Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Stability of the planetary three-body problem

II. Kam theory and existence of quasiperiodic motions

  • 146 Accesses

  • 56 Citations

Abstract

Using new expansions of the planetary Hamiltonian in Poincaré canonical elliptic variables, Arnold's theorem for the existence of quasiperiodic orbits in degenerated cases is applied to the general spatial planetary three body problem. The existence of quasiperiodic motion is demonstrated for almost all values of the ratio of semi-major axis α in ]0, 0.8] and almost all values of the mutual inclination up to about 1 degree. This extends the previous result of Arnold (1963).

This is a preview of subscription content, log in to check access.

References

  1. Arnold, V.: 1963a, ‘Proof of Kolmogorov's theorem on the preservation of quasi-periodic motions under small perturbations of the Hamiltonian’Russ. Math. Survey 18, N6 9–36.

  2. Arnold, V.: 1963b, ‘Small denominators’,Russ. Math. Survey 18, N6 85–192.

  3. Celletti, A.: 1990a, ‘Analysis of resonances in the spin-orbit problem in Celestial Mechanics: The synchronous resonance (Part I)’Journal of Applied Mathematics and Physics 41, 175–204.

  4. Celletti, A.: 1990b, ‘Analysis of resonances in the spin-orbit problem in Celestial Mechanics: Higher order resonances and some numerical experiments (Part II)’, ‘Journal of Applied Mathematics and Physics’41, 453–478.

  5. Celletti, A. and Chierchia, L.: 1988, ‘Construction of analytic KAM surfaces and effective stability bounds’Commun. Math. Phys. 188, 119–161.

  6. Chenciner, A.: 1989, ‘Séries de Lindstedt,Notes scientifiques et techniques du B.D.L. S028.

  7. Deprit, A. and Deprit-Bartholomé, A.: 1967, ‘Stability of the triangular Lagrangian points’Astron. J. 72, 173–179.

  8. Giorgilli, A., Delshams, A., Fontich, E., Galgani, L. and Simò, C.: 1989, ‘Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem’J. Diff. Eq. 77, 167–198.

  9. Hénon, M.: 1966, ‘Exploration numérique du problème restreint’,Bulletin Astronomique 1, 3e série.

  10. Jefferys, W.H. and Moser, J.: 1966, ‘Quasi-Periodic Solutions for the three-Boby Problem’Astron. J. 71 568–578.

  11. Kolmogorov, A.N.: 1954, ‘The conservation of conditionally periodic motion with a small variation in the hamiltonian’Dokl. Akad. Nauk SSSR 98, 527–530.

  12. Laskar, J.: 1989, ‘A numerical experiment on the chaotic behaviour of the solar system’Nature 338, 237–238.

  13. Laskar, J.: 1990a, ‘The chaotic motion of the solar system: a numerical estimate of the size of the chaotic zones’,Icarus 88, 266–291.

  14. Laskar, J.: 1990b, ‘Manipulation des séries’, inLes méthodes modernes de la mécanique celeste. D. Benest, C. Froeschele (eds).Frontières C36, 89–107.

  15. Laskar, J.: 1990c, ‘Systèmes de Variables et Eléments’, inLes méthodes modernes de la mécanique celeste. D. Benest, C. Froeschele (eds).Frontières C36, 63–87.

  16. Laskar, J.: 1991, ‘Analytical framework in Poincaré variables for the motion of the solar system’, in Predictability, Stability, and Chaos in N-Body Dynamical Systems. A.Roy (ed),Kluwer Publ B272, 93–114.

  17. Laskar, J.: 1992, ‘La stabilité du système solaire’, inChaos et déterminisme. A. Dahan Dalmedico, J.L. Chabert, K. Chemla (eds),Seuil,S80, 170–211.

  18. Laskar, J.: 1994a, ‘Description des routines utilisateur de TRIP 0.8’, preprint.

  19. Laskar, J.: 1994b, ‘Large scale chaos in the solar system’,Astron. Astrophys. 278, L9-L12.

  20. Laskar, J., Robutel P.: 1995, ‘Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian’,Celestial Mechanics 62, 193–217 (this issue).

  21. Léontovitch, A. M.: 1962, ‘On the stability of Lagrange's periodic solutions of the restricted three-body problem’Soviet Math. 3, 425–429.

  22. Le Verrier, U.J.: 1855,Annales de l'Observatoire Impérial de Paris, Tome I, Mallet-Bachelier, Paris.

  23. Lieberman, B.B.: 1971, ‘Existence of quasi-periodic solutions to the three-body problem’Celestial Mechanics 3, 408–426.

  24. Meyer, K.R., Hall, G.R.: 1991, ‘Introduction to Hamiltonian dynamical systems and N-body problem’, Springer-VerlagAp. Math. Sc. 90.

  25. Meyer, K.R. and Schmidt, D.S.: 1986, ‘The stability of the Lagrange triangular point and the theorem of Arnold’J. Diff. Eq. 62, 222–236.

  26. Moser, J.: 1962, ‘On invariant curves of area-preserving mapping of an annulus’Nachr. Akad. Wiss. Göttingen Math. Phys. K1, 1–20.

  27. Nejshtadt, A. I.: 1982, ‘Estimates in the Kolmogorov theorem on conservation of conditionally periodic motion’J. Appl. Math. Mech. 45, 766–772.

  28. Nejshtadt, A. I.: 1984, ‘The separtion of motions in systems with rapidly rotating phase,J. Appl. Math. Mech. 48, 133–139.

  29. Niederman, L.: 1993, ‘Résonances et stabilité dans le probléme planétaire’, Thèse, Univ. Paris VI.

  30. Poincaré, H.: 1892,‘Méthodes Nouvelles de la Mécanique Céleste’,t.I. Gauthier Villars, Paris, reprinted by Blanchard, 1987.

  31. Poincaré, H.: 1907, ‘Leçons de mécanique céleste’ Gauthier Villars, Paris.

  32. Press, W., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T.: 1988, ‘Numerical recipes’, Cambridge University Press.

  33. Robutel, P.: 1993, ‘Contribution à l'étude de la stabilité du problème planétaire des trois corps’ Thèse, Observatoire de Paris.

  34. Sussman, G.J. and Wisdom, J.: 1992, ‘Chaotic evolution of the solar system’,Science 57 56–62.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Robutel, P. Stability of the planetary three-body problem. Celestial Mech Dyn Astr 62, 219–261 (1995). https://doi.org/10.1007/BF00692089

Download citation

Key words

  • Planetary motion
  • perturbation theory
  • KAM theory
  • celestial mechanics
  • stability of planetary system