The 4-Body Problem in a (1+1)-Dimensional Self-Gravitating System

TL;DR

This study explores the complex dynamics of a four-particle one-dimensional self-gravitating system, revealing chaotic and quasiperiodic behaviors through phase space analysis and Lyapunov exponents.

Contribution

It introduces a visualization method for the four-body problem using a single particle in a pyramid-shaped potential and generalizes state classification via Braid Group operators.

Findings

Identification of chaotic and quasiperiodic regions in phase space

Calculation of Lyapunov exponents showing stochasticity

Largest visualizable N-body system in this framework

Abstract

We report on the results of a study of the motion of a four particle non-relativistic one-dimensional self-gravitating system. We show that the system can be visualized in terms of a single particle moving within a potential whose equipotential surfaces are shaped like a box of pyramid-shaped sides. As such this is the largest $N$-body system that can be visualized in this way. We describe how to classify possible states of motion in terms of Braid Group operators, generalizing this to $N$ bodies. We find that the structure of the phase\textcolor{black}{{} space of each of these systems yields a large variety of interesting dynamics, containing regions of quasiperiodicity and chaos. Lyapunov exponents are calculated for many trajectories to measure stochasticity and previously unseen phenomena in the Lyapunov graphs are observed.