Choreographies and braids in the $n$-body problem

Carlos García Azpeitia
Universidad Nacional Autónoma de México, Mexico City

In this talk, we present a summary of results about the existence of periodic solutions for $n$ bodies that are known as choreographies and braids. In the second part of the talk, we explain how to obtain the existence of a special type of braid solutions where one body in a central configuration of $n$ bodies is replaced by a pair of bodies rotating uniformly around its center of mass. In these solutions $n-1$ bodies and the center of mass of the pair winds around the origin $q$ times, while the pair of bodies winds around its center of mass $p$ times. The proof uses blow-up techniques to separate the $\left( n+1\right) $-body problem as 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 the use of the equivariant Lyusternik-Schnirelmann category.