On existence of periodic solutions for Kepler type problems

TL;DR

This paper establishes the existence and multiplicity of periodic solutions in Kepler-type problems using topological methods, relating solutions to geometric and topological invariants, with applications to the restricted n-body problem.

Contribution

It introduces new topological criteria for guaranteeing multiple periodic solutions in Kepler-type problems, extending to the restricted n-body problem.

Findings

Proves existence of multiple periodic solutions based on winding numbers and knot types.

Provides lower bounds for the number of solutions using topological invariants.

Applies results to the restricted n-body problem.

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.