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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Polygonal homographic orbits of the curved $ n$-body problem

Author: Florin Diacu
Journal: Trans. Amer. Math. Soc. 364 (2012), 2783-2802
MSC (2010): Primary 70F10; Secondary 34C25, 37J45
Published electronically: December 1, 2011
MathSciNet review: 2888228
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Abstract: In the $ 2$-dimensional $ n$-body problem, $ n\ge 3$, in spaces of constant curvature, $ \kappa \ne 0$, we study polygonal homographic solutions. We first provide necessary and sufficient conditions for the existence of these orbits and then consider the case of regular polygons. We further use this criterion to show that, for any $ n\ge 3$, the regular $ n$-gon is a polygonal homographic orbit if and only if all masses are equal. Then we prove the existence of relative equilibria of nonequal masses on the sphere of curvature $ \kappa >0$ for $ n=3$ in the case of scalene triangles. Such triangular relative equilibria occur only along fixed geodesics and are generated from fixed points of the sphere. Finally, through a classification of the isosceles case, we prove that not any choice of three masses can form a triangular relative equilibrium.

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Florin Diacu
Affiliation: Pacific Institute for the Mathematical Sciences – and – Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, British Columbia, Canada V8W 3R4

Received by editor(s): October 26, 2010
Received by editor(s) in revised form: January 20, 2011
Published electronically: December 1, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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