Nearly-circular periodic solutions of perturbed relativistic Kepler problems: the fixed-period and the fixed-energy problems

TL;DR

This paper investigates the existence of nearly-circular periodic solutions in a perturbed relativistic Kepler problem, focusing on fixed-period and fixed-energy cases, and demonstrates bifurcation from circular solutions when perturbations are small.

Contribution

It provides new results on bifurcation of periodic solutions in relativistic Kepler problems under small perturbations, covering both fixed-period and fixed-energy scenarios.

Findings

Existence of bifurcating periodic solutions for small perturbations.

Analysis of solutions in both time-periodic and autonomous perturbation cases.

Extension of classical Kepler problem results to relativistic and perturbed settings.

Abstract

The paper studies the existence of periodic solutions of a perturbed relativistic Kepler problem of the type \begin{equation*} \dfrac{\mathrm{d}}{\mathrm{d}t}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right) = -\alpha\frac{x}{|x|^{3}} + \varepsilon \, \nabla_{x} U(t,x), \qquad x \in \mathbb{R}^d\setminus\{0\}, \end{equation*} with $d=2$ or $d=3$, bifurcating, for $\varepsilon$ small enough, from the set of circular solutions of the unperturbed system. Both the case of the fixed-period problem (assuming that $U$ is $T$-periodic in time) and the case of the fixed-energy problem (assuming that $U$ is independent of time) are considered.