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

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.& COPY; 2023 Elsevier B.V. All rights reserved.