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

2025/05/03 by Kajihara, Yuika, Kin, Eiko, Shibayama, Mitsuru
#57K10 #57K20 #70F10 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2505.01867

Abstract

We study periodic solutions of the planar Newtonian N-body problem with equal masses. Each periodic solution traces out a braid with N strands in 3-dimensional space. When the braid is of pseudo-Anosov type, it has an associated stretch factor greater than 1, which reflects the complexity of the corresponding periodic solution. For each N ≥ 3, Guowei Yu established the existence of a family of simple choreographies to the planar Newtonian N-body problem. We prove that braids arising from Yu's periodic solutions are of pseudo-Anosov types, except in the special case where all particles move along a circle. We also identify the simple choreographies whose braid types have the largest and smallest stretch factors, respectively.

Citations

• Braids, metallic ratios and periodic solutions of the 2n-body problem
• An asymptotic behavior of the dilatation for a family of pseudo-Anosov braids
• A family of pseudo-Anosov braids with small dilatation
• The minimum dilatation of pseudo-Anosov 5-braids
• Train-tracks for surface homeomorphisms

Related

• Braids, metallic ratios and periodic solutions of the 2n-body problem
• Braids of the N-body problem II: carousel solutions by cabling central configurations
• Simple choreographies of the planar Newtonian N-body Problem
• Choreographies with Dihedral Symmetry in the Planar n-Body Problem
• New Periodic Solutions for Some Planar N+3-Body Problems with Newtonian Potentials