Periodic solutions to relativistic Kepler problems: a variational approach

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 noncircular periodic solutions of the unforced problem, via Morse index theory and level estimates of the action functional.