Periodic solutions of a class of singular Hamiltonian systems. (English) Zbl 0648.34048

The author investigates the existence of periodic solutions, with a prescribed period, of the system of N second order differential equations \(d^ 2u/dt^ 2=-V^ 1(t,u)\) where \(V^ 1(t,.)\) denotes the gradient of the function V(t,.) defined on \({\mathbb{R}}^ N-\{0\}\). V(t,x) is supposed to be T-periodic int. With suitable restriction on V, basically related to its singular properties at \(x=0\) and its behaviour at \(| x| \to \infty\), theorems are proved pertaining to the existence of at least one non-constant T-periodic \(C^ 2\) solution, as well as the existence of infinitely many non-constant T-periodic \(C^ 2\) solutions. The proofs are heavily based on functional analysis, in particular of the nature and behaviour of the critical points. The paper mostly deals with \(N>2\) when the set of singularities of V is simple.
Reviewer: N.D.Sengupta


34C25 Periodic solutions to ordinary differential equations
70H05 Hamilton’s equations
34A34 Nonlinear ordinary differential equations and systems
Full Text: DOI


[1] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. funct. Analysis, 14, 349-381 (1973) · Zbl 0273.49063
[2] Benci, V., A geometrical index for the group \(S^1\) and some applications to the study of periodic solutions of ordinary differential equations, Communs pure appl. math., 34, 393-432 (1981) · Zbl 0447.34040
[4] Gordon, W. B., Conservative dynamical systems involving strong forces, Trans. Am. math. Soc., 204, 113-135 (1975) · Zbl 0276.58005
[6] Palais, S. R., Lusternik-Schnirelman theory on Banach manifolds, Topology, 5, 115-132 (1966) · Zbl 0143.35203
[7] Rabinowitz, P. H., Variational methods for nonlinear eigenvalue problems, (Prodi, G., Eigenvalue of Nonlinear Problems (1974), Edizioni Cremonese: Edizioni Cremonese Roma), 141-195
[8] Schwartz, J. T., Nonlinear Functional Analysis (1969), Gordon and Breach: Gordon and Breach New York · Zbl 0203.14501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.