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 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 \(T/2\pi \) 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.

我们考虑一类自然的时间周期无限维非线性哈密顿系统，模拟经典力学粒子系统与标量波场的相互作用。当场定义在空间环面上\({\mathbb {T}}^d={\mathbb {R}}^d/(2\pi {\mathbb {Z}})^d\) 和坐标的粒子被约束到一个子流形\(Q\subset {\mathbb {T}}^d\) ，我们证明了耦合哈密顿粒子场系统的可收缩T周期解的数量从下面受到限制\({\mathbb {Z}}_2\) - Q中可收缩环的空间\(\Lambda ^{{\text {contr}}} Q\)的立方长度，前提是比率的平方时间周期T和空间周期\(X= 2\pi \)的\(T /2\pi \)是丢番图无理数。后一个条件是必要的，因为对于 Gromov–Floer 紧致性的无限维版本以及\(C^0\) -边界，我们需要处理小除数。