Skip to main content
SpringerLink
Log in

Variational construction of unbounded orbits in Lagrangian systems

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We show the existence of unbounded orbits in perturbations of generic geodesic flow in \( \mathbb{T}^2 \) by a generic periodic potential. Different from previous work such as in Mather (1997), the initial values of the orbits obtained here are not required sufficiently large.

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.

Similar content being viewed by others

References

  1. Arnol’d V I. Instability of dynamical systems with several degrees of freedom. Sov Math Dokl, 1964, 5: 581–585

    MATH  Google Scholar 

  2. Bolotin S, Treschev D. Unbounded growth of energy in nonautonomous Hamiltonian systems. Nonlinearity, 1999, 12: 365–388

    Article  MATH  MathSciNet  Google Scholar 

  3. Cheng C-Q, Yan J. Existence of diffusion orbits in a priori unstable Hamiltonian systems. J Differ Geom, 2004, 67: 457–517

    MATH  MathSciNet  Google Scholar 

  4. Cheng C-Q, Yan J. Arnold diffusion in Hamiltonian systems: a priori unstable case. J Differ Geom, 2009, 82: 229–277

    MATH  MathSciNet  Google Scholar 

  5. Delshama A, de la Llave R, Seara T M. A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of T2. Comm Math Phys, 2000, 209: 353–392

    MathSciNet  Google Scholar 

  6. Delshams A, de la Llave R, Seara T M. Geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristic and rigorous verification on a model. Mem Amer Math Soc, 2006, 179: 1–141

    Google Scholar 

  7. Delshama A, de la Llave R, Seara T M. Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows. Adv Math, 2006, 202: 64–188

    Article  MathSciNet  Google Scholar 

  8. Hirsch M W, Pugh C C, Shub M. Invariant Manifolds. In: Lect Notes Math, Vol. 583. Berlin: Springer-Verlag, 1997

    Google Scholar 

  9. Mather J N. Variational construction of connecting orbits. Ann Inst Fourier (Grenoble), 1993, 43: 1349–1386

    MATH  MathSciNet  Google Scholar 

  10. Mather J N. Variational construction of trajectories for time periodic Lagrangian systems on the two torus. Manuscript, 1997

  11. Moser J K. On the volume elements on a manifold. Trans Amer Math Soc, 1965, 120: 286–294

    Article  MATH  MathSciNet  Google Scholar 

  12. Treschev D V. Evolution of slow variables in a priori unstable Hamiltonian systems. Nonlinearity, 2004, 17: 1803–1841

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Chong-Qing Cheng

  2. Department of Mathematics, Suzhou University of Science and Technology, Suzhou, 215001, China

    Xia Li

Authors
  1. Chong-Qing Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xia Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chong-Qing Cheng.

Additional information

Dedicated to Professor Yang Lo on the Occasion of his 70th Birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cheng, CQ., Li, X. Variational construction of unbounded orbits in Lagrangian systems. Sci. China Math. 53, 617–624 (2010). https://doi.org/10.1007/s11425-010-0033-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-010-0033-7

Keywords

MSC(2000)

Search

Navigation