Abstract
We study relative equilibria (\(RE\)) for the three-body problem on \(\mathbb{S}^{2}\), under the influence of a general potential which only depends on \(\cos\sigma_{ij}\) where \(\sigma_{ij}\) are the mutual angles among the masses. Explicit conditions for masses \(m_{k}\) and \(\cos\sigma_{ij}\) to form relative equilibrium are shown. Using the above conditions, we study the equal masses case under the cotangent potential. We show the existence of scalene, isosceles, and equilateral Euler \(RE\), and isosceles and equilateral Lagrange \(RE\). We also show that the equilateral Euler \(RE\) on a rotating meridian exists for general potential \(\sum_{i<j}m_{i}m_{j}U(\cos\sigma_{ij})\) with any mass ratios.
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29 October 2023
The numbering issue has been changed to 4-5.
References
Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Spatial Problem of \(2\) Bodies on a Sphere: Reduction and Stochasticity, Regul. Chaotic Dyn., 2016, vol. 21, no. 5, pp. 556–580.
Borisov, A. V., Mamaev, I. S., and Kilin, A. A., Two-Body Problem on a Sphere: Reduction, Stochasticity, Periodic Orbits, Regul. Chaotic Dyn., 2004, vol. 9, no. 3, pp. 265–279.
Diacu, F., Pérez-Chavala, E., and Santoprete, M., The \(n\)-Body Problem in Spaces of Constant Curvature: Part 1. Relative Equilibria, J. Nonlinear Sci., 2012, vol. 22, no. 2, pp. 247–266.
Diacu, F., Relative Equilibria of the Curved \(N\)-Body Problem, Atlantis Studies in Dynamical Systems, vol. 1, Amsterdam: Atlantis, 2012.
Diacu, F. and Pérez-Chavala, E., Homographic Solutions of the Curved \(3\)-Body Problem, J. Differential Equations, 2011, vol. 250, no. 1, pp. 340–366.
Diacu, F. and Zhu, Sh., Almost All \(3\)-Body Relative Equilibria on \(\mathbb{S}^{2}\) and \(\mathbb{H}^{2}\) Are Inclined, Discrete Contin. Dyn. Syst. Ser. S, 2020, vol. 13, no. 4, pp. 1131–1143.
Euler, L., De motu rectilineo trium corporum se mutuo attrahentium, Novi Comm. Acad. Sci. Imp. Petrop., 1767, vol. 11, pp. 144–151.
Goldstein, H., Poole, Ch. P., Jr., and Safko, J. L., Classical Mechanics, 3rd ed., Reading, Mass.: Addison-Wesley, 2001.
Hestenes, D., New Foundations for Classical Mechanics, 2nd ed., Fundam. Theor. Phys., vol. 99, Dordrecht: Kluwer, 1999.
Landau, L. D. and Lifshitz, E. M., Course of Theoretical Physics: Vol. 1. Mechanics, 3rd ed., Oxford: Pergamon, 1976.
Marsden, J. E., Lectures on Mechanics, London Math. Soc. Lecture Note Ser., vol. 174, Cambridge: Cambridge Univ. Press, 1992.
Martínez, R. and Simó, C., On the Stability of the Lagrangian Homographic Solutions in a Curved Three-Body Problem on \(\mathbb{S}^{2}\), Discrete Contin. Dyn. Syst. Ser. A, 2013, vol. 33, no. 3, pp. 1157–1175.
Moeckel, R., Notes on Celestial Mechanics: Especially Central Configurations, http://www.math.umn.edu/~rmoeckel/notes/Notes.html (2014).
Pérez-Chavela, E. and Reyes-Victoria, J. G., An Intrinsec Approach in the Curved \(n\)-Body Problem. The Positive Curvature Case, Trans. Amer. Math. Soc., 2012, vol. 364, no. 7, pp. 3805–3827.
Pérez-Chavela, E. and Sánchez-Cerritos, J. M., Euler-Type Relative Equilibria and Their Stability in Spaces of Constant Curvature, Canad. J. Math., 2018, vol. 70, no. 2, pp. 426–450.
Routh, E. J., An Elementary Treatise on the Dynamics of a System of Rigid Bodies, Cambridge: Cambridge Univ. Press, 1860.
Shchepetilov, A. V., Nonintegrability of the Two-Body Problem in Constant Curvature Spaces, J. Phys. A, 2006, vol. 39, no. 20, pp. 5787–5806.
Tibboel, P., Polygonal Homographic Orbits in Spaces of Constant Curvature, Proc. Amer. Math. Soc., 2013, vol. 141, no. 4, pp. 1465–1471.
Wintner, A., The Analytical Foundations of Celestial Mechanics, Princeton Math. Ser., vol. 5, Princeton, N.J.: Princeton Univ. Press, 1941.
Zhu, Sh., Eulerian Relative Equilibria of the Curved \(3\)-Body Problems in \(\mathbb{S}^{2}\), Proc. Amer. Math. Soc., 2014, vol. 142, no. 8, pp. 2837–2848.
ACKNOWLEDGMENTS
Thanks to our friend Florin Diacu, who was the inspiration of this work. We are grateful to the anonymous referees for several helpful comments which have helped us to improve this manuscript.
Funding
The second author (EPC) has been partially supported by Asociación Mexicana de Cultura A.C. and Conacyt-México Project A1S10112.
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MSC2010
70F07, 70F10, 70F15
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Fujiwara, T., Pérez-Chavela, E. Three-Body Relative Equilibria on \(\mathbb{S}^{2}\). Regul. Chaot. Dyn. 28, 690–706 (2023). https://doi.org/10.1134/S1560354723040111
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DOI: https://doi.org/10.1134/S1560354723040111