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Symmetric Central Configurations and the Inverse Problem

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Abstract

We consider the problem: given a configuration, find the masses which make it central. This is the inverse problem for central configurations. The main result is a symmetry theorem (see Theorem 1) that allows simplifying the equations for symmetric central configurations, with any symmetry group, at any dimension, using linear representations of finite groups. Using this theorem, one can study the existence of central configurations with a given symmetry group and also study the masses allowed in this central configuration. As an application, we determine the masses when the configurations are the Platonic solids. The simplifications significantly reduce the equations allowing to use of a computer algebra system to make exact computations. In this case, we prove that a central configuration is possible if and only if the masses are equal (see Theorem 5). Except for the Tetrahedron that is known to be a central configuration for any masses.

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Notes

  1. This value for c coincides with the one found in [24] for the square. Compare with Theorem 1, in that reference, and note that by our Proposition 2.1, we have \(c=\frac{\lambda }{M}\), so in the notation of [24], \(c=\frac{\omega ^ 2}{M}=\frac{\gamma }{N}\).

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Acknowledgements

The author would like to thank Eduardo S. Leandro for suggesting that representation theory could be helpful in studying central configurations, as well as many insightful comments and suggestions. Also, thanks to Thiago Dias and Leon D. Silva for the helpful comments and suggestions. And thanks to the anonymous reviewers whose suggestions improved an earlier version of this paper and the Department of Mathematics at Universidade Federal Rural de Pernambuco for their assistance.

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  1. Depto de Matemática, Universidade Federal Rural de Pernambuco, Rua Dom Manoel de Medeiros, Recife, PE, Brasil

    Marcelo P. Santos

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Correspondence to Marcelo P. Santos.

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Santos, M.P. Symmetric Central Configurations and the Inverse Problem. J Dyn Diff Equat 36, 209–229 (2024). https://doi.org/10.1007/s10884-021-10123-0

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