ON THE KINEMATIC GEOMETRY OF MANY BODY SYSTEMS

1.IntroductionTheconfigurationspaceofagivenn-bodysystemconsistsofthetotalityofallpossibleconfigurationswhichcanbeconvenientlyrepresentedbythen-tuplesofpositionvectorswithrespecttoasuitablychosenorigin.Letujbethemassofthej-thparticle,bethetotalmassand...

In mechanics, both classical and quantum, one studies the profound interaction betweentwo types of energy, namely, the kinetic energy and the potential energy. The former can beorganized as the kinematic metric on the configuration space while the latter can be representedby a suitable potential function, such as the Newtonian potential in celestial mechanics andthe Coulomb potential in quantum mechanics of atomic and molecular physics. In this paper