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

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.