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Eulerian relative equilibria of the curved $ 3$-body problems in $ \mathbf{S}^2$

Author: Shuqiang Zhu
Journal: Proc. Amer. Math. Soc. 142 (2014), 2837-2848
MSC (2010): Primary 70-XX; Secondary 70F07, 70F15
Published electronically: May 7, 2014
MathSciNet review: 3209337
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Abstract: We consider the gravitational motion of $ n$ point particles with masses $ m_1$, $ m_2$, $ \ldots $, $ m_n>0$ on surfaces of constant Gaussian curvature. Based on the work of Diacu and his co-authors, we derive the law of universal gravitation in spaces of constant curvature. Using the results, we examine all possible $ 3$-body configurations that can generate geodesic relative equilibria. We prove the existence of all acute triangle Eulerian relative equilibria and get a necessary and sufficient condition for the existence of obtuse triangle Eulerian relative equilibria. We also show that any three positive masses can generate Eulerian relative equilibria.

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Additional Information

Shuqiang Zhu
Affiliation: College of Mathematics and Yangtze Center, Sichuan University, Chengdu 610064, People’s Republic of China

Keywords: $3$-body problem, constant curvature spaces, law of universal gravitation, Eulerian relative equilibria, acute triangle configuration, obtuse triangle configuration, equilateral triangle configuration, isosceles triangle configuration
Received by editor(s): July 23, 2012
Published electronically: May 7, 2014
Communicated by: Walter Craig
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.