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CLASSIFICATION OF SYMMETRY GROUPS FOR PLANAR $n$-BODY CHOREOGRAPHIES

Part of: Smooth dynamical systems: general theory Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  13 December 2013

JAMES MONTALDI  and
KATRINA STECKLES
JAMES MONTALDI
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UKj.montaldi@manchester.ac.uk
KATRINA STECKLES
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UKj.montaldi@manchester.ac.uk

Abstract

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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 into two 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.

MSC classification

Secondary: 37C80: Symmetries, equivariant dynamical systems 70F10: $n$-body problems 58E40: Group actions
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 1 , 2013 , e5
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2013.

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