Periodic solutions of a class of singular Hamiltonian systems. (English) Zbl 0648.34048

The author investigates the existence of periodic solutions, with a prescribed period, of the system of N second order differential equations \(d^ 2u/dt^ 2=-V^ 1(t,u)\) where \(V^ 1(t,.)\) denotes the gradient of the function V(t,.) defined on \({\mathbb{R}}^ N-\{0\}\). V(t,x) is supposed to be T-periodic int. With suitable restriction on V, basically related to its singular properties at \(x=0\) and its behaviour at \(| x| \to \infty\), theorems are proved pertaining to the existence of at least one non-constant T-periodic \(C^ 2\) solution, as well as the existence of infinitely many non-constant T-periodic \(C^ 2\) solutions. The proofs are heavily based on functional analysis, in particular of the nature and behaviour of the critical points. The paper mostly deals with \(N>2\) when the set of singularities of V is simple.
Reviewer: N.D.Sengupta


34C25 Periodic solutions to ordinary differential equations
70H05 Hamilton’s equations
34A34 Nonlinear ordinary differential equations and systems
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