Skip to main content
SpringerLink
Log in

A minimizing property of hyperbolic Keplerian orbits

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

In this short note, we characterize hyperbolic Keplerian orbits as minimizing paths of the Keplerian action functional in the space of curves from a ray emanating from the attractive focus to a point in space. Variants of this result have been previously proved by different methods. Our proof based on hyperbolic anomaly is simple and informative.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition. AIAA Education Series, USA (1999)

    Book  MATH  Google Scholar 

  2. Barutello, V., Terracini, S., Verzini, G.: Entire minimal parabolic trajectories: the planar anisotropic Kepler problem. Arch. Ration. Mech. Anal. 207, 583–609 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barutello, V., Terracini, S., Verzini, G.: Entire parabolic trajectories as minimal phase transitions. Cal. Var. Partial Differ. Equations 49, 329–491 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Chenciner, A.: Symmetries and “simple” solutions of the classical \(n\)-body problem. XIVth International Congress on Mathematical Physics, pp. 4–20 (2005)

  5. Chen, K.-C.: On Chenciner-Montgomery’s orbit in the three-body problem. Discrete Contin. Dynam. Syst. 7, 85–90 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, K.-C.: Binary decompositions for the planar \(N\)-body problem and symmetric periodic solutions. Arch. Ration. Mech. Anal. 170, 247–276 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, K.-C.: Removing collision singularities from action minimizers for the \(N\)-body problem with free boundaries. Arch. Ration. Mech. Anal. 181, 311–331 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, K.-C.: Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses. Ann. Math. 167, 325–348 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

  10. Chen, K.-C., Lin, Y.-C.: On action-minimizing retrograde and prograde orbits of the three-body problem. Commun. Math. Phys. 1291, 403–441 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, K.-C., Yu, G.: Syzygy sequences of the \(N\)-center problem. Ergodic Theory Dynam. Systems (2016) (to appear)

  12. Chenciner, A., Montgomery, R.: A remarkable periodic solution of the three body problem in the case of equal masses. Ann. Math. 152, 881–901 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chenciner, A., Venturelli, A.: Minima de l’intégrale d’action du Problème newtonien de 4 corps de masses égales dans \(\mathbb{R}^3\): orbites ‘hip-hop’. Celest. Mech. Dynam. Astron. 77(2), 139–151 (2000)

    Article  Google Scholar 

  14. Ferrario, D., Terracini, S.: On the existence of collisionless equivariant minimizers for the classical \(n\)-body problem. Invent. Math. 155, 305–362 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fitzpatrick, R.: An Introduction to Celestial Mechanics. Cambridge University Press, Cambridge (2012)

  16. Fusco, G., Gronchi, G.F., Negrini, P.: Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem. Invent. Math. 185, 283–332 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gordon, W.: A minimizing property of Keplerian orbits. Am. J. Math. 99, 961–971 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  18. Maderna, E.: On weak KAM theory of N-body problems. Ergodic Theory Dynam. Syst. 32, 1019–1041 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Maderna, E.: Venturelli, A, Globally minimizing parabolic motions in the Newtonian N-body problem. Arch. Ration. Mech. Anal. 194, 283–313 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Soave, N., Terracini, S.: Symbolic dynamics for the N-centre problem at negative energies. Discrete Contin. Dyn. Syst. Ser. A 32, 3245–3301 (2012)

  21. Terracini, S., Venturelli, A.: Symmetric trajectories for the \(2N\)-body problem with equal masses. Arch. Ration. Mech. Anal. 184, 465–493 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yu, G.: Periodic solutions of the planar \(N\)-center problem with topological constraints. Discrete Contin. Dynam. Syst. A 36, 5131–5162 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, S., Zhou, Q.: Variational methods for the choreography solution to the three-body problem. Sci. China Ser. A 45, 594–597 (2002)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

I feel privileged and honored to be invited to contribute in this special volume in honor of Professor Paul Rabinowitz. 16 years ago, on 2000/05/28 to be exact, I delivered a contributed talk on my analytic proof for the existence of figure-8 orbit [5] at University of Wisconsin, during the “Madison Conference on Nonlinear Analysis” in honor of Paul. That was the debut of my academic career. Ever since, I have kept working on variational methods for the n-body problem and have received continuous support and encouragements from Paul. I became regular participant for related conferences and workshops, such as the conference series in Nankai, and was invited by Paul to deliver lectures multiple times at Postech. I sincerely thank him for the his generous help and recognition, which were particularly important at the beginning stage of my career. I also thank Yiming Long for his invitation, and Guowei Yu for showing me related results. This work is supported in parts by the Ministry of Science and Technology (Grant NSC 102-2628-M-007-004-MY4) in Taiwan.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Tsing Hua University, Hsinchu, 30013, Taiwan

    Kuo-Chang Chen

Authors
  1. Kuo-Chang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kuo-Chang Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, KC. A minimizing property of hyperbolic Keplerian orbits. J. Fixed Point Theory Appl. 19, 281–287 (2017). https://doi.org/10.1007/s11784-016-0353-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11784-016-0353-5

Keywords

Mathematics Subject Classification

Search

Navigation