Braids with the symmetries of Platonic polyhedra in the Coulomb (N+1)-body problem

TL;DR

This paper investigates symmetric periodic orbits in a Coulomb (N+1)-body problem with particles arranged according to Platonic symmetries, using numerical methods rather than variational minimization.

Contribution

It introduces a numerical approach to compute symmetric periodic orbits in Coulombic N-body systems constrained by Platonic symmetries, where traditional variational methods are ineffective.

Findings

Computed periodic orbits for N=12, 24, 60.

Orbits are not minimizers of the action functional.

Numerical methods successfully find symmetric solutions.

Abstract

We take into account the Coulomb (N + 1)-body problem with N = 12, 24, 60. One of the particles has positive charge Q > 0, and the remaining N have all the same negative charge q < 0. These particles move under the Coulomb force, and the positive charge is assumed to be at rest at the center of mass. Imposing a symmetry constraint, given by the symmetry group of the Platonic polyhedra, we were able to compute periodic orbits, using a shooting method and continuation with respect to the value Q of the positive charge. In the setting of the classical N -body problem, the existence of such orbits is proved with Calculus of Variation techniques, by minimizing the action functional. Here this approach does not seem to work, and numerical computations show that the orbits we compute are not minimizers of the action.