Cuplength estimates for periodic solutions of Hamiltonian particle-field systems

TL;DR

This paper establishes a lower bound on the number of periodic solutions in Hamiltonian particle-field systems, linking it to the cuplength of loop spaces under Diophantine conditions, addressing small divisor issues.

Contribution

It introduces a novel lower bound for periodic solutions in infinite-dimensional Hamiltonian systems with particle-field interactions, using topological and Diophantine analysis.

Findings

Number of T-periodic solutions is bounded below by the cuplength of loop space in Q.

The bound holds when T/2π squared is a Diophantine irrational number.

Addresses small divisor problems in infinite-dimensional Hamiltonian 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 $\mathbb{T}^d=\mathbb{R}^d/(2\pi\mathbb{Z})^d$ 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.