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Variational methods for Hamiltonian systems. (English) Zbl 1048.37055

Hasselblatt, B. (ed.) et al., Handbook of dynamical systems. Volume 1A. Amsterdam: North-Holland (ISBN 0-444-82669-6/hbk). 1091-1127 (2002).
Variational methods have been used very successfully to find periodic, homoclinic, or heteroclinic orbits of Hamiltonian systems. The author, a most influential contributor to this area for three decades, surveys some of the representative problems, methods and results. He treats first-order Hamiltonian systems (HS) \(J\dot{z}+H_z(t,z)=0\) as well as second-order systems (HS2) \(\ddot{q}+V_q(t,q)=0\), both autonomous or nonautonomous. In the nonautonomous case it is required that \(H,V\) depend periodically on \(t\).
The paper consists of two parts: Part 1 is concerned with periodic solutions, Part 2 with homoclinic and heteroclinic orbits. After formulating without details a technical framework for periodic solutions, the following topics are discussed: superquadratic autonomous Hamiltonian systems, fixed energy results, brake orbits, time dependent superquadratic fixed period problems, perturbations from symmetry, subquadratic Hamiltonian systems, asymptotically quadratic Hamiltonians, singular potentials. Part 2 starts with the variational formulation for homoclinics to \(0\), and contains some results for homoclinics, basic heteroclinic results, multibump solutions in the time dependent case, and multibump solutions in the autonomous case.
The author states a number of selected theorems precisely and gives sometimes ideas of proofs or of essential ingredients of the proofs. Other results are discussed informally. For all results mentioned, references to the literature are given.
For the entire collection see [Zbl 1013.00016].

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C25 Periodic solutions to ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
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