Classification of symmetry groups for planar n-body choreographies
    James Montaldi, Katrina Steckles
    (Submitted on 2 May 2013)

    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.