Periodische Lösungen von Hamiltonschen Systemen. (Periodic solutions of Hamiltonian systems). (German) Zbl 0633.58033

The author reviews a large number of results obtained during the last decade on the existence of periodic solutions for Hamiltonian systems. The emphasis is on global results obtained by variational methods (Morse theory, Lyusternik-Schnirel’man theory, mini-max methods, dual variational principles, etc.). The first part of the paper concentrates on fixed point results for symplectic mappings and, connected to this, results on the Arnold conjecture that a periodic Hamiltonian vector field on a compact symplectic manifold has at least as many periodic solutions as the minimal number of critical points of a function on the manifold; related problems which are briefly discussed are the intersection problem of Lagrangian manifolds and the existence of subharmonic solutions. In the second part the author discusses the existence of periodic solutions with given energy or with given period for autonomous Hamiltonian systems in \({\mathbb{R}}^{2n}\), and the existence of periodic and subharmonic solutions in the periodic nonautonomous case. Part III is concerned with local periodic solutions near an equilibrium point (the bifurcation case), and the paper concludes with an impressive list of references, containing about 200 items.
Reviewer: A.Vanderbauwhede


37G99 Local and nonlocal bifurcation theory for dynamical systems
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58-02 Research exposition (monographs, survey articles) pertaining to global analysis