Periodic solutions to relativistic Kepler problems: a variational approach

TL;DR

This paper investigates the existence and properties of periodic solutions in relativistic Kepler problems using variational methods, revealing multiple solution families and analyzing their minimality through Morse theory.

Contribution

It introduces a variational approach employing non-smooth critical point theory to find multiple periodic solutions in relativistic Kepler problems with external forces.

Findings

Two infinite families of periodic solutions identified

First family consists of local minima, second from mountain pass geometry

Analysis of minimality of solutions using Morse index theory

Abstract

We study relativistic Kepler problems in the plane. At first, using non-smooth critical point theory, we show that under a general time-periodic external force of gradient type there are two infinite families of T-periodic solutions, parameterized by their winding number around the singularity: the first family is a sequence of local minima, while the second one comes from a mountain pass-type geometry of the action functional. Secondly, we investigate the minimality of the circular and non-circular periodic solutions of the unforced problem, via Morse index theory and level estimates of the action functional.