The Variational Structure of Potential Systems and Stability of Periodic Solutions

Abstract

In this paper, we first survey variational approaches to potential systems to show the existence of periodic solutions. As simple examples, we consider a periodically forced pendulum and the Keplertype problem. Next we focus on the n-body problem and show the existence of symmetric periodicsolutions. To show the existence of perodic solutions by variational method, the most difficult part is to eliminate the possibility that an obtained minimizer has collisions. We introduce known methods for it. As recent progresses, we show the existence of orbits realizing given symbolic sequences in the n-body and the n-center problem. We also discuss the stability of minimizing periodic solutions.