Exactly Integrable Analogue of a One-dimensional Gravitating System

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.