Scholar
  23. 23.
    Maneff, G.: Le principe de la moindre action et la gravitation. C. R. 190, 963–965 (1930)zbMATHGoogle Scholar
  24. 24.
    McGehee, R.: Singularities in classical celestial mechanics. In: Proceedings of the International Congress of Mathematicians 1978, pp. 827–834. Helsinki (1980)Google Scholar
  25. 25.
    Mioc, V.: Elliptic-type motion in Fock’s gravitational field. Astron. Nachr. 315, 175–180 (1994)CrossRefzbMATHGoogle Scholar
  26. 26.
    Mioc, V., Radu, E.: Orbits in an anisotropic radiation field. Astron. Nachr. 313, 353–357 (1992)CrossRefzbMATHGoogle Scholar
  27. 27.
    Mioc, V., Stoica, C.: Discussion et résolution compléte du probléme des deux corps dans le champ gravitationnel de Maneff. C. R. Acad. Sci. 320, 645–648 (1995)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mioc, V., Stoica, C.: Discussion et résolution compléte du probl‘eme des deux corps dans le champ gravitationnel de Maneff. II. C. R. Acad. Sci. 321, 961–964 (1995)zbMATHGoogle Scholar
  29. 29.
    Moulton, F.R.: An Introduction to Celestial Mechanics, 2nd edn. Dover, New York (1970)zbMATHGoogle Scholar
  30. 30.
    Ollé, M.: To and fro motion for the hydrogen atom in a circularly polarized microwave field. Commun. Nonlinear Sci. Numer. Simul. 54, 286–301 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ollé, M., Pacha, J.R.: Hopf bifurcation for the hydrogen atom in a circularly polarized microwave field. Commun. Nonlinear Sci. Numer. Simul. 62, 27–60 (2018)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Poincaré, H.: Mèthodes Nouvelles, vol. I. Gauthier-Villars, Paris (1892)zbMATHGoogle Scholar
  33. 33.
    Rakovic, M.J., Chu, S.I.: Approximate dynamical symmetry of hydrogen atomsin circularly polarized microwave fields. Phys. Rev. A 50(6), 5077–5080 (1994)CrossRefGoogle Scholar
  34. 34.
    Rakovic, M.J., Chu, S.I.: New integrable systems: hydrogen atom in external fields. Phys. D 81(3), 207–316 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Roy, A.E.: Orbit Motion, 4th edn. CRC Press, Boco Raton (2004)CrossRefGoogle Scholar
  36. 36.
    Saslaw, W.C.: Motions around a source whose luminosity changes. Astrophys. J. 226, 240–252 (1978)CrossRefGoogle Scholar
  37. 37.
    Schwarzschild, K.: Über eine classe periodischer losungen des dreikorperproblems. Astron. Nachr. 147, 17–24 (1896)CrossRefzbMATHGoogle Scholar
  38. 38.
    Selaru, D., Cucu-Dumitrescu, C., Mioc, V.: On a two-body problem with periodically changing equivalent gravitational parameter. Astron. Nachr. 313, 257–263 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Stoica, C., Mioc, V.: On the two-body problem in Maneff-type fields. Bull. Astron. Belgrade 154, 1–8 (1996)Google Scholar
  40. 40.
    Ureche, V.: Free-fall collapse of a homogeneous sphere in Maneff’s gravitational field. Rom. Astron. J. 5, 145–148 (1995)Google Scholar
  41. 41.
    Zakrzewski, J., Delande, D., Gay, J.C.: Ionization of hoghly excited hydrogen atoms by a circularly polarized microwave field. Phys. Rev. A 47(4), 2468–2471 (1993)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBarcelonaSpain
  2. 2.School of Mathematics and StatisticsSouthwest UniversityChongqingChina

Personalised recommendations