Braids, metallic ratios and periodic solutions of the 2n-body problem

2022/04/04 by Yuika Kajihara, Kajihara, Yuika, Eiko Kin +3 · 1 citation
Engineering · Physics and Astronomy · #57K10 #57K20 #70F10 #Astro and Planetary Science #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Nuclear physics research studies #Spacecraft Dynamics and Control

paper · pdf · doi:10.48550/arxiv.2204.01420

openalex publication_date 2022/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Periodic solutions of the planar N-body problem determine braids through the trajectory of N bodies. Braid types can be used to classify periodic solutions. According to the Nielsen-Thurston classification of surface automorphisms, braids fall into three types: periodic, reducible and pseudo-Anosov. To a braid of pseudo-Anosov type, there is an associated stretch factor greater than 1, and this is a conjugacy invariant of braids. In 2006, the third author discovered a family of multiple choreographic solutions of the planar 2n-body problem. We prove that braids obtained from the solutions in the family are of pseudo-Anosov type, and their stretch factors are expressed in metallic ratios. New numerical periodic solutions of the planar 2n-body problem are also provided.

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• A study of braids arising from simple choreographies of the planar Newtonian N-body problem

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