On the existence of bi--pyramidal central configurations of the$n+2$--body problem   with an $n$--gon base

Montserrat Corbera; Jaume Llibre

doi:10.3934/dcds.2013.33.1049

Article Contents

2013, Volume 33, Issue 3: 1049-1060. Doi: 10.3934/dcds.2013.33.1049

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On the existence of bi--pyramidal central configurations of the $n+2$--body problem with an $n$--gon base

Montserrat Corbera^1, and
Jaume Llibre^2,

1.
Departament de Tecnologies Digitals i de la Informació, Universitat de Vic, C/. Laura, 13, 08500 Vic, Barcelona, Catalonia
2.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia

Received: March 31, 2011

Revised: November 30, 2011

Published: February 2013

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Abstract

In this paper we prove the existence of central configurations of the $n+2$--body problem where $n$ equal masses are located at the vertices of a regular $n$--gon and the remaining $2$ masses, which are not necessarily equal, are located on the straight line orthogonal to the plane containing the $n$--gon passing through its center. Here this kind of central configurations is called bi--pyramidal central configurations. In particular, we prove that if the masses $m_{n+1}$ and $m_{n+2}$ and their positions satisfy convenient relations, then the configuration is central. We give explicitly those relations.

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Mathematics Subject Classification: Primary: 70F10; Secondary: 70F15.

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References

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