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Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the n-body problem

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Abstract

We use numerical continuation and bifurcation techniques in a boundary value setting to follow Lyapunov families of periodic orbits and subsequently bifurcating families. The Lyapunov families arise from the polygonal equilibrium of n bodies in a rotating frame of reference. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship, then the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, correspond to choreographies. We present a sample of the many choreographies that we have determined numerically along the Lyapunov families and along bifurcating families, namely for the cases \(n=3\), 4, and 6–9. We also present numerical results for the case where there is a central body that affects the choreography, but that does not participate in it. Animations of the families and the choreographies can be seen at the link below.

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Notes

  1. We say that \(q\prime \) is the n-modular inverse of q if \(q\prime q\) is congruent to 1 modulus n.

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Acknowledgements

We thank R. Montgomery, J. Montaldi, D. Ayala, and L. García-Naranjo for many interesting discussions. We also acknowledge the assistance of Ramiro Chavez Tovar with the preparation of figures and animations. This research was also supported by NSERC (Canada) Grant N00138. R. C. was partially supported by PAPIIT Project IA102818.

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Authors and Affiliations

  1. Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Mexico, Mexico

    Renato Calleja

  2. Department of Computer Science, Concordia University, Montreal, Canada

    Eusebius Doedel

  3. Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico, Mexico

    Carlos García-Azpeitia

Authors
  1. Renato Calleja
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  2. Eusebius Doedel
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  3. Carlos García-Azpeitia
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Correspondence to Renato Calleja.

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http://mym.iimas.unam.mx/renato/choreographies/index.html

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Calleja, R., Doedel, E. & García-Azpeitia, C. Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the n-body problem. Celest Mech Dyn Astr 130, 48 (2018). https://doi.org/10.1007/s10569-018-9841-9

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