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 Regul. Chaotic Dyn., 2016, Volume 21, Issue 5, Pages 556–580 (Mi rcd205)

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

Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev

Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: 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

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work was supported by the Russian Scientific Foundation (project No. 14–50–00005).

DOI: https://doi.org/10.1134/S1560354716050075

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Document Type: Article
MSC: 70F15, 01A85
Language: English

Citation: Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity”, Regul. Chaotic Dyn., 21:5 (2016), 556–580

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/rcd/v21/i5/p556

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. V. Borisov, I. S. Mamaev, “Rigid body dynamics in non-euclidean spaces”, Russ. J. Math. Phys., 23:4 (2016), 431–454
2. Miguel A. Gonzalez Leon, Juan Mateos Guilarte, Marina de la Torre Mayado, “Orbits in the Problem of Two Fixed Centers on the Sphere”, Regul. Chaotic Dyn., 22:5 (2017), 520–542
3. A. A. Oshemkov, P. E. Ryabov, S. V. Sokolov, “Explicit determination of certain periodic motions of a generalized two-field gyrostat”, Russ. J. Math. Phys., 24:4 (2017), 517–525
4. A. V. Borisov, L. C. Garcia-Naranjo, I. S. Mamaev, J. Montaldi, “Reduction and relative equilibria for the two-body problem on spaces of constant curvature”, Celest. Mech. Dyn. Astron., 130:6 (2018), UNSP 43
5. Barry K. Carpenter, Gregory S. Ezra, Stavros C. Farantos, Zeb C. Kramer, Stephen Wiggins, “Dynamics on the Double Morse Potential: A Paradigm for Roaming Reactions with no Saddle Points”, Regul. Chaotic Dyn., 23:1 (2018), 60–79
6. Jaime Andrade, Claudio Vidal, “Stability of the Polar Equilibria in a Restricted Three-body Problem on the Sphere”, Regul. Chaotic Dyn., 23:1 (2018), 80–101