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A note on the Dziobek central configurations

Author: Jaume Llibre
Journal: Proc. Amer. Math. Soc. 143 (2015), 3587-3591
MSC (2010): Primary 70F07; Secondary 70F15
Published electronically: February 16, 2015
MathSciNet review: 3348799
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Abstract: For the Newtonian $ n$-body problem in $ \mathbb{R}^{n-2}$ with $ n\geq 3$ we prove that the following two statements are equivalent.

Let $ x$ be a Dziobek central configuration having one mass located at the center of mass.

Let $ x$ be a central configuration formed by $ n-1$ equal masses located at the vertices of a regular $ (n-2)$-simplex together with an arbitrary mass located at its barycenter.

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Jaume Llibre
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

Keywords: Central configuration, $n$-body problem
Received by editor(s): November 27, 2011
Received by editor(s) in revised form: March 16, 2014
Published electronically: February 16, 2015
Additional Notes: The first author is partially supported by a MINECO/FEDER grant MTM2008-03437 and MTM2013-40998-P, an AGAUR grant number 2014SGR-568, an ICREA Academia, the grants FP7-PEOPLE-2012-IRSES 318999 and 316338, grant UNAB13-4E-1604.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2015 American Mathematical Society

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