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arXiv · 2511.14170

Choreographies with Dihedral Symmetry in the Planar $n$-Body Problem

Juan Manuel Sánchez Cerritos
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Abstract

We prove the existence of planar $D_n$--equivariant choreographies in the $n$--body problem with homogeneous potential of degree $-\alpha$, $0<\alpha<2$. 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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Juan Manuel Sánchez Cerritos. 2025-11-18. Choreographies with Dihedral Symmetry in the Planar $n$-Body Problem. https://arxiv.org/abs/2511.14170

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