Celestial Mechanics and Dynamical Astronomy

, Volume 96, Issue 1, pp 1–17 | Cite as

The restricted two-body problem in constant curvature spaces

  • Alexey V. Borisov
  • Ivan S. Mamaev
Research Article


We perform the bifurcation analysis of the Kepler problem on \(\mathbb{S}^{3}\) and \(\mathbb{H}^{3}\). An analog of the Delaunay variables is introduced. We investigate the motion of a point mass in the field of a Newtonian center moving along a geodesic on \(\mathbb{S}^{2}\) and \(\mathbb{H}^{2}\) (the restricted two-body problem). For the case of a small curvature, the pericenter shift is computed using the perturbation theory. We also present the results of numerical analysis based on an analogy with the motion of a rigid body.


Kepler problem Bifurcation analysis Perihelion shift Delaunay variables Restricted two-body problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Appell P. (1891) Sur les lois de forces centrales faisant décrire à leur point d’application une conique quelles que soient les conditions initiales. Am. J. Math. 13, 153–158CrossRefGoogle Scholar
  2. Arnold V.I., Kozlov V.V., Neishtadt A.I. (1993) Mathematical Aspects of Classical and Celestial Mechanics. Springer-Verlag, BerlinGoogle Scholar
  3. Borisov, A.V., Mamaev, I.S.: Poisson Structures and Lie Algebras in Hamiltonian Mechanics. Izhevsk, SPC “RCD” (in Russian) (1999)Google Scholar
  4. Borisov, A.V., Mamaev, I.S.: Rigid Body Dynamics. Izhevsk, SPC “RCD” (in Russian) (2001)Google Scholar
  5. Born M. (1925) Vorlesungen über Atommechanik. Berlin, SpringerzbMATHGoogle Scholar
  6. Chernikov N.A. (1992) The relativistic Kepler problem in the Lobachevsky space. Acta Phys. Polonica 24, 927–950Google Scholar
  7. Chernoïvan V.A., Mamaev I.S. (1999) The restricted two-body problem and the Kepler problem in the constant curvature spaces. Reg. Chaot. Dyn. 4(2): 112–124zbMATHCrossRefGoogle Scholar
  8. Demin, V.G., Kosenko, I.I., Krasilnikov P.S.: Selected problems of Celestial Mechanics. Izhevsk, SPC “RCD” (in Russian) (1999)Google Scholar
  9. Eddington A.S. (1963) Mathematical Theory of Relativity, 3rd edn. Cambridge University Press, Cambridge, EnglandGoogle Scholar
  10. Grebennikov E.A. (1986) Averaging Method in Applicative Problems. Nauka, Moscow (in Russian)Google Scholar
  11. Killing W. (1885) Die Mechanik in den nicht-euklidischen Raumformen. J. Reine Angew. Math. 98, 1–48Google Scholar
  12. Kozlov, V.V.: On dynamics in constant curvature spaces. Vestnik MGU, Ser. Math. Mech. 28–35 (1994)Google Scholar
  13. Liebmann H. (1903) Über die Zentralbewegung in der nichteuklidische Geometrie. Leipzig Berichte 55, 146–153Google Scholar
  14. Maciejewski A., Przybylska M. (2003) Non-integrability of restricted two body problem in constant curvature spaces. Reg. Chaot. Dyn. 8, 413–430zbMATHCrossRefGoogle Scholar
  15. Markeev A.P. (1990) Theoretical Mechanics. Nauka, Moscow, (in Russian)zbMATHGoogle Scholar
  16. Ziglin S.L. (2001) On non-integrability of the restricted two-body problem on a sphere. Doklady RAN 379(4): 477–478 (in Russian)Google Scholar
  17. Ziglin S.L. (2003) Non-integrability of a restricted two-body for an elastic potential on a sphere. Phys. Doklady 48(7): 353–354CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia

Personalised recommendations