Prescribed energy periodic solutions of Kepler problems with relativistic corrections (Q6643878)

The authors consider the problem of finding periodic solutions with a prescribed energy of two equations related to the Kepler problem. The first one involves the relativistic differential operator and reads as \N\[\N\frac{d}{dt}\left(\frac{m\dot x}{\sqrt{1-|\dot x|^2/c^2}}\right)=-\alpha\frac{x}{|x|^3}+\varepsilon\nabla U(x). \N\]\NHere \(U\) is an external potential, and \(\varepsilon\) is a small parameter. The second equation reads as \N\[\Nm\ddot x=-\kappa\frac{x}{|x|^3}-2\lambda\frac{x}{|x|^4}+\varepsilon\nabla U(x). \N\]\NIn both cases, they prove the existence, multiplicity and localization of periodic solutions winding around the singularity.\N\NFor the entire collection see [Zbl 1544.34003].