Skip to main content
SpringerLink
Log in

Symplectic invariants and Hamiltonian dynamics

  • Chapter
The Floer Memorial Volume

Part of the book series: Progress in Mathematics ((PM,volume 133))

  • 1930 Accesses

Abstract

There is a mysterious relation between rigidity phenomena of symplectic geometry and global periodic solutions of Hamiltonian dynamics. One of the links is provided by a special class of symplectic invariants discovered by I. Ekeland and H. Hofer in [2], [3] called symplectic capacities. We first recall this concept in a more general setting from [26] and consider the class of all symplectic manifolds (M, ω) possibly with boundary, but of fixed dimension 2n. Here ω is a symplectic structure, i.e. a two-form on M which is closed and nondegenerate.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V.I. Arnold: Mathematical methods of classical mechanics, Appendix 9, Springer (1978).

    MATH  Google Scholar 

  2. I. Ekeland and H. Hofer: Symplectic topology and Hamiltonian dynamics I. Math. Zeit. 200 (1990), 355–378.

    Article  MathSciNet  Google Scholar 

  3. I. Ekeland and H. Hofer: Symplectic topology and Hamiltonian dynamics II. Math. Zeit. 203 (1990), 553–567.

    Article  MathSciNet  MATH  Google Scholar 

  4. Ya. M. Eliashberg: A Theorem on the structure of wave fronts and its application in symplectic topology. Functional Analysis and its Applications, Vol. 21 (1987), 227–232.

    Article  MathSciNet  Google Scholar 

  5. Ya. Eliashberg: Topological characterization of Stein manifolds of dimension > 2. Intern. J. Math. (1990), 29–46.

    Google Scholar 

  6. Ya. Eliashberg, H. Hofer: Unseen Symplectic Boundaries, Preprint Stanford University 1993.

    Google Scholar 

  7. A. Floer: Morse theory for Lagrangian intersections. J. Diff. Geom. 28 (1988), 513–547.

    MathSciNet  MATH  Google Scholar 

  8. A. Floer: Witten’s complex and infinite dimensional Morse theory. J. Diff. Geom. 30 (1989), 207–221.

    MathSciNet  MATH  Google Scholar 

  9. A. Floer: Symplectic fixed points and holomorphic spheres. Comm. Math. Phys. 120(1989), 575–611.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Floer, H. Hofer and C. Viterbo: The Weinstein conjecture in P × ℂd . Math. Zeitschrift 203 (1989), 355–378.

    MathSciNet  Google Scholar 

  11. A. Floer and H. Hofer: Coherent orientations for periodic orbit problems in symplectic geometry. Math. Zeit. 212 (1993), 13–38.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Floer and H. Hofer: Symplectic homology I: Open sets in ℂn. Math. Zeit. 215 (1994), 37–88.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Floer and H. Hofer: Symplectic homology II: General symplectic manifolds, to appear Math. Zeit.

    Google Scholar 

  14. A. Floer, H. Hofer and D. Salamon: Transversality results in the elliptic Morse theory for the action functional, to appear Duke.

    Google Scholar 

  15. A. Floer, H. Hofer and K. Wysocki: Applications of symplectic homology I, to appear Math. Zeit.

    Google Scholar 

  16. A. Floer, H. Hofer and K. Wysocki: Applications of symplectic homology II, to appear Math. Zeit.

    Google Scholar 

  17. V Ginzburg: An embedding S2n-1 → ℝ2n , 2n - 1 ≥ 7, whose Hamiltonian flow has no periodic trajectories, to appear Int. Math. Res. Notes.

    Google Scholar 

  18. M. Gromov: Pseudo holomorphic curves in symplectic manifolds. Inv. Math. 82 (1985), 307–347.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Gromov: Partial Differential Relations. Springer-Verlag (1986).

    MATH  Google Scholar 

  20. M. Gromov: Soft and hard differential geometry. In: Proceedings of the ICM at Berkeley 1986, (1987), 81–89.

    Google Scholar 

  21. M. Herman: Examples of compact hypersurfaces in ℝ2p, 2p ≥ 6, with no periodic orbits, preprint.

    Google Scholar 

  22. H. Hofer: On the topological properties of symplectic maps. Proceeding of the Royal Society of Edinburgh 115 A (1990), 25–38.

