Eulerian relative equilibria of the curved 3-body problems in 𝐒²

Zhu, Shuqiang

doi:10.1090/S0002-9939-2014-11995-2

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

Proc. Amer. Math. Soc. 142 (2014), 2837-2848

DOI: https://doi.org/10.1090/S0002-9939-2014-11995-2

Published electronically: May 7, 2014

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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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