Classification of symmetry groups for planar n-body choreographies

Montaldi, James and Steckles, Katrina (2013) Classification of symmetry groups for planar n-body choreographies. Forum of Mathematics: Sigma, 1 (2013, e5). pp. 1-55. ISSN 1749-9097

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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.

Item Type: Article
Uncontrolled Keywords: Equivariant dynamics, n-body problem, variational problems, loop space, equivariant topology, braid group
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory
MSC 2010, the AMS's Mathematics Subject Classification > 58 Global analysis, analysis on manifolds
MSC 2010, the AMS's Mathematics Subject Classification > 70 Mechanics of particles and systems
Depositing User: Dr James Montaldi
Date Deposited: 22 May 2013
Last Modified: 28 Apr 2018 10:33
URI: https://eprints.maths.manchester.ac.uk/id/eprint/1982

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