Subharmonics for convex nonautonomous Hamiltonian systems. (English) Zbl 0601.58035

Ekeland, I.; Hofer, H.

Commun. Pure Appl. Math. 40, No. 1, 1-36 (1987).

Denote by J the standard symplectic matrix in \({\mathbb{R}}^{2n}\) and let \(H\in C^ 2({\mathbb{R}}\times {\mathbb{R}}^{2n}, {\mathbb{R}})\) be T-periodic in the first variable. In this paper we investigate the existence of kT-periodic solutions of the time dependent Hamiltonian system \[ (HS)_ k\quad - J\dot x=H'(t,x),\quad x(0)=x(kT), \] where \(k\in {\mathbb{N}}\). A solution of \((HS)_ k\) for \(k\geq 2\) is called a subharmonic. Clearly, a solution of \((HS)_ k\) will also be a solution of \((HS)_{2k}\), \((HS)_{3k}\), etc. We show under a convexity assumption on H and a suitable asymptotic behaviour of H that for every \(k\in {\mathbb{N}}\) there is a solution \(x_ k\) of \((HS)_ k\) such that the \(x_ k\), \(k\in {\mathbb{N}}\), are pairwise geometrically distinct.

MSC:

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

Keywords:

time dependent Hamiltonian system

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