The Direct Method in the Study of Periodic Solutions of Hamiltonian Systems with Prescribed Period

  • V. Benci
Part of the Annals of CEREMADE book series (CEREMADE, volume 2)


The periodic solutions of Hamiltonian systems correspond to the critical points of the “action” functional which is indefinite in a “strong” sense. In this paper, we show the advantages and the difficulties of studying such indefinite functionals directly in an infinite dimensional function space. The outlines of some proofs are presented.


Periodic Solution Hamiltonian System Critical Point Theory Closed Invariant Subset Equivariant Homeomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ A ]
    H. Amann, Saddle points and multiple solutions of differential equations, Math., Z. 169, (1979), 127 - 166.MathSciNetMATHCrossRefGoogle Scholar
  2. [ AZ1 ]
    H. Amann E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Sc. Norm. Sup. Pisa, in press.Google Scholar
  3. [ AZ2 ]
    H. Amann E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems, Preprint.Google Scholar
  4. [ AM]
    A.Ambrosetti G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems, Preprint.Google Scholar
  5. [ AR ]
    A. Ambrosetti P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14, (1973), 349 - 381.MathSciNetMATHCrossRefGoogle Scholar
  6. [ BI ]
    V. Benci, A geometrical Index for the group S1 and some applications to the study of periodic solutions of ordinary differential equations, Comm. Pure Appl. Math., 34 (1981), 393 - 432.MathSciNetMATHCrossRefGoogle Scholar
  7. [ B2 ]
    V. Benci, On the critical point theory for indefinite functionals in the presence of symmetries, to appear in Trans. Amer. Math. Soc.Google Scholar
  8. [ BCE]
    V. Benci A. Capozzi D. Fortunado, Periodic solutions of Hamiltonian systems with a prescribed period, Preprint.Google Scholar
  9. [ BCF2 ]
    V. Benci A. Capozzi D. Fortunado, Periodic solutions for a class of Hamiltonian systems, to appear.Google Scholar
  10. [ BFI ]
    V. Benci D. Fortunado, Un teorema di molteplicità per un'equazione ellittica non lineare su varietà simmetriche, Proceedings of the Symposium "Metodi asintotici e topologici in problem diff. non lineari", L’Aquila (1981).Google Scholar
  11. [ BF2 ]
    V. Benci D. Fortunado, The dual method in critical point theory. Multiplicity results for indefinite functionals, to appear in Ann. Mat. Pura e Applicata.Google Scholar
  12. [ BR ]
    V. Benci P.H. Rabinowitz, Critical point theorems for indefinite functionals, Inv. Math., 52 (1979), 336 - 352.MathSciNetGoogle Scholar
  13. [ BCN ]
    H. Brezis J.M. Coron L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Preprint.Google Scholar
  14. [ C ]
    J Clark B.C., A variant of Ljiusternik Schnirelmann theory, to appear in J. Diff. Eq.Google Scholar
  15. [ CE ]
    F.H. Clarke I. Ekeland, Hamiltonian trajectories having prescribed minimal period, Comm. Pure Appl. Math., 33, (1980), 103 - 116.MathSciNetMATHCrossRefGoogle Scholar
  16. [ E ]
    I. Ekeland, Periodic solutions of Hamiltonian equations and a theorem of P. Rabinowitz, J; Diff. Eq., 34, (1979), 523 - 534.MathSciNetMATHCrossRefGoogle Scholar
  17. [ FHR ]
    E.R. Fadell S. Husseini P.H. Rabinowitz, Borsuk-Ulam theorems for arbitrary S1 actions and applications, Math. Research Center Technical Summary Report, University of Wisconsin-Madison, 1981.Google Scholar
  18. [ FR ]
    E.R. Fadell P.H. Rabinowitz, Generalized cohomological index theories for Lie "group actions with an application to bifurcation questions for Hamiltonian systems, Inv. Math., 45, (1978), 139 - 174.MathSciNetMATHCrossRefGoogle Scholar
  19. [ R1 ]
    P.H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31, (1978), 157 - 184.MathSciNetCrossRefGoogle Scholar
  20. R2 ] P.H. Rabinowitz, Periodic solutions of Hamiltonian systems: a survey, Math. Research Center Technical Summary Report, University of Wisconsin-Madison.Google Scholar
  21. [ R3 ]
    P.H. Rabinowitz, Periodic solutions of large norm of Hamiltonian systems, Math. Resaerch Center Technical Summary Report, University of Wisconsin-Madison (1981).Google Scholar

Copyright information

© Birkhäuser Boston, Inc. 1983

Authors and Affiliations

  • V. Benci
    • 1
  1. 1.Instituto di Matematica ApplicataUniversità di BariItaly

Personalised recommendations