Continuous Measure of Symmetry as a Dynamic Variable: a New Glance on the Three-Body Problem

TL;DR

This paper investigates how a continuous measure of symmetry evolves over time in three-body systems with gravitational and electrostatic interactions, revealing different behaviors such as collapse or stabilization.

Contribution

It introduces a dynamic analysis of symmetry in three-body problems, highlighting how the measure behaves under different interaction types and initial conditions.

Findings

In gravitational interactions, the system collapses with increasing symmetry measure.

In electrostatic interactions, the symmetry measure stabilizes over time.

The measure's behavior depends on initial configurations and charge-to-mass ratios.

Abstract

The time evolution of the continuous measure symmetry for the system built of the three bodies interacting via the potential U(r)~1/r is reported. Gravitational and electrostatic interactions between the point bodies were addressed. In the case of the pure gravitational interaction the three-body-system deviated from its initial symmetrical location, described by the Lagrange equilateral triangle, comes to collapse, accompanied by the growth of the continuous measure of symmetry. When three point bodies interact via the Coulomb repulsive interaction, the time evolution of CMS is quite different. CMS calculated for all of studied initial configurations of the point charges and all of their charge-to-mass ratios always comes with time to its asymptotic value, evidencing the stabilization of the shape of the triangle, constituted by the interacting bodies.