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