Regular and Chaotic Dynamics

, Volume 21, Issue 5, pp 556–580 | Cite as

The spatial problem of 2 bodies on a sphere. Reduction and stochasticity

  • Alexey V. Borisov
  • Ivan S. Mamaev
  • Ivan A. Bizyaev
Article

Abstract

In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature S2 and S3. We perform a reduction (analogous to that in rigid body dynamics) after which the problem reduces to analysis of a two-degree-of-freedom system. In the general case, in canonical variables the Hamiltonian does not correspond to any natural mechanical system. In addition, in the general case, the absence of an analytic additional integral follows from the constructed Poincaré section. We also give a review of the historical development of celestial mechanics in spaces of constant curvature and formulate open problems.

Keywords

celestial mechanics space of constant curvature reduction rigid body dynamics Poincaré section 

MSC2010 numbers

70F15 01A85 

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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • Alexey V. Borisov
    • 1
  • Ivan S. Mamaev
    • 1
  • Ivan A. Bizyaev
    • 1
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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