The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity

    2016, Volume 21, Number 5, pp.  556-580

    Author(s): Borisov A. V., Mamaev I. S., Bizyaev I. A.

    In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature $S^2$ and $S^3$. 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
    Citation: Borisov A. V., Mamaev I. S., Bizyaev I. A., The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity, Regular and Chaotic Dynamics, 2016, Volume 21, Number 5, pp. 556-580

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