2013/01/01 by James Montaldi, Katrina Steckles, Katrina Mary Steckles · 12 citations
Engineering · Mathematics · Physics and Astronomy · #Astro and Planetary Science #Combinatorics #Control and Dynamics of Mobile Robots #Coset #Equivariant map #Geometry #Global symmetry #Group (periodic table) #Mathematics #One-dimensional symmetry group #Physics #Plane symmetry #Pure mathematics #Quantum mechanics #Rotational symmetry #Spacecraft Dynamics and Control #Spontaneous symmetry breaking #Symmetry (geometry) #Symmetry breaking #Symmetry group #Symmetry operation #Theoretical physics #Topology (electrical circuits) #math.DS #math.GT #msc:37C80 #msc:58E40 #msc:70F10
paper · pdf · doi:10.1017/fms.2013.5
published in Forum of Mathematics Sigma 1 (Cambridge University Press) · 45 pages, many figures. Minor changes from v.1
openalex publication_date 2013/01/01 · arxiv created 2013/11/14 · arxiv updated 2013/11/15 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 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.