Exactly integrable analogue of a one-dimensional gravitating system

Abstract

Exchange symmetry in acceleration partitions the configuration space of an N particle, onedimensional, 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.