    Google Scholar 

  23. H. Hofer: Estimates for the energy of a symplectic map. Commentarii Math. Helv. 68 (1993), 48–72.

    Article  MathSciNet  MATH  Google Scholar 

  24. H. Hofer and C. Viterbo: The Weinstein Conjecture in the Presence of Holomorphic Spheres. Comm. Pure Appl. Math., Vol. XLV (1992), 583–622.

    Article  MathSciNet  Google Scholar 

  25. H. Hofer and E. Zehnder: Periodic solutions on hypersurfaces and a result of C. Viterbo. Inv. Math. 90 (1987), 1–7.

    Article  MathSciNet  MATH  Google Scholar 

  26. H. Hofer and E. Zehnder: A new capacity for symplectic manifolds. In: Analysis et cetera, Academic Press (1990), 405–428. Edited by P. Rabinowitz and E. Zehnder.

    Google Scholar 

  27. Mei-Yue Jiang: Hofer-Zehnder capacity for 2-dimensional manifolds. Preprint, Peking University (1992)

    Google Scholar 

  28. D. McDuff: Elliptic methods in symplectic geometry. Bull. A.M.S. 23 (1990), 311–358.

    Article  MathSciNet  Google Scholar 

  29. P. Rabinowitz: Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math. 31 (1978), 157–184.

    Article  MathSciNet  Google Scholar 

  30. Renyi Ma: A remark on the Hofer-Zehnder symplectic capacity in M × ℝ2n . Preprint (1991) Tsinghua University, Beijing.

    Google Scholar 

  31. D. Salamon and E. Zehnder: In: Analysis et cetera, Floer homology, the Maslov index and periodic orbits of Hamiltonian equations. Academic Press (1990), 573–600. Edited by P. Rabinowitz and E. Zehnder

    Google Scholar 

  32. D. Salamon and E. Zehnder: Morse Theory for Periodic Solutions of Hamiltonian Systems and the Maslov-Index. Comm. on Pure and Appl. Math., Vol. XLV (1992), 1303–1360.

    Article  MathSciNet  Google Scholar 

  33. K.F. Siburg: Symplectic Capacities in Two Dimensions. Manuscripta math., 78 (1993), 149–164.

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Struwe: Existence of periodic solutions of Hamiltonian systems on almost every energy surface. Bol. Soc. Bras. Mat. 20 (1990), 49–58.

    Article  MathSciNet  MATH  Google Scholar 

  35. C. Viterbo: A proof of the Weinstein conjecture in ℝ2n . Ann. Inst. Henri Poincaré, Analyse nonlinéaire 4 (1987), 337–357.

    MathSciNet  MATH  Google Scholar 

  36. A. Weinstein: Periodic orbits for convex Hamiltonian systems. Ann. Math. 108 (1978), 507–518.

    Article  MATH  Google Scholar 

  37. A. Weinstein: On the hypothesis of Rabinowitz’s periodic orbit theorems. J. Diff. Equations 33 (1979), 353–358.

    Article  MATH  Google Scholar 

  38. A. Weinstein: Contact surgery and symplectic handlebodies. Preprint UC Berkeley (1990).

    Google Scholar 

  39. J.C. Yoccoz: Travaux de Herman sur les tores invariants. Séminaire BOUR-BAKI, 44ème année, 1991–92, no. 754, février 1992.

    Google Scholar 

  40. E. Zehnder: Remarks on Periodic Solutions on Hypersurfaces, in: NATO ASI, Series C Vol 209 (1987), 267–279.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics, ETH-Zentrum, CH-8092, Zürich, Switzerland

    H. Hofer & E. Zehnder

Authors
  1. H. Hofer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Zehnder
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. of Mathematics, ETH-Zentrum, 8092, Zürich, Switzerland

    Helmut Hofer  & Eduard Zehnder  & 

  2. Dept. of Mathematics, Harvard University, 02138, Cambridge, MA, USA

    Clifford H. Taubes

  3. Dept. of Mathematics, University of California, 94720, Berkeley, CA, USA

    Alan Weinstein

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Birkhäuser Verlag

About this chapter

Cite this chapter

Hofer, H., Zehnder, E. (1995). Symplectic invariants and Hamiltonian dynamics. In: Hofer, H., Taubes, C.H., Weinstein, A., Zehnder, E. (eds) The Floer Memorial Volume. Progress in Mathematics, vol 133. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9217-9_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9217-9_21

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9948-2

  • Online ISBN: 978-3-0348-9217-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation