Cuplength estimates for periodic solutions of Hamiltonian particle-field systems

Abstract

We 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/(2pi {mathbb {Z}})^d and the coordinates of the particles are constrained to a submanifold Qsubset {mathbb {T}}^d, we prove that the number of contractible T-periodic solutions of the coupled Hamiltonian particle-field system is bounded from below by the {mathbb {Z}}_2-cuplength of the space Lambda ^{{text {contr}}} Q of contractible loops in Q, provided that the square of the ratio T/2pi of time period T and space period X=2pi is 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-bounds we need to deal with small divisors.