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Classifying four-body convex central configurations

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Abstract

We classify the full set of convex central configurations in the Newtonian planar four-body problem. Particular attention is given to configurations possessing some type of symmetry or defining geometric property. Special cases considered include kite, trapezoidal, co-circular, equidiagonal, orthodiagonal, and bisecting-diagonal configurations. Good coordinates for describing the set are established. We use them to prove that the set of four-body convex central configurations with positive masses is three-dimensional, a graph over a domain D that is the union of elementary regions in \({\mathbb {R}}^{+^3}\).

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Notes

  1. Thanks to Richard Montgomery for suggesting this idea to the third author at the 2018 Joint Math Meetings.

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Acknowledgements

M. Corbera and J. M. Cors were partially supported by MINECO Grant MTM2016-77278-P(FEDER); J. M. Cors was also supported by AGAUR Grant 2017 SGR 1617. We also wish to thank John Little, Richard Montgomery, and the two referees for insightful discussions regarding this work.

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Authors and Affiliations

  1. Departament de Tecnologies Digitals i de la Informació, Universitat de Vic, Vic, Spain

    Montserrat Corbera

  2. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

    Josep M. Cors

  3. Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, USA

    Gareth E. Roberts

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  1. Montserrat Corbera
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  2. Josep M. Cors
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  3. Gareth E. Roberts
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Correspondence to Gareth E. Roberts.

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Corbera, M., Cors, J.M. & Roberts, G.E. Classifying four-body convex central configurations. Celest Mech Dyn Astr 131, 34 (2019). https://doi.org/10.1007/s10569-019-9911-7

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