Forced vibrations of superquadratic Hamiltonian systems. (English) Zbl 0592.70027

The authors consider the Hamiltonian system (\(\circ)\) \(\dot z=gH'(z)+f(t)\) where g is the standard skewsymmetric matrix, \(z=z(t)=(p,q): {\mathbb{R}}\to {\mathbb{R}}^{2N}\), \(H: {\mathbb{R}}^{2N}\to {\mathbb{R}}\) is a given Hamiltonian and \(f:{\mathbb{R}}\to {\mathbb{R}}^{2N}\) is an assigned T-periodic function. It is supposed that H satisfies some suitable conditions; in particular the Hamiltonian is required to be superquadratic as \(| z| \to +\infty\). Under these circumstances it is proved that the system (\(\circ)\) has infinitely many distinct T- periodic solutions \(\{z_ k\}_{k\in N}\) and \(\| z_ k\|_ L\to +\infty\) as \(k\to +\infty\). This is the main result of the paper. The method to prove it lies in constructing critical topologically stable values for the Lagrangian functional associated with the autonomous system. Then by using Morse theory the authors show that there exist infinitely many critical values for some perturbations of the autonomous functional. In the last part of this lengthy paper the authors prove a more general version of a result obtained by Rabinowitz [one periodic solution, for a system of the type \(\dot z=gH'_ z(t,z)\), exists; P. H. Rabinowitz, Commun. Pure Appl. Math. 31, 156-184 (1978; Zbl 0358.70014)].
Reviewer: A.Muracchini


70K40 Forced motions for nonlinear problems in mechanics
70H05 Hamilton’s equations
34C25 Periodic solutions to ordinary differential equations
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