Choreographies with Dihedral Symmetry in the Planar n-Body Problem

2025/11/18 by Cerritos, Juan Manuel Sánchez
Computer Science · Engineering · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Optimization and Variational Analysis #Quantum chaos and dynamical systems #Spacecraft Dynamics and Control

paper · doi:10.48550/arxiv.2511.14170

openalex publication_date 2025/11/18 · openalex created_date 2025/11/20 · openalex updated_date 2026/07/28

Abstract

We prove the existence of planar D_n--equivariant choreographies in the n--body problem with homogeneous potential of degree -α, 0<α<2. Each body follows the same closed path, rotated and time-shifted, forming a choreography whenever the winding number W is coprime with n. Using Mawhin's coincidence degree, we establish collision-free periodic solutions under a simple nonresonance condition. The proof relies on the spectral structure of the linearized operator, symmetry-induced separation of the bodies, and uniform energy bounds ensuring compactness of the nonlinear term. This provides a topological route to choreographies beyond variational and numerical frameworks.

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