## 2013.26: Classification of symmetry groups for planar n-body choreographies

2013.26:
James Montaldi and Katrina Steckles
(2013)
*Classification of symmetry groups for planar n-body choreographies.*
Forum of Mathematics, Sigma, 1.
ISSN 2050-5094

*This is the latest version of this eprint.*

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DOI: 10.1017/fms.2013.5

## 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.

Item Type: | Article |
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Uncontrolled Keywords: | Equivariant dynamics, n-body problem, variational problems, loop space, equivariant topology, braid group |

Subjects: | MSC 2000 > 37 Dynamical systems and ergodic theory MSC 2000 > 58 Global analysis, analysis on manifolds MSC 2000 > 70 Mechanics of particles and systems |

MIMS number: | 2013.26 |

Deposited By: | Dr James Montaldi |

Deposited On: | 14 November 2013 |

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