Abstract
AbstractWe consider a natural class of time-periodic infinite-dimensional nonlinear Hamiltonian systems modelling the interaction of a classical mechanical system of particles with a scalar wave field. When the field is defined on a space torus$${\mathbb {T}}^d={\mathbb {R}}^d/(2\pi {\mathbb {Z}})^d$$Td=Rd/(2πZ)dand the coordinates of the particles are constrained to a submanifold$$Q\subset {\mathbb {T}}^d$$Q⊂Td, we prove that the number of contractibleT-periodic solutions of the coupled Hamiltonian particle-field system is bounded from below by the$${\mathbb {Z}}_2$$Z2-cuplength of the space$$\Lambda ^{{\text {contr}}} Q$$ΛcontrQof contractible loops inQ, provided that the square of the ratio$$T/2\pi $$T/2πof time periodTand space period$$X=2\pi $$X=2πis a Diophantine irrational number. The latter condition is necessary since for the infinite-dimensional version of Gromov–Floer compactness as well as for the$$C^0$$C0-bounds we need to deal with small divisors.
Funder
nederlandse organisatie voor wetenschappelijk onderzoek
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Geometry and Topology,Modeling and Simulation
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