Exact General Solutions to Extraordinary N-body Problems

TL;DR

This paper presents exact solutions for a class of N-body problems where the potential depends on the system's radius, revealing persistent breathing modes and a novel statistical equilibrium.

Contribution

It provides the first exact solutions for N-body systems with potentials based on the radius, including cases with inverse-square scaling, and describes their unique vibrational and equilibrium properties.

Findings

Fundamental breathing mode vibrates non-linearly forever.

Statistical equilibrium persists despite continual radius change.

Solutions apply to potentials depending on the radius and inverse-square scaling.

Abstract

We solve the N-body problems in which the total potential energy is any function of the mass-weighted root-mean-square radius of the system of N point masses. The fundamental breathing mode of such systems vibrates non-linearly for ever. If the potential is supplemented by any function that scales as the inverse square of the radius there is still no damping of the fundamental breathing mode. For such systems a remarkable new statistical equilibrium is found for the other coordinates and momenta, which persists even as the radius changes continually.