Classification of symmetry groups for planar n-body choreographies

TL;DR

This paper systematically classifies all symmetry groups of planar n-body choreographies, analyzes their topological properties, and demonstrates the existence of many new choreographies, advancing the understanding of symmetric periodic solutions.

Contribution

It provides a comprehensive classification of symmetry groups for planar n-body choreographies and links these symmetries to the topology of the solution space.

Findings

Classified all symmetry groups into two infinite families and three exceptional groups for odd n.

Developed the equivariant fundamental group to analyze the topology of loop spaces with symmetry.

Proved the existence of many new choreographies based on the symmetry and topological analysis.

Abstract

Since the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of the n-body problem: periodic motions where the n bodies all follow one another at regular intervals along a closed path. The principal approach combines variational methods with symmetry properties. In this paper, we give a systematic treatment of the symmetry aspect. In the first part we classify all possible symmetry groups of planar n-body, collision-free choreographies. These symmetry groups fall in to 2 infinite families and, if n is odd, three exceptional groups. In the second part we develop the equivariant fundamental group and use it to determine the topology of the space of loops with a given symmetry, which we show is related to certain cosets of the pure braid group in the full braid group, and to centralizers of elements of the corresponding coset. In particular, we refine the symmetry classification by classifying the connected components of the set of loops with any given symmetry. This leads to the existence of many new choreographies in $n$-body systems governed by a strong force potential.