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

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.