Analytic Harmonic Tunable Closed Orbits of Trapped N-Body Systems

TL;DR

This paper presents an analytic solution for the motion of particles in N-body systems with linear interactions, showing conditions for closed orbits and providing trajectories for various force configurations.

Contribution

It introduces a fully analytic, separable solution for N-body systems with linear forces, detailing conditions for closed orbits in different trap and swarm interactions.

Findings

All particle motions are analytically solvable and separable.

Closed periodic orbits occur under specific force constant ratios.

Trajectories are illustrated for various force configurations.

Abstract

The motion of each particle in an N body system of identical masses interacting via an attractive or repulsive pairwise linear force law, the "Swarm," and with an external attractive or repulsive linear force law, the "Trap," is considered. In all Swarm and Trap combinations the motion of all N particles is completely separable and positions are found as a function of time in simple analytic form. in all attractive Traps the center of mass of the Swarm is bound to the center of the trap. For attractive or weakly repulsive Swarms in an attractive Trap the particles are bound to the center of mass of the Swarm and an infinite set of Swarm to Trap force constant ratios result in every particle executing a closed periodic orbit. Figures are provided of trajectories for various combinations of Swarm and Trap force constants. The derivations are suitable for the advanced undergraduate classroom.