N-body choreographies on a p-limacon curve

2025/04/30 by M. Fernández‐Guasti, Fernandez-Guasti, Manuel, Toshiaki Fujiwara +5
Engineering · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Spacecraft Dynamics and Control

paper · pdf · doi:10.48550/arxiv.2504.21713

openalex publication_date 2025/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider an N--body problem under a harmonic potential of the form (1)/(2)∑ κ_jl |q_j-q_l|². A p-limaçon curve is a planar curve parametrized by t given by a(cos t,sin t)+b(cos pt, sin pt), where a,b∈ ℝ, p ∈ ℤ, and t ∈ [0,2π]. We study N-body choreographic motions constrained to a p-limaçon curve and establish necessary and sufficient conditions for their existence. Specifically, we prove that choreographic motions exist if and only if p/N, (p ± 1)/N ∉ ℤ. Under an additional symmetry assumption on the force coefficients, we further refine these conditions. We also analyze the occurrence of collisions, showing that for given p and N, at most 2(N-1) choices of a/b lead to collisions. Furthermore, we find additional conserved quantities.

Citations

• New Periodic Solutions for Newtonian n-Body Problems with Dihedral Group Symmetry and Topological Constraints
• Variational proof of the existence of the super-eight orbit in the four-body problem
• On the dihedral n-body problem
• Symmetry groups and non-planar collisionless action-minimizing solutions of the three-body problem in three-dimensional space
• Action minimizing orbits in the n-body problem with simple choreography constraint
• On the Existence of Collisionless Equivariant Minimizers for the Classical n-body Problem
• Choreographic Three Bodies on the Lemniscate

Related

• On the four-body limaçon choreography: maximal superintegrability and choreographic fragmentation
• Simple choreographies of the planar Newtonian N-body Problem
• Choreographies with Dihedral Symmetry in the Planar n-Body Problem
• A Variational Approach to Planar Choreographies via Ekeland's Principle
• On the n-body problem on surfaces of revolution