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Hamiltonian systems with periodic nonlinearities. (English) Zbl 0721.58010
Differential equations and their applications, Proc. 7th Conf., Equadiff 7, Prague/Czech. 1989, Teubner-Texte Math. 118, 131-134 (1990).
[For the entire collection see Zbl 0704.00019.]
Several results, which will appear in the journal Nonlinear Anal., are announced in this paper. The number of \(2\pi\)-periodic solutions of the Hamiltonian system (1): \(\dot z=JH_ z(z,t)\), \(z=(p,q)\in {\mathbb{R}}^{2n}\), (J is the standard symplectic matrix), is investigated with help of the functional \[ \Phi (z)=(1/2)\int^{2\pi}_{0}(-J\dot z\cdot z)dt-\int^{2\pi}_{0}H(z,t)dt, \] whose critical points correspond to \(2\pi\)-periodic solutions of (1). The properties of \(\Phi\) are used to the generalization of earlier results of several authors (C. C. Conley, E. Zehnder, K. C. Chang).
Reviewer: A.Klíč (Praha)
MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37G99 Local and nonlocal bifurcation theory for dynamical systems
70H05 Hamilton’s equations