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Equilateral Chains and Cyclic Central Configurations of the Planar Five-Body Problem

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Abstract

Central configurations and relative equilibria are an important facet of the study of the N-body problem, but become very difficult to rigorously analyze for \(N>3\). In this paper, we focus on a particular but interesting class of configurations of the five-body problem: the equilateral pentagonal configurations, which have a cycle of five equal edges. We prove a variety of results concerning central configurations with this property, including a computer-assisted proof of the finiteness of such configurations for any positive five masses with a range of rational-exponent homogeneous potentials (including the Newtonian case and the point-vortex model), some constraints on their shapes, and we determine some exact solutions for particular N-body potentials.

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All data generated or analyzed during this study are included in this published article.

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Acknowledgements

The authors would like to thank Manuele Santoprete for the suggestion to study this class of configuration. Yiyang Deng was partially supported by the Mathematics and Statistics Team from Chongqing Technology and Business University (ZDPTTD201906).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China

    Yiyang Deng

  2. Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN, 55812, USA

    Marshall Hampton

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  1. Yiyang Deng
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  2. Marshall Hampton
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Correspondence to Marshall Hampton.

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Deng, Y., Hampton, M. Equilateral Chains and Cyclic Central Configurations of the Planar Five-Body Problem. J Nonlinear Sci 33, 4 (2023). https://doi.org/10.1007/s00332-022-09864-z

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