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Families of Symmetric Exchange Orbits in the Planar \((1+2n)\)-Body Problem

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Abstract

We study a particular \((1+2n)\)-body problem, conformed by a massive body and 2n equal small masses, since this problem is related with Maxwell’s ring solutions, we call planet to the massive body, and satellites to the other 2n masses. Our goal is to obtain doubly-symmetric orbits in this problem. By means of studying the reversing symmetries of the equations of motion, we reduce the set of possible initial conditions that leads to such orbits, and compute the 1-parameter families of time-reversible invariant tori. The initial conditions of the orbits were determined as solutions of a boundary value problem with one free parameter, in this way we find analytically and explicitly a new involution, until we know this is a new and innovative result. The numerical solutions of the boundary value problem were obtained using pseudo arclength continuation. For the numerical analysis we have used the value of \(3.5 \times 10 ^{-4}\) as mass ratio of some satellite and the planet, and it was done for \(n=2,3,4,5,6\). We show numerically that the succession of families that we have obtained approach the Maxwell solutions as n increases, and we establish a simple proof why this should happen in the configuration.

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Notes

  1. We have chosen this value in order to compare the results of this work with our previous studies.

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Acknowledgements

We thank to the anonymous referee for pointing out some references that we had missed, which help us in the drafting of the new manuscript. The first and third author have been partially supported from Asociación Mexicana de Cultura A.C., the National System of Researchers (SNI), and Conacyt-México Project A1S10112. The second author wishes to acknowledge financial support from the Spanish government through grants PGC2018-096265-B-I00 and PGC2018-100680-B-C21. We would like to thank Rafael Ortega for valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, ITAM, Río Hondo 1, 01080, Ciudad de México, Mexico

    Abimael Bengochea & Ernesto Pérez-Chavela

  2. Departamento de Matemática Aplicada II, Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Sevilla, Spain

    Jorge Galán-Vioque

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  1. Abimael Bengochea
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  3. Ernesto Pérez-Chavela
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Correspondence to Ernesto Pérez-Chavela.

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Bengochea, A., Galán-Vioque, J. & Pérez-Chavela, E. Families of Symmetric Exchange Orbits in the Planar \((1+2n)\)-Body Problem. Qual. Theory Dyn. Syst. 20, 34 (2021). https://doi.org/10.1007/s12346-021-00473-x

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