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

Alvarez, Martha; Corbera, Montserrat; Delgado, Joaquin; Llibre, Jaume

doi:10.1090/S0002-9939-04-07590-2

The number of planar central configurations for the $4$–body problem is finite when $3$ mass positions are fixed
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by Martha Alvarez, Montserrat Corbera, Joaquin Delgado and Jaume Llibre PDF

Proc. Amer. Math. Soc. 133 (2005), 529-536 Request permission

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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