# Prescribed energy periodic solutions of Kepler problems with relativistic corrections

## TL;DR

This paper investigates the existence of prescribed energy periodic solutions in relativistic Kepler problems with perturbations, using bifurcation theory and KAM conditions to extend classical results to relativistic settings.

## Contribution

It introduces a framework for finding periodic solutions in relativistic Kepler problems with perturbations, applying Weinstein's bifurcation theory to nearly integrable Hamiltonian systems.

## Key findings

- Existence of periodic solutions bifurcating from invariant tori.
- Application of Weinstein's bifurcation theory to relativistic Kepler problems.
- Validation of KAM non-degeneracy conditions in relativistic contexts.

## Abstract

We consider two different relativistic versions of the Kepler problem in the plane: the first one involves the relativistic differential operator, the second one involves a correction for the usual gravitational potential due to Levi-Civita. When a small external perturbation is added into such equations, we investigate the existence of periodic solutions with prescribed energy bifurcating from periodic invariant tori of the unperturbed problems. Our main tool is an abstract bifurcation theory from periodic manifolds developed by Weinstein, which is applied in the case of nearly integrable Hamiltonian systems satisfying the usual KAM isoenergetic non-degeneracy condition.