Skip to main content
SpringerLink
Log in

Indices and Stability of the Lagrangian System on Riemannian Manifold

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, let m ≥ 1 be an integer, M be an m-dimensional compact Riemannian manifold. Firstly the linearized Poincaré map of the Lagrangian system at critical point x

$${d \over {dt}}{L_q}\left( {t,x,\dot x} \right) - {L_p}\left( {t,x,\dot x} \right) = 0$$

is explicitly given, then we prove that Morse index and Maslov-type index of x are well defined whether the manifold M is orientable or not via the parallel transport method which makes no appeal to unitary trivialization and establish the relation of Morse index and Maslov-type index, finally derive a criterion for the instability of critical point and orientation of M and obtain the formula for two Maslov-type indices.

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.: Morse theory for asymptotically linear Hamiltonian systems. Nonlinear Anal., 39(8), 997–1049 (2000)

    Article  MathSciNet  Google Scholar 

  2. An, T., Long, Y.: On the index theories for second order Hamiltonian systems. Nonlinear Anal., 34(4), 585–592 (1998)

    Article  MathSciNet  Google Scholar 

  3. Atiyah, M., Patodi, V., Singer, I.: Spectral asymmetry and Riemannian geometry. I. Math. Proc. Cambridge Philos. Soc., 77, 43–69 (1975)

    Article  MathSciNet  Google Scholar 

  4. Atiyah, M., Patodi, V., Singer, I.: Spectral asymmetry and Riemannian geometry. II. Math. Proc. Cambridge Philos. Soc., 78(3), 405–432 (1975)

    Article  MathSciNet  Google Scholar 

  5. Bott, R.: On the iteration of closed geodesics and the Sturm intersection theory. Comm. Pure Appl. Math., 9, 171–206 (1956)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Dong, Y., Long, Y.: Closed characteristics on partially symmetric compact convex hypersurfaces in ℝ2n. J. Differential Equations, 196, 226–248 (2004)

    Article  MathSciNet  Google Scholar 

  8. Dong, Y., Long, Y.: Stable closed characteristics on partially symmetric compact convex. J. Differential Equations, 206, 265–279 (2004)

    Article  MathSciNet  Google Scholar 

  9. Duistermaat, J.: On the Morse index in variational calculus. Adv. Math., 21(2), 173–195 (1976)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Hu, X., Sun, S.: Morse index and the stability of closed geodesics. Sci. China Math., 53, 1207–1212 (2010)

    Article  MathSciNet  Google Scholar 

  12. Liu, C.: The relation of the Morse index of closed geodesics with the Maslov-type index of symplectic paths. Acta Math. Sin., Engl. Series, 21, 237–248 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Liu, H.: Stability of symmetric closed characteristics on symmetric compact convex hypersurfaces in ℝ2n under a pinching condition. Acta Math. Sin., Engl. Series, 28, 885–900 (2012)

    Article  MathSciNet  Google Scholar 

  14. Liu, H.: Multiple P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in ℝ2n. Cal. Variations and PDEs, 49, 1121–1147 (2014)

    Article  MathSciNet  Google Scholar 

  15. Liu, H., Zhang, D.: On the number of P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in ℝ2n. Science China Mathematics, 58, 1771–1778 (2015)

    Article  MathSciNet  Google Scholar 

  16. Long, Y.: Index Theory for Symplectic Paths with Applications. Progress in Math., Vol. 207, Birkhäuser, Basel, 2002

    Google Scholar 

  17. Long, Y., An, T.: Indexing domains of instability for Hamiltonian systems. NoDEA Nonlinear Differential Equations Appl., 5(4), 461–478 (1998)

    Article  MathSciNet  Google Scholar 

  18. Long, Y., Zhu, C.: Maslov-type index theory for symplectic paths and spectral flow. II. Chinese Ann. Math.Ser.B, 21(1), 89–108 (2000)

    Article  MathSciNet  Google Scholar 

  19. Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villars, Paris, 1899

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Robbin, J., Salamon, D.: The spectral flow and Maslov index. Bull. London Math. Soc., 27, 1–33 (1995)

    Article  MathSciNet  Google Scholar 

  22. Salamon, D., Zehnder, E.: Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Comm. Pure and Appl. Math., 45, 1303–1360 (1992)

    Article  MathSciNet  Google Scholar 

  23. Viterbo, C.: A new obstruction to embedding Lagrangian tori. Invent. Math., 100, 301–320 (1990)

    Article  MathSciNet  Google Scholar 

  24. Weber, J.: Perturbed closed geodesics are periodic orbits: index and transversality. Math. Z., 241(1), 45–82 (2002)

    Article  MathSciNet  Google Scholar 

  25. Zhang, D.: P-cyclic symmetric closed characteristics on compact convex P-cyclic symmetric hypersurface in ℝ2n. Discrete Contin. Dyn. Syst., 33(2), 947–964 (2013)

    Article  MathSciNet  Google Scholar 

  26. Zhu, C., Long, Y.: Maslov-type index theory for symplectic paths and spectral flow. I. Chinese Ann. Math. Ser. B., 20(4), 413–424 (1999)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I would like to sincerely thank Professor Hui Liu for his valuable discussions and careful reading of the manuscript.

Author information

Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, 300072, P. R. China

    Gao Sheng Zhu

Authors
  1. Gao Sheng Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gao Sheng Zhu.

Additional information

Supported by NSFC (Grant Nos. 11871356, 11871368)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, G.S. Indices and Stability of the Lagrangian System on Riemannian Manifold. Acta. Math. Sin.-English Ser. 37, 565–580 (2021). https://doi.org/10.1007/s10114-020-9311-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-020-9311-7

Keywords

MR(2010) Subject Classification

Search

Navigation