Periodic solution for perturbed Hamiltonian systems
The author studies how the existence of infinitely many periodic solutions for second-order perturbed Hamiltonian systems might occur. The system the author considers is \(\ddot{q}(t) + A\dot{q}(t) + \nabla F(q(t) = \nabla_q V(t, q(t))\) with \(q(0) - q(T) = \dot{q}(0) -\dot{q}(T) = 0\) for \(t \in \mathbb{R}\) and \(T > 0\). Here \(A\) is a skew-symmetric matrix, \(F(q) = -K(q) + W(q)\), and where \(K\) and \(W\) are in \(C^2(\mathbb{R}^N, \mathbb{R})\), \(V \in C^2(\mathbb{R} \times \mathbb{R}^N, \mathbb{R})\), \(V\) is \(T\)-periodic in the first variable and \(T > 0\). In addition, \(K\) and \(W\) are even functions.\N\NThe goal of this paper is to establish the existence of an infinite number of periodic solutions for this system using variational methods and \textit {P. Bolle}'s perturbation method [J. Diff. Eq. 152, 274--288 (1999; Zbl 0960.76093] (see also [\textit{P. Bolle} et al., Manuscripta Math. 101, 325--350 (2000; Zbl 0963.35001)]). Under certain technical conditions on \(K\), \(W\), and \(V\), the author proves that the above system has a sequence of periodic solutions \(q_n\) with \(||q_n|| \rightarrow \infty\) as \(n \rightarrow \infty\).
scientific article; zbMATH DE number 7868210
scientific article