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Hamiltonian systems: Periodic and homoclinic solutions by variational methods. (English) Zbl 1090.37045
Cañada, A.(ed.) et al., Ordinary differential equations. Vol. II. Amsterdam: Elsevier/North Holland (ISBN 0-444-52027-9/hbk). Handbook of Differential Equations, 77-146 (2005).
Variational treatment of Hamiltonian systems goes back to Poincaré, who investigated periodic solutions of conservative systems with two degrees of freedom using a version of the least action principle. The Hamiltonian action functional $$\Phi(p,q)=\int_0^{2\pi}(p,\dot{q})dt-\int_0^{2\pi}H(p,q,t)$$ on a suitable space of $$2\pi$$-periodic functions $$(p,q):\mathbb R\to \mathbb R^N\times \mathbb R^N$$ is unbounded from below and above, so that the classical method of variational calculus and also the refined variational methods of Morse or a Ljusternik-Schnirelmann theory are unapplicable. Beginning with articles of P. H. Rabinowitz [in: Nonlinear evolution equations, Proc. Symp., Madison/Wis. 1977, 225–251 (1978; Zbl 0486.35009)] and V. Benci and P. H. Rabinowitz [Invent. Math. 52, 241–273 (1979; Zbl 0465.49006)], variational methods for strongly indefinite functionals with applications to Hamiltonian systems (HS) are intensively developed.
The goal of this Chapter 2 is to present an introduction to such variational methods. After presentation of the basic critical point theory and critical point theory for strongly indefinite functionals (Section 1), the authors consider in Section 2 periodic solutions of HS. A unified approach is presented via a finite-dimensional reduction in order to show the existence of one solution, and via a Galerkin-type method for finding more solutions. Then, the existence of periodic solutions near equilibria and a fixed energy problem, i.e., the finding of solutions of a priori unknown period lying on a prescribed (energy surface), are investigated. Furthermore, the questions about the existence and a number of periodic solutions are considered under different growth conditions on the Hamiltonian and for spatially symmetric Hamiltonians. Section 3 deals with homoclinic solutions for HS with time-periodic Hamiltonian. Here, basic existence and multiplicity results are presented and relations to the Bernoulli shift and complicated dynamics are discussed. Note that the investigations on subjects of Section 3 now are still rapidly developing.
For the entire collection see [Zbl 1074.34003].

##### MSC:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 34C25 Periodic solutions to ordinary differential equations 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations