  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 $\mathbb{T}^d=\mathbb{R}^d/(2\pi\mathbb{Z})^d$ and the coordinates of the
particles are constrained to a submanifold $Q\subset\mathbb{T}^d$, we prove
that the number of $T$-periodic solutions of the coupled Hamiltonian
particle-field system is bounded from below by the $\mathbb{Z}_2$-cuplength of
the space of contractible loops in $Q$, provided that the square of the ratio
$T/2\pi$ of time period $T$ and space period $X=2\pi$ 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.
