Montaldi, James and Steckles, Katrina (2013) Classification of symmetry groups for planar nbody choreographies. Forum of Mathematics, Sigma, 1. ISSN 20505094 (In Press)
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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 nbody 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 nbody, collisionfree 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 nbody systems governed by a strong force potential.
Item Type:  Article 

Uncontrolled Keywords:  Equivariant dynamics, nbody 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:  14 Nov 2013 
Last Modified:  20 Oct 2017 14:13 
URI:  https://eprints.maths.manchester.ac.uk/id/eprint/2063 
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Classification of symmetry groups for planar nbody choreographies. (deposited 22 May 2013)
 Classification of symmetry groups for planar nbody choreographies. (deposited 14 Nov 2013) [Currently Displayed]
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