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Distance Estimates for Action-Minimizing Solutions of the \(N\)-Body Problem

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Abstract

In this paper we provide estimates for mutual distances of periodic solutions for the Newtonian \(N\)-body problem. Our estimates are based on masses, total variations of turning angles for relative positions, and predetermined upper bounds for action values. Explicit formulae will be proved by iterative arguments. We demonstrate some applications to action-minimizing solutions for three- and four-body problems.

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Change history

  • 29 October 2023

    The numbering issue has been changed to 4-5.

References

  1. Chen, K.-Ch., Action-Minimizing Orbits in the Parallelogram Four-Body Problem with Equal Masses, Arch. Ration. Mech. Anal., 2001, vol. 158, no. 4, pp. 293–318.

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, K.-Ch., Existence and Minimizing Properties of Retrograde Orbits to the Three-Body Problem with Various Choices of Masses, Ann. of Math. (2), 2008, vol. 167, no. 2, pp. 325–348.

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, K.-Ch., Keplerian Action, Convexity Optimization, and the \(4\)-Body Problem, Anal. Theory Appl., 2021, vol. 37, no. 1, pp. 24–58.

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, K.-Ch., Variational Aspects of the Two-Center Problem, Arch. Ration. Mech. Anal., 2022, vol. 244, no. 2, pp. 225–252.

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, K.-Ch. and Lin, Y.-Ch., On Action-Minimizing Retrograde and Prograde Orbits of the Three-Body Problem, Comm. Math. Phys., 2009, vol. 291, no. 2, pp. 403–441.

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, K.-Ch. and Pan, B.-Y., Distance Estimates for Simple Choreographic Solutions of the Three-Body Problem, Acta Math. Sin. (Engl. Ser.), 2022, vol. 38, no. 10, pp. 1823–1830.

    Article  MathSciNet  MATH  Google Scholar 

  7. Chenciner, A., Action Minimizing Solutions of the Newtonian \(n\)-Body Problem: From Homology to Symmetry, in Proc. of the Internat. Congr. of Math. (Beijing, 2002): Vol. 3, Beijing: Higher Educ. Press, 2002, pp. 279–294. Errata, in Proc. of the Internat. Congr. of Math. (Beijing, 2002): Vol. 1, Beijing: Higher Educ. Press, 2002, pp. 651–653.

    Google Scholar 

  8. Chenciner, A., Gerver, J., Montgomery, R., and Simó, C., Simple Choreographic Motions of \(N\) Bodies: A Preliminary Study, in Geometry, Mechanics, and Dynamics: Volume in Honor of the 60th Birthday of J. E. Marsden, P. Newton, Ph. Holmes, A. Weinstein (Eds.), New York: Springer, 2002, pp. 287–308.

    Chapter  MATH  Google Scholar 

  9. Chenciner, A. and Montgomery, R., A Remarkable Periodic Solution of the Three-Body Problem in the Case of Equal Masses, Ann. of Math. (2), 2000, vol. 152, no. 3, pp. 881–901.

    Article  MathSciNet  MATH  Google Scholar 

  10. Chenciner, A. and Venturelli, A., Minima de l’intégrale d’action du problème newtonien de \(4\) corps de masses égales dans \(\mathbb{R}^{3}\): orbites “hip-hop”, Celestial Mech. Dynam. Astronom., 2000, vol. 77, no. 2, pp. 139–152.

    Article  MathSciNet  MATH  Google Scholar 

  11. Ferrario, D. L. and Terracini, S., On the Existence of Collisionless Equivariant Minimizers for the Classical \(n\)-Body Problem, Invent. Math., 2004, vol. 155, no. 2, pp. 305–362.

    Article  MathSciNet  MATH  Google Scholar 

  12. Gordon, W. B., A Minimizing Property of Keplerian Orbits, Amer. J. Math., 1977, vol. 99, no. 5, pp. 961–971.

    Article  MathSciNet  MATH  Google Scholar 

  13. Kapela, T. and Zgliczyński, P., The Existence of Simple Choreographies for the \(N\)-Body Problem: A Computer-Assisted Proof, Nonlinearity, 2003, vol. 16, no. 6, pp. 1899–1918.

    Article  MathSciNet  MATH  Google Scholar 

  14. Montgomery, R., \(N\)-Body Choreographies, Scholarpedia, 2010, vol. 5, no. 11, Art. 10666.

    Article  Google Scholar 

  15. Nickalls, R. W. D., Viète, Descartes and the Cubic Equation, Math. Gaz., 2006, vol. 90, no. 518, pp. 203–208.

    Article  Google Scholar 

  16. Qiu, S.-L. and Vamanamurthy, M. K., Sharp Estimates for Complete Elliptic Integrals, SIAM J. Math. Anal., 1996, vol. 27, no. 3, pp. 823–834.

    Article  MathSciNet  MATH  Google Scholar 

  17. Simó, C., Choreographic Solutions of the Planar Three Body Problem, http://www.maia.ub.es/dsg/3body.html (2002).

  18. Shibayama, M., Variational Proof of the Existence of the Super-Eight Orbit in the Four-Body Problem, Arch. Ration. Mech. Anal., 2014, vol. 214, no. 1, pp. 77–98.

    Article  MathSciNet  MATH  Google Scholar 

  19. Yu, G., Simple Choreographies of the Planar Newtonian \(N\)-Body Problem, Arch. Ration. Mech. Anal., 2017, vol. 225, no. 2, pp. 901–935.

    Article  MathSciNet  MATH  Google Scholar 

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ACKNOWLEDGMENTS

(K.C.) It is my great honor and pleasure to contribute to this special issue dedicated to Alain Chenciner on the occasion of his 80th birthday. Alain’s research works greatly inspired my research directions, his constant support and vigorous communications greatly influenced my research career, and his versatility (in languages, caricatures, etc) greatly changed the stereotype about mathematicians (at least among people in my neighborhood). I also thank the RCD editorial team for their invitation. The second author (B.P.) wishes to express his gratitude to Ya-Lun Tsai for his generous support through his grant at the National Science and Technology Council in Taiwan.

Funding

This work was supported in part by the National Science and Technology Council in Taiwan. Grant number MOST 110-2115-M-007-004-MY3.

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Authors and Affiliations

  1. Department of Mathematics, National Tsing Hua University, 101, Sec 2, Kuangfu Rd., 300, Hsinchu, Taiwan R.O.C.

    Kuo-Chang Chen

  2. Department of Applied Mathematics, National Chung Hsing University, 145 Xingda Rd., South Dist., 402, Taichung, Taiwan R.O.C.

    Bo-Yu Pan

Authors
  1. Kuo-Chang Chen
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  2. Bo-Yu Pan
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Correspondence to Kuo-Chang Chen or Bo-Yu Pan.

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The authors declare that they have no conflicts of interest.

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MSC2010

70F10, 70F15, 70F07

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Chen, KC., Pan, BY. Distance Estimates for Action-Minimizing Solutions of the \(N\)-Body Problem. Regul. Chaot. Dyn. 28, 561–577 (2023). https://doi.org/10.1134/S1560354723040044

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