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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The number of planar central configurations for the $4$-body problem is finite when $3$ mass positions are fixed


Authors: Martha Alvarez, Montserrat Corbera, Joaquin Delgado and Jaume Llibre
Journal: Proc. Amer. Math. Soc. 133 (2005), 529-536
MSC (2000): Primary 70F07; Secondary 70F15
Published electronically: August 26, 2004
MathSciNet review: 2093077
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Abstract: In the $n$-body problem a central configuration is formed when the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration vector. Lindstrom showed for $n=3$ and for $n>4$ that if $n-1$masses are located at fixed points in the plane, then there are only a finite number of ways to position the remaining $n$th mass in such a way that they define a central configuration. Lindstrom leaves open the case $n=4$. In this paper we prove the case $n=4$using as variables the mutual distances between the particles.


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Additional Information

Martha Alvarez
Affiliation: Departamento de Matemáticas, UAM–Iztapalapa, A.P. 55–534, 09340 Iztapalapa, Mexico, D.F., Mexico
Email: mar@xanum.uam.mx

Montserrat Corbera
Affiliation: Departament d’Informàtica i Matemàtiques, Universitat de Vic, 08500 Vic, Barcelona, Spain
Email: montserrat.corbera@uvic.es

Joaquin Delgado
Affiliation: Departamento de Matemáticas, UAM–Iztapalapa, A.P. 55–534, 09340 Iztapalapa, Mexico, D.F., Mexico
Email: jdf@xanum.uam.mx

Jaume Llibre
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Email: jllibre@mat.uab.es

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07590-2
PII: S 0002-9939(04)07590-2
Keywords: $4$--body problem, central configurations
Received by editor(s): November 12, 2002
Received by editor(s) in revised form: February 6, 2003, and July 16, 2003
Published electronically: August 26, 2004
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2004 American Mathematical Society