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On the central configurations in the spatial 5-body problem with four equal masses

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Abstract

We analyze the families of central configurations of the spatial 5-body problem with four masses equal to 1 when the fifth mass m varies from 0 to \(+\infty \). In particular we continue numerically, taking m as a parameter, the central configurations (which all are symmetric) of the restricted spatial (\(4+1\))-body problem with four equal masses and \(m=0\) to the spatial 5-body problem with equal masses (i.e. \(m=1\)), and viceversa we continue the symmetric central configurations of the spatial 5-body problem with five equal masses to the restricted (\(4+1\))-body problem with four equal masses. Additionally we continue numerically the symmetric central configurations of the spatial 5-body problem with four equal masses starting with \(m=1\) and ending in \(m=+\infty \), improving the results of Alvarez-Ramírez et al. (Discrete Contin Dyn Syst Ser S 1: 505–518, 2008). We find four bifurcation values of m where the number of central configuration changes. We note that the central configurations of all continued families varying m from 0 to \(+\infty \) are symmetric.

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Acknowledgments

The first author is supported by the project grant Red de cuerpos académicos Ecuaciones Diferenciales. Proyecto sistemas dinámicos y estabilización. PROMEP 2011-SEP, Mexico. The second author is partially supported by MINECO Grant Number MTM2013-40998-P. The third author is partially supported by MINECO grant number MTM2013-40998-P, by AGAUR Grant Number 2014SGR 568, two FP7-PEOPLE-2012-IRSES Numbers 316338 and 318999, and MINECO/FEDER Grant Number UNAB10-4E-378. We thank to the anonymous reviewer for his/her valuable comments and suggestions.

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Authors and Affiliations

  1. Departamento de Matemáticas, UAM-Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, Iztapalapa, 09340, Mexico, D.F., Mexico

    Martha Alvarez-Ramírez

  2. Departament de Tecnologies Digitals i de la Informació, U Science Tech, Universitat de Vic-Universitat Cental de Catalunya (UVic-UCC), C/. Laura, 13, 08500, Vic, Barcelona, Spain

    Montserrat Corbera

  3. Departament de Matemàtiques, Edifici C, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain

    Jaume Llibre

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  1. Martha Alvarez-Ramírez
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  2. Montserrat Corbera
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  3. Jaume Llibre
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Correspondence to Montserrat Corbera.

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Alvarez-Ramírez, M., Corbera, M. & Llibre, J. On the central configurations in the spatial 5-body problem with four equal masses. Celest Mech Dyn Astr 124, 433–456 (2016). https://doi.org/10.1007/s10569-015-9670-z

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  • DOI: https://doi.org/10.1007/s10569-015-9670-z

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