Periodic Solutions of Non-Autonomous Second Order Hamiltonian Systems

TL;DR

This paper extends existing results on periodic solutions of non-autonomous Hamiltonian systems by relaxing the periodicity assumption on the potential, using variational methods to establish the existence of solutions.

Contribution

It generalizes previous work by only requiring the potential's integral over time to be periodic, enabling analysis of more general forced pendulum equations.

Findings

Established existence of periodic solutions under new assumptions

Applied variational methods and saddle point theorem effectively

Extended results to forced pendulum systems

Abstract

We try to generalize a result of M. Willem on forced periodic oscillations which required the assumption that the forced potential is periodic on spatial variables. In this paper, we only assume its integral on the time variable is periodic, and so we extend the result to cover the forced pendulum equation. We apply the direct variational minimizing method and Rabinowtz's saddle point theorem to study the periodic solution when the integral of the potential on the time variable is periodic.