On a two-particles system associated to the one spatial dimensional Galilei group

TL;DR

This paper explores a two-particle system in one-dimensional Galilei spacetime, analyzing its dynamics through coadjoint orbits and distinguishing between isolated and non-isolated cases, revealing differences in barycenter motion.

Contribution

It extends the study of Galilean two-particle systems to one spatial dimension using coadjoint orbits, highlighting the behavior of barycenter coordinates in different physical scenarios.

Findings

Barycenter accelerates in non-isolated systems.

Barycenter moves with constant velocity in isolated systems.

Provides a framework for analyzing two-particle Galilean systems in 1D.

Abstract

In [8], M.Daumens and M.Perroud studied a two Galilean free particles system by realizing the three dimensional Galilei group on its maximal coadjoint orbit. In this paper we realize a similar study for the one spatial dimesnional Galilei group. As its maximal coadjoint orbit describes a non free massive particle [6], this gives us a two non free galilean partcles system. Use of the barycenter coordinates gives rise to the notions of a total force and a relative force similar to those of a total linaer momentum and a relative linear momentum ([]8),[9],[10]). We will distinguish the case of a non isolated system from that of an isolated one. We will show that the barycenter is accelerated in the first case while it moves with a constant velocity in the second one.