Symmetric Periodic Orbits in the Anisotropic Schwarzschild-Type Problem

  • Vasile Mioc
  • Mira-Cristiana Anisiu
  • Michael Barbosu


Studying the two-body problem associated to an anisotropic Schwarzschild-type field, Mioc et al. (2003) did not succeed in proving the existence or non-existence of periodic orbits. Here we answer this question in the affirmative. To do this, we start from two basic facts: (1) the potential generates a strong force in Gordon’s sense; (2) the vector field of the problem exhibits the symmetries S i , \(i =\overline {1, 7} \), which form, along with the identity, an Abelian group of order 8 with three generators of order 2. Resorting to S2 and S3, in connection with variational methods (particularly the classical lower-semicontinuity method), we prove the existence of infinitely many S2- or S3-symmetric periodic solutions. The symmetries S2 and S3 constitute an indicator of the robustness of the classical isotropic Schwarzschild-type system to perturbations (as the anisotropy may be considered).


Schwarzschild-type problems nonlinear particle dynamics symmetries periodic orbits variational methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Amann, H. 1990Ordinary Differential Equations: An Introduction to Nonlinear AnalysisWalter de GruyerBerlin, New YorkGoogle Scholar
  2. Ambrosetti, A., Coti Zelati, V. 1993Periodic Solutions of Singular Lagrangian Systems. Progresses in Nonlinear Differential Equations and their Applications, No. 10BirkhäuserBostonGoogle Scholar
  3. Anisiu, M.-C. 1998Methods of Nonlinear Analysis Applied to Celestial MechanicsCluj University PressCluj-Napoca(Romanian)Google Scholar
  4. Bertotti, M. L. 1991‘Forced oscillations of singular dynamical systems with an application to the restricted three-body problem’J. Diff. Eq.93102141CrossRefGoogle Scholar
  5. Casasayas, J. and Llibre, J.: 1984, ‘Qualitative analysis of the anisotropic Kepler problem’, Mem. Amer. Math. Soc., Vol. 52, No. 312, AMS, Providence, RI.Google Scholar
  6. Chenciner, A.: 2002, ‘Action minimizing periodic orbits in the Newtonian n-body problem’, Celestial Mechanics (Evanston, IL, 1999), Contemporary Mathematics 292, Amer. Math. Soc., Providence, RI, pp. 71–90.Google Scholar
  7. Chenciner, A., Montgomery, R. 2000‘A remarkable periodic solution of the three-body problem in the case of equal masses’Ann. Math.152881901Google Scholar
  8. Coti Zelati, V.: 1994, Introduction to Variational Methods and Singular Lagrangian Systems, School and Workshop on Variational and Local Methods in the Study of Hamiltonian Systems, International Centre for Theoretical Physics, Trieste, Italy, 10–28 October 1994, SMR 779/4.Google Scholar
  9. Craig, S., Diacu, F. N., Lacomba, E. A., Perez, E. 1999‘On the anisotropic Manev problem’J. Math. Phys.4013591375CrossRefGoogle Scholar
  10. Devaney, R. L. 1978‘Collision orbits in the anisotropic Kepler problem’Invent. Math.45221251CrossRefGoogle Scholar
  11. Devaney R. L. 1981, ‘Singularities in classical mechanical systems’, in Ergodic Theory and Dynamical Systems, Vol. 1, Birkhäuser, Boston, pp. 211–333.Google Scholar
  12. Diacu, F. N. 1996‘Near-collision dynamics for particle systems with quasihomogeneous potentials’J. Diff. Eq.1285877CrossRefGoogle Scholar
  13. Diacu, F., Santoprete, M. 2001‘Nonintegrability and chaos in the anisotropic Manev problem’Physica D1563952CrossRefGoogle Scholar
  14. Diacu, F. and Santoprete, M.: 2002, ‘On the global dynamics of the anisotropic Manev problem’, Scholar
  15. Gordon, W. B. 1975‘Conservative dynamical systems involving strong forces’Trans. Amer. Math. Soc.204113135Google Scholar
  16. Gutzwiller, M. C. 1971‘Periodic orbits and classical quantization conditions’J. Math. Phys.12343358CrossRefGoogle Scholar
  17. Gutzwiller, M. C. 1973‘The anisotropic Kepler problem in two dimensions’J. Math. Phys.14139152CrossRefGoogle Scholar
  18. Gutzwiller, M. C. 1977‘Bernoulli sequences and trajectories in the anisotropic Kepler problem’J. Math. Phys.18806823CrossRefGoogle Scholar
  19. Hénon, M., Heiles, C. 1964‘The applicability of the third integral of motion: some numerical experiments’Astron. J.697379CrossRefGoogle Scholar
  20. McGehee, R. 1973‘A stable manifold theorem for degenerate fixed points with applications to celestial mechanics’J. Diff. Eq.147088CrossRefGoogle Scholar
  21. McGehee, R. 1974‘Triple collision in the collinear three-body problem’Invent. Math.27191227CrossRefGoogle Scholar
  22. Mioc, V. 2002‘Symmetries in the Schwarzschild problem’Baltic Astron.11393407Google Scholar
  23. Mioc, V., Pérez-Chavela, E., Stavinschi, M. 2003‘The anisotropic Schwarzschild-type problem, main features’Celest. Mech. Dyn. Astron.8681106(Paper I)CrossRefGoogle Scholar
  24. Mioc, V., Radu, E. 1992‘Orbits in an anisotropic radiation field’Astron. Nachr.313353357Google Scholar
  25. Palais, R. 1979‘The principle of symmetric criticality’Comm. Math. Phys.691930CrossRefGoogle Scholar
  26. Poincaré, H. 1896‘Sur les solutions périodiques et Ie principe de la moindre action’C. R. Acad. Sci. Paris123915918Google Scholar
  27. Santoprete, M. 2002‘Symmetric periodic solutions of the anisotropic Manev problem’J. Math. Phys.4332073219CrossRefGoogle Scholar
  28. Saslaw, W. C. 1978‘Motion around a source whose luminosity changes’Astrophys. J.226240252CrossRefGoogle Scholar
  29. Schwarzschild, K.: 1916, ‘Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie’, Sitzber. Preuss. Akad. Wiss. Berlin, 189–196.Google Scholar
  30. Stoica, C., Mioc, V. 1997‘The Schwarzschild problem in astrophysics’Astrophys. Space Sci.249161173CrossRefGoogle Scholar
  31. Struwe, M. 1996Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian SystemsSpringer-VerlagBerlinGoogle Scholar
  32. Tonelli, L. 1915‘Sur une méthode directe du calcul des variations’Rend. Circ. Mat. Palermo39233263Google Scholar
  33. Vinti, J. P. 1972‘Possible effects of anisotropy of G on celestial orbits’Celest. Mech.6198207CrossRefGoogle Scholar
  34. Will, C. 1971‘Relativistic gravity in the solar system. II. Anisotropy in the Newtonian gravitational constant’Astrophys. J.169141155CrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Vasile Mioc
    • 1
  • Mira-Cristiana Anisiu
    • 2
  • Michael Barbosu
    • 3
  1. 1.Astronomical Institute of the Romanian AcademyBucharestRomania
  2. 2.T. Popoviciu Institute of Numerical Analysis of the Romanian AcademyCluj-NapocaRomania
  3. 3.SUNY Brockport, Department of MathematicsBrockportU.S.A.

Personalised recommendations