Cuplength estimates for periodic solutions of Hamiltonian particle-field systems

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 T^d= R^d/ (2 πZ) ^d and the coordinates of the particles are constrained to a submanifold Q⊂ 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 Z₂-cuplength of the space Λ ^contrQ of contractible loops in Q, provided that the square of the ratio T/ 2 π of time period T and space period X= 2 π 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-bounds we need to deal with small divisors.