Braids of the N-body problem by cabling a body in a central configuration

TL;DR

This paper proves the existence of special periodic braid solutions in the N-body problem by combining variational methods, blow-up techniques, and topological tools, extending previous numerical findings to rigorous mathematical results.

Contribution

It introduces a new analytical approach to construct braid solutions in the N-body problem by replacing a body with a pair's center of mass, using advanced reduction and topological methods.

Findings

Existence of periodic braid solutions near central configurations.

Application of Lyapunov-Schmidt reduction to the N-body problem.

Use of equivariant Lyusternik-Schnirelmann category to prove results.

Abstract

We prove the existence of periodic solutions of the N=(n+1)-body problem starting with n bodies whose reduced motion is close to a non-degenerate central configuration and replacing one of them by the center of mass of a pair of bodies rotating uniformly. When the motion takes place in the standard Euclidean plane, these solutions are a special type of braid solutions obtained numerically by C. Moore. The proof uses blow-up techniques to separate the problem into the n-body problem, the Kepler problem, and a coupling which is small if the distance of the pair is small. The formulation is variational and the result is obtained by applying a Lyapunov-Schmidt reduction and by using the equivariant Lyusternik-Schnirelmann category.