Abstract
Clarke has shown that the problem of findingT-periodic solutions for a convex Hamiltonian system is equivalent to the problem of finding critical points to a certain functional, dual to the classical action functional. In this paper, we relate the Morse index of the critical point to the minimal period of the correspondingT-periodic solution. In particular, we show that if the critical point is obtained by the Ambrosetti-Rabinowitz mountain-pass theorem the corresponding solution has minimal periodT, that is, it cannot beT/k-periodic withk integer,k≧2. As a consequence, we prove that if the Hamiltonian is flat near an equilibrium and superquadratic near infinity, then for anyT>0, the corresponding Hamiltonian system has a periodic solution with minimal periodT.
Similar content being viewed by others
References
Amann, H., Zehnder, E.: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Sc. Norm. Super. Pisa7, 539–603 (1980)
Amann, H., Zehnder, E.: Periodic solutions of asymptotically linear Hamiltonian equations. Manuscr. Math.32, 149–189 (1980)
Ambrosetti, A., Mancini, G.: Solutions of minimal period for a class of convex Hamiltonian systems. Math. Ann.255, 405–421 (1981)
Ambrosetti, A., Rabinowitz, P.: Dual variational methods in critical point theory and applications. J. Funct. Anal.14, 349–381 (1973)
Aubin, J.P., Ekeland, I.: Applied nonlinear analysis. New York: Wiley 1984
Cambini, A.: Sul lemma di Morse. Boll. Unione Met. Ital.7, 87–93 (1973)
Clarke, F.: Solutions périodiques des équations hamiltoniennes. C.R. Acad. Sci., Paris287, 951–952 (1978)
Clarke, F.: Periodic solutions of Hamiltonian inclusions. J. Differ. Equations40, 1–6 (1981)
Clarke, F.: Periodic solutions of Hamiltonian's equations and local minima of the dual action (To appear)
Clarke, F., Ekeland, I.: Hamiltonian trajectories having prescribed minimal period. Comm. Pure Appl. Math.33, 103–116 (1980)
Conley, C., Zehnder, E.: Morse type index theory for flows and periodic solutions for Hamiltonian equations. Comm. Pure Appl. Math. (To appear)
Ekeland I.: Nonconvex minimization problems. Bull. Am. Math. Soc.1, New Series 443–474 (1979)
Ekeland, I.: Une théorie de Morse pour les systèmes hamiltoniens convexes. Ann. Inst. Henri Poincaré: Analyse non linéaire.1, 19–78 (1984)
Ekeland, I.: Periodic solutions to Hamiltonian equations and a theorem of P. Rabinowitz. Differ. Equations34, 523–534 (1979)
Ekeland, I.: An index theory for periodic solutions of convex Hamiltonian systems. Proc. Am. Math. Soc. Summer Institute on Nonlinear Functional Analysis (Berkeley, 1983, (To appear)
Ekeland, I.: Hypersurfaces pincées et systèmes hamiltoniens. Note C.R. Acad. Sci. Paris (à paraître 1984)
Ekeland, I., Teman, R.: Analyse convexe et problèmes variationnels, Dunod-Gauthier-Villars, 1974; English translation, “Convex analysis and variational problems”. North-Holland-Elsevier, 1976
Gromoll, D., Meyer, W.: On differentiable functions with isolated critical points. Topology8, 361–369 (1969)
Girardi, M., Matzeu, M.: Some results on solutions of minimal period to superquadratic Hamiltonian equations. Nonlinear Anal., Theory Methods Appl.7, 475–482 (1983)
van Groesen, E.: Existence of multiple normal mode trajectories on convex energy surfaces of even, classical Hamiltonian systems. J. Differ. Equations (To appear)
Hofer, H.: A geometric description of the neighbourhood of a critical point given by the mountain pass theorem. J. Lond. Math. Soc. (To appear)
Hofer, H.: The topological degree at a critical point of mountain pass type. Proc. Am. Math. Soc. Summer Institute on Nonlinear Functional Analysis (Berkely, 1983) (To appear)
Krasnoselskii, M.A.: Topological methods in the theory of nonlinear integral equations. English translation, Pergamon press, 1963
Palais, R.: Ljusternik-Schnirelman theory on Banach manifolds. Topology5, 115–132 (1966)
Rabinowitz, P.: Periodic solutions of Hamiltonian systems Comm. Pure Appl. Math.31, 157–184 (1978)
Rockafellar, R.T.: Convex analysis. Princeton: University Press (1970)
Takens, F.: A note on sufficiency of jets. Invent. Math. 225–231 (1971)
Yakubovich, V., Starzhinskii, V.: Linear differential equations with periodic coefficients. New York: Halsted Press, Wiley
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ekeland, I., Hofer, H. Periodic solutions with prescribed minimal period for convex autonomous hamiltonian systems. Invent Math 81, 155–188 (1985). https://doi.org/10.1007/BF01388776
Issue Date:
DOI: https://doi.org/10.1007/BF01388776