Periodic oscillations in a 2N-body problem

TL;DR

This paper proves the existence of periodic solutions called hip-hop solutions in the 2N-body problem, explores their symmetry properties, and identifies bifurcations leading to new solution branches, including choreographies.

Contribution

It establishes the existence of a family of periodic hip-hop solutions with specific symmetry properties and analyzes bifurcations leading to solutions with different symmetries.

Findings

Existence of periodic hip-hop solutions for all N and m.

Bifurcation analysis reveals branches with broken symmetry.

Explicit examples of choreographies among the solutions.

Abstract

Hip-Hop solutions of the $2N$-body problem are solutions that satisfy at every instance of time, that the $2N$ bodies with the same mass $m$, are at the vertices of two regular $N$-gons, each one of these $N$-gons are at planes that are equidistant from a fixed plane $\Pi_0$ forming an antiprism. In this paper, we first prove that for every $N$ and every $m$ there exists a family of periodic hip-hop solutions. For every solution in these families the oriented distance to the plane $\Pi_0$, which we call $d(t)$, is an odd function that is also even with respect to $t=T$ for some $T>0.$ For this reason we call solutions in these families, double symmetric solutions. By exploring more carefully our initial set of periodic solutions, we numerically show that some of the branches stablished in our existence theorem have bifurcations that produce branches of solutions with the property that the oriented distance function $d(t)$ is not even with respect to any $T>0$, we call these solutions single symmetry solutions. We prove that no single symmetry solution is a choreography. We also display explicit double symmetric solutions that are choreographies.