Exactly Integrable Analogue of a One-dimensional Gravitating System

TL;DR

This paper introduces an exactly integrable analogue of a one-dimensional gravitational system by exploiting exchange symmetry, enabling the construction of a related central force problem with comparable dynamics.

Contribution

The authors develop an integrable model of a 1D gravitational system using symmetry properties, providing a new approach for analyzing such systems.

Findings

Excellent agreement between the integrable model and the original system for certain initial conditions

Construction of trajectories in the integrable analogue

Analysis of continuum limits of the model

Abstract

Exchange symmetry in acceleration partitions the configuration space of an N particle, one-dimensional, gravitational system into N! equivalent cells. We take advantage of the resulting small angular extent of each cell to construct a related integrable version of the system that takes the form of a central force problem in N-1 dimensions. The properties of the latter, including the construction of trajectories, as well as several continuum limits, are developed. Dynamical simulation is employed to compare the two models. For a class of initial conditions, excellent agreement is observed.