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.
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
\Bibitem{BorMamBiz16}
\by Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev
\paper The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity
\jour Regul. Chaotic Dyn.
\yr 2016
\vol 21
\issue 5
\pages 556--580
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Linking options:
https://www.mathnet.ru/eng/rcd205
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Philip Arathoon, “Relative equilibria of mechanical systems with rotational symmetry”, Nonlinearity, 37:9 (2024), 095001
Toshiaki Fujiwara, Ernesto Pérez-Chavela, “Continuations and Bifurcations of Relative Equilibria for the Positively Curved Three-Body Problem”, Regul. Chaotic Dyn., 29:6 (2024), 803–824
Alain Albouy, “Note on the Attraction of an Ellipsoid in a Spherical Universe”, Proc. Steklov Inst. Math., 327 (2024), 12–19
Bengochea A., Garcia-Azpeitia C., Perez-Chavela E., Roldan P., “Continuation of Relative Equilibria in the N-Body Problem to Spaces of Constant Curvature”, J. Differ. Equ., 307 (2022), 137–159
Juan Manuel Sánchez-Cerritos, Liang Ding, Jinlong Wei, “Equilibrium points in restricted problems on S2 and H2”, Journal of Mathematical Physics, 63:6 (2022)
Juan Manuel Sánchez-Cerritos, Ernesto Pérez-Chavela, “Hyperbolic regularization of the restricted three–body problem on curved spaces”, Anal.Math.Phys., 12:1 (2022)
Garcia-Naranjo L.C. Montaldi J., “Attracting and Repelling 2-Body Problems on a Family of Surfaces of Constant Curvature”, J. Dyn. Differ. Equ., 33:4 (2021), 1579–1603
Tomasz Stachowiak, Andrzej J. Maciejewski, “Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group”, SIGMA, 17 (2021), 074, 12 pp.
Jackman C., “Secular Dynamics For Curved Two-Body Problems”, J. Dyn. Differ. Equ., 2021
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E. A. Malkov, A. A. Bekov, S. B. Momynov, I. B. Beckmuhamedov, D. M. Kurmangaliyev, A. M. Mukametzhan, I. S. Orynqul, “Investigation of two fixed centers problem and Henon-Heiles potential based on the Poincare section”, News Natl. Acad. Sci. Rep. Kazakhstan-Ser. Phys.-Math., 1:329 (2020), 55–61
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Philip Arathoon, “Singular Reduction of the 2-Body Problem on the 3-Sphere and the 4-Dimensional Spinning Top”, Regul. Chaotic Dyn., 24:4 (2019), 370–391
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D. G. Dritschel, “Point mass dynamics on spherical hypersurfaces”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 377:2158 (2019), 20180349
E. Perez-Chavela, J. M. Sanchez-Cerritos, “Regularization of the restricted (n+1)-body problem on curved spaces”, Astrophys. Space Sci., 364:10 (2019), 170
W. Szuminski, “On certain integrable and superintegrable weight-homogeneous Hamiltonian systems”, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 600–616
Miguel A. González León, Juan Mateos Guilarte, Marina de la Torre Mayado, Integrability, Supersymmetry and Coherent States, 2019, 359
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