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Three topologically nontrivial choreographic motions of three bodies

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Abstract

Choreographies are periodic orbits in which all bodies move on the same trajectory with equal time delay. The best known three-body choreography is figure-eight orbit. Here we introduce a search method specialized for choreographies and present three new orbits with vanishing angular momentum that are the first clear examples of choreographies that cannot be described as k-th powers of the figure-eight solution, according to the topological classification of orbits. We have also found 17 new “powers of the eight” choreographies. According to our numerical computation, one of two distinct \(k=7\) choreographies is linearly stable.

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Acknowledgments

The authors would like to thank V. Dmitrašinović for his help in the initial and the final stages of this work, and to Dr. Ayumu Sugita (Osaka City University) for alerting them to the fourth-significant-digit error in the period of Moore’s figure-8 orbit published in Table 2 of Šuvakov and Dmitrašinović (2014), as well as for lending us his program for the evaluation of periods. This work was supported by the Serbian Ministry of Education, Science and Technological Development under grant numbers OI 171037 and III 41011, and the Japan Society for the Promotion of Science (JSPS), Grant-in-Aid for Young Scientists (B) No. 26800059. The computing cluster Zefram (zefram.ipb.ac.rs) at the Institute of Physics Belgrade has been extensively used for calculations.

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Author notes

  1. Mitsuru Shibayama

    Present address: Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

Authors and Affiliations

  1. Institute of Physics, University of Belgrade, Pregrevica 118, Zemun-Beograd, 11080, Serbia

    Milovan Šuvakov

  2. Division of Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka, 560-8531, Japan

    Mitsuru Shibayama

Authors
  1. Milovan Šuvakov
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  2. Mitsuru Shibayama
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Correspondence to Mitsuru Shibayama.

Additional information

Milovan Šuvakov—On leave of absence, presently at Serbian Ministry of Education, Science and Technological Development, Nemanjina 22, 11050 Beograd, Serbia.

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Šuvakov, M., Shibayama, M. Three topologically nontrivial choreographic motions of three bodies. Celest Mech Dyn Astr 124, 155–162 (2016). https://doi.org/10.1007/s10569-015-9657-9

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  • DOI: https://doi.org/10.1007/s10569-015-9657-9

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