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The inverse problem for homothetic polygonal central configurations

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Abstract

We prove that, for some potentials (including the Newtonian one and the potential of Helmholtz vortices in the plane), central configurations with nonzero total mass consisting of two homothetic polygons of arbitrary size can only occur if the masses at each polygon are equal. The same result is true for many polygons as long as the ratios between the radii of the polygons are sufficiently large.

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Notes

  1. Note that \(\mathbf {v}_{N-1}=\omega _{N-1}\mathbf {V}_{N}\), where \(\mathbf {V}_{N}\) is the vector on the referred lemma, and \(\tilde{a}\mathbf {V}_{N}\in \mathbb {R}^N\) only if \(\tilde{a}=0\) or \(N=2\).

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Acknowledgements

The author would like to thank Eduardo S. G. Leandro for being the advisor on this work and Thiago Dias for the helpful comments, as well as the Department of Mathematics at Universidade Federal Rural de Pernambuco for their assistance. We would like to thank the anonymous referees for their helpful comments and suggestions that improved an earlier version of this paper.

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  1. Depto de Matemática, Universidade Federal Rural de Pernambuco, Rua Dom Manoel de Medeiros, S/N, Dois Irmãos, Recife, PE, 52171-900, Brazil

    Marcelo P. Santos

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Correspondence to Marcelo P. Santos.

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Santos, M.P. The inverse problem for homothetic polygonal central configurations. Celest Mech Dyn Astr 131, 17 (2019). https://doi.org/10.1007/s10569-019-9896-2

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  • DOI: https://doi.org/10.1007/s10569-019-9896-2

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