Skip to main content
SpringerLink
Log in

An Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In the variational study of singular Lagrange systems, the zero energy solutions play an important role. The anisotropic Kepler problem is such a singular system introduced by physicist M. Gutzwiller to reveal connections between classic and quantum chaos. In this paper we find a simple way of computing the Morse indices of the zero solutions in this problem. In particular, we show the Morse indices are connected with the oscillating behaviors of these solutions. Our results can also be applied to some problems in celestial mechanics.

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. Abbondandolo A., Majer P.: Ordinary differential operators in Hilbert spaces and Fredholm pairs. Math. Z. 243(3), 525–562 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosetti, A., Coti Zelati, V.: Periodic solutions of singular Lagrangian systems, volume 10 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA, (1993)

  3. Ambrosetti A., Coti Zelati V.: Non-collision periodic solutions for a class of symmetric 3-body type problems. Topol. Methods Nonlinear Anal. 3(2), 197–207 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnold V. I.: On a characteristic class entering into conditions of quantization. Funkcional. Anal. i Priložen. 1, 1–14 (1967)

    Article  MathSciNet  Google Scholar 

  5. Bahri A., Rabinowitz P. H.: A minimax method for a class of Hamiltonian systems with singular potentials. J. Funct. Anal. 82(2), 412–428 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bahri A., Rabinowitz P. H.: Periodic solutions of Hamiltonian systems of 3-body type. Ann. Inst. H. Poincaré Anal. Non Linéaire 8(6), 561–649 (1991)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  7. Barutello, V., Hu, X., Portaluri, A., Terracini, S.: An index theory for asymptotic motions under singular potentials. Preprint, (2017), arXiv:1705.01291

  8. Barutello V., Secchi S.: Morse index properties of colliding solutions to the N-body problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(3), 539–565 (2008)

    Article  MathSciNet  MATH  ADS  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Barutello V., Terracini S., Verzini G.: Entire parabolic trajectories as minimal phase transitions. Calc. Var. Partial Differ. Equ. 49(1-2), 391–429 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Boscaggin A., Dambrosio W., Terracini S.: Scattering parabolic solutions for the spatial N-centre problem. Arch. Ration. Mech. Anal. 223(3), 1269–1306 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cappell S. E., Lee R., Miller E. Y.: On the Maslov index. Commun. Pure Appl. Math. 47(2), 121–186 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen K.-C.: Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses. Ann. Math. (2) 167(2), 325–348 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chenciner A., Montgomery R.: A remarkable periodic solution of the three-body problem in the case of equal masses. Ann. Math. (2) 152(3), 881–901 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Courant R., Hilbert D.: Methods of mathematical Physics. Vol. I. Interscience Publishers Inc., New York (1953)

    MATH  Google Scholar 

  16. da Luz A., Maderna E.: On the free time minimizers of the Newtonian N-body problem. Math. Proc. Camb. Philos. Soc. 156(2), 209–227 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Devaney R. L.: Collision orbits in the anisotropic Kepler problem. Invent. Math. 45(3), 221–251 (1978)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  18. Devaney, R. L.: Singularities in classical mechanical systems. In: Ergodic theory and dynamical systems, I (College Park, Md., 1979–80), volume 10 of Progr. Math., pages 211–333. Birkhäuser, Boston, Mass. (1981)

  19. Ferrario D. L., Terracini S.: On the existence of collisionless equivariant minimizers for the classical n-body problem. Invent. Math. 155(2), 305–362 (2004)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  20. Gutzwiller M. C.: The anisotropic Kepler problem in two dimensions. J. Math. Phys. 14, 139–152 (1973)

    Article  MathSciNet  ADS  Google Scholar 

  21. Gutzwiller M. C.: Bernoulli sequences and trajectories in the anisotropic Kepler problem. J. Math. Phys. 18(4), 806–823 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  22. Hu X., Ou Y.: Collision index and stability of elliptic relative equilibria in planar n-body problem. Commun. Math. Phys. 348(3), 803–845 (2016)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  23. Hu, X., Portaluri, A.: Index theory for heteroclinic orbits of Hamiltonian systems. Calc. Var. Partial Differ. Equ., 56(6):Art. 167, 24, (2017)

  24. Hu X., Sun S.: Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit. Commun. Math. Phys. 290(2), 737–777 (2009)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  25. Long, Y.: Index theory for symplectic paths with applications, volume 207 of Progress in Mathematics. Birkhäuser Verlag, Basel, (2002)

  26. McGehee R.: Triple collision in the collinear three-body problem. Invent. Math. 27, 191–227 (1974)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  27. Moeckel R.: Orbits of the three-body problem which pass infinitely close to triple collision. Am. J. Math. 103(6), 1323–1341 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  28. Moeckel R.: Chaotic dynamics near triple collision. Arch. Ration. Mech. Anal. 107(1), 37–69 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  29. Moeckel R., Montgomery R.: Realizing all reduced syzygy sequences in the planar three-body problem. Nonlinearity 28(6), 1919–1935 (2015)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  30. Moeckel, R., Montgomery, R., Sánchez Morgado, H.: Free time minimizers for the three-body problem. Celest. Mech. Dyn. Astron. 130(3), 130–28 (2018)

  31. Robbin J., Salamon D.: The Maslov index for paths. Topology 32(4), 827–844 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  32. Siegel, C. L., Moser, J. K.: Lectures on celestial mechanics. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Translated from the German by C. I. Kalme, Reprint of the 1971 translation. (1995)

  33. Tanaka K.: Noncollision solutions for a second order singular Hamiltonian system with weak force. Ann. Inst. H. Poincaré Anal. Non Linéaire 10(2), 215–238 (1993)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  34. Yu, G.: Application of morse index in weak force n-body problem. preprint, (2017), arXiv:1711.05077

  35. Yu G.: Simple choreographies of the planar Newtonian N-body problem. Arch. Ration. Mech. Anal. 225(2), 901–935 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhang, Z. F., Ding, T. R., Huang, W. Z., Dong, Z. X.: Qualitative theory of differential equations, volume 101 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1992. Translated from the Chinese by Anthony Wing Kwok Leung. (1992)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shandong University, Jinan, People’s Republic of China

    Xijun Hu

  2. Dipartimento di Matematica “Giuseppe Peano”, Università degli Studi di Torino, Turin, Italy

    Guowei Yu

Authors
  1. Xijun Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guowei Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guowei Yu.

Additional information

Communicated by C. Liverani

Both authors thank the support of NSFC(No.11425105). The second author also acknowledges the support of Fondation Sciences Mathématiques de Paris and the ERC Advanced Grant 2013 No. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, X., Yu, G. An Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem. Commun. Math. Phys. 361, 709–736 (2018). https://doi.org/10.1007/s00220-018-3184-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-018-3184-y

Search

Navigation