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

TL;DR

This paper proves the existence of symmetric, collision-free choreographies in the planar $n$-body problem with homogeneous potential, using topological methods and symmetry considerations, extending beyond variational and numerical approaches.

Contribution

It introduces a topological approach to establish symmetric choreographies with dihedral symmetry in the $n$-body problem, under nonresonance conditions, without relying on variational methods.

Findings

Existence of $D_n$-equivariant choreographies proven

Collision-free solutions established under nonresonance conditions

Spectral analysis and symmetry used to ensure compactness

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.