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Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair


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Volume 18, Issue 4

Issues

Existence of Prograde Double-Double Orbits in the Equal-Mass Four-Body Problem

Wentian Kuang / Duokui Yan
Published Online: 2018-03-08 | DOI: https://doi.org/10.1515/ans-2018-0009

Abstract

By introducing simple topological constraints and applying a binary decomposition method, we show the existence of a set of prograde double-double orbits for any rotation angle θ(0,π/7] in the equal-mass four-body problem. A new geometric argument is introduced to show that for any θ(0,π/2), the action of the minimizer corresponding to the prograde double-double orbit is strictly greater than the action of the minimizer corresponding to the retrograde double-double orbit. This geometric argument can also be applied to study orbits in the planar three-body problem, such as the retrograde orbits, the prograde orbits, the Schubart orbit and the Hénon orbit.

Keywords: Four-Body Problem; Variational Method; Topological Constraint; Geometric Method

MSC 2010: 70F10; 70F16; 70F07

References

  • [1]

    X. Chang, T. Ouyang and D. Yan, Linear stability of the criss-cross orbit in the equal-mass three-body problem, Discrete Contin. Dyn. Syst. 36 (2016), no. 11, 5971–5991. CrossrefWeb of ScienceGoogle Scholar

  • [2]

    K.-C. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses, Arch. Ration. Mech. Anal. 158 (2001), no. 4, 293–318. CrossrefGoogle Scholar

  • [3]

    K. Chen, Keplerian action functional, convex optimization, and an application to the four-body problem, preprint (2013).

  • [4]

    K.-C. Chen, Binary decompositions for planar N-body problems and symmetric periodic solutions, Arch. Ration. Mech. Anal. 170 (2003), no. 3, 247–276. CrossrefGoogle Scholar

  • [5]

    K.-C. Chen, Variational methods on periodic and quasi-periodic solutions for the N-body problem, Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1691–1715. CrossrefGoogle Scholar

  • [6]

    K.-C. Chen, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Ann. of Math. (2) 167 (2008), no. 2, 325–348. CrossrefWeb of ScienceGoogle Scholar

  • [7]

    K.-C. Chen and Y.-C. Lin, On action-minimizing retrograde and prograde orbits of the three-body problem, Comm. Math. Phys. 291 (2009), no. 2, 403–441. CrossrefGoogle Scholar

  • [8]

    A. Chenciner, Action minimizing solutions of the Newtonian N-body problem: From homology to symmetry, Proceedings of the International Congress of Mathematicians. Vol. III (Beijing 2002), Higher Education Press, Beijing (2002), 279–294. Google Scholar

  • [9]

    A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math. (2) 152 (2000), no. 3, 881–901. CrossrefGoogle Scholar

  • [10]

    D. L. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical N-body problem, Invent. Math. 155 (2004), no. 2, 305–362. CrossrefGoogle Scholar

  • [11]

    W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math. 99 (1977), no. 5, 961–971. CrossrefGoogle Scholar

  • [12]

    C. Marchal, How the method of minimization of action avoids singularities, Celestial Mech. Dynam. Astronom. 83 (2002), no. 1–4, 325–353. CrossrefGoogle Scholar

  • [13]

    R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), no. 1, 19–30. CrossrefGoogle Scholar

  • [14]

    R. Vanderbei, New orbits for the N-body problem, Ann. N.Y. Acad. Sci. 1017 (2004), 422–433. CrossrefGoogle Scholar

  • [15]

    G. Yu, Simple choreographies of the planar Newtonian N-body problem, Arch. Ration. Mech. Anal. 225 (2017), no. 2, 901–935. CrossrefWeb of ScienceGoogle Scholar

  • [16]

    S. Zhang and Q. Zhou, Nonplanar and noncollision periodic solutions for N-body problems, Discrete Contin. Dyn. Syst. 10 (2004), no. 3, 679–685. CrossrefGoogle Scholar

About the article


Received: 2017-12-22

Revised: 2018-02-01

Published Online: 2018-03-08

Published in Print: 2018-11-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11131004

Award identifier / Grant number: 11671215

Award identifier / Grant number: 11432001

The first author was partially supported by the Ph.D. Candidate Research Innovation Fund of Nankai University and NSFC (no. 11131004 and 11671215). The second author was supported by NSFC (no. 11432001) and the China Scholarship Council.


Citation Information: Advanced Nonlinear Studies, Volume 18, Issue 4, Pages 819–843, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2018-0009.

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