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The central configurations of four masses x, -x, y, -y

Martin Celli
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Abstract

The configuration of a homothetic motion in the N-body problem is called a central configuration. In this paper, we prove that there are exactly three planar non-collinear central configurations for masses x, -x, y, -y with x different from y (a parallelogram and two trapezoids) and two planar non-collinear central configurations for masses x, -x, x, -x (two diamonds). Except the case studied here, the only known case where the four-body central configurations with non-vanishing masses can be listed is the case with equal masses (Albouy, 1996), which requires the use of a symbolic computation program. Thanks to a lemma used in the proof of our result, we also show that a co-circular four-body central configuration has non-vanishing total mass or vanishing multiplier.

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Martin Celli. 2006-09-20. The central configurations of four masses x, -x, y, -y. https://arxiv.org/abs/math/0609578

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