Superintegrability and choreographic obstructions in dihedral $n$-body Hamiltonian systems

TL;DR

This paper investigates the symmetry constraints and conditions for choreographic solutions in planar dihedral $n$-body Hamiltonian systems, revealing how superintegrability and phase-matching influence periodic motions.

Contribution

It identifies the symmetry obstructions to choreographies and distinguishes conditions for superintegrability, periodicity, and choreography in dihedral $n$-body systems.

Findings

Choreographies are collision-free solutions with bodies on a single closed curve.

Superintegrability and choreography are governed by distinct frequency and phase-matching conditions.

True choreographies occur only on phase-matched loci or through degeneracy in specific sectors.

Abstract

We analyze planar $n$-body Hamiltonian systems with quadratic $D_n$-invariant interactions and identify the symmetry obstruction to choreographic motion. Choreographies are taken throughout to be collision-free solutions of the equations of motion in which all bodies traverse one closed curve with uniform time shifts. By diagonalizing the dynamics into discrete Fourier sectors, we show that superintegrability, periodicity, and choreography are governed by distinct conditions: commensurability of the active frequencies closes bounded motions, whereas a sectorwise $C_n$ phase-matching condition is required for full equivariance. At the configuration level this equivariance is already equivalent to a genuine simple choreography. Thus generic resonant multi-sector motions are periodic but multi-trace, while true choreographies occur only on phase-matched loci, in single irreducible sectors, or through effective one-sector reductions produced by exact degeneracy. The cases $n=4,5,6$ exhibit this mechanism explicitly, with $n=6$ marking the first distinction between nondegenerate commensurability and additional exact degeneracy.