On existence of periodic solutions for Kepler type problems
Pablo Amster, Julián Haddad
TL;DR
This work proves existence and multiplicity results for periodic solutions of forced Kepler-type problems with singular central forces by blending variational/topological methods. The authors reformulate the problem via a Leray-Schauder degree framework, analyze a family of fixed-point operators $\mathcal{G}_{\epsilon}$, and connect zeros to a finite-dimensional map $F$ whose degree governs solution existence for small $\epsilon$ (large $\lambda$). They derive lower bounds linked to topology: the winding number of the forcing second primitive $H$ in the planar case, and the homology of the curve complement in higher dimensions; they also show generic forcing yields nondegenerate, multiple solutions. In dimension 3, the knot type of $H$ further refines the lower bounds through Morse-theoretic considerations and tunnel numbers, while in the restricted $n$-body problem, the geometry of the union of forcing curves governs solution multiplicity. Overall, the paper builds a comprehensive topological framework connecting singular celestial mechanics with knot theory, homology, and genericity to guarantee multiple periodic orbits across dimensions and configurations.
Abstract
We prove existence and multiplicity of periodic motions for the forced 2-body problem under conditions of topological character. In the different cases, the lower bounds obtained for the number of solutions are related to the winding number of a curve in the plane, the homology of a space in $\R^3$, the knot type of a curve and the link type of a set of curves. Also, the results are applied to the restricted $n$-body problem.