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Convex but not Strictly Convex Central Configurations

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Abstract

Central configurations of the n-body problem have been studied for more than 200 years since the pioneer works of Euler and Lagrange. In this article we study convex central configurations which are not strictly convex. We give explicit examples of such configurations in both planar and spatial n-body problems. Particularly, in the spatial case, we consider regular polyhedra with bodies of same mass m at the vertices and bodies of same mass M at the middle points of each edge. In this setting we prove that the cube is the unique regular polyhedron such that this construction leads to a convex central configuration which is not strictly convex.

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Acknowledgements

The authors are partially supported by FAPEMIG Grant APQ-001082/14. The third author is partially supported by FAPEMIG Grant PPM-00516-15.

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Authors and Affiliations

  1. Instituto de Matemática e Computação, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, Itajubá, MG, CEP 37500–903, Brazil

    Antonio Carlos Fernandes, Braulio Augusto Garcia & Luis Fernando Mello

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  1. Antonio Carlos Fernandes
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  2. Braulio Augusto Garcia
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  3. Luis Fernando Mello
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Correspondence to Antonio Carlos Fernandes.

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This article is dedicated to Professor Jorge Sotomayor on his 75th birthday.

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Fernandes, A.C., Garcia, B.A. & Mello, L.F. Convex but not Strictly Convex Central Configurations. J Dyn Diff Equat 30, 1427–1438 (2018). https://doi.org/10.1007/s10884-017-9596-0

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  • DOI: https://doi.org/10.1007/s10884-017-9596-0

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