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This article is cited in 6 scientific papers (total in 6 papers)
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
 References:
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Document Type:
Article
MSC: 70F15, 01A85
Received: 17.08.2016
Accepted:13.09.2016
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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\paper The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity
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\issue 5
\pages 556--580
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Citing articles on Google Scholar:
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This publication is cited in the following articles:
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A. V. Borisov, I. S. Mamaev, “Rigid body dynamics in non-euclidean spaces”, Russ. J. Math. Phys., 23:4 (2016), 431–454
-
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
 -
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
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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
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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
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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
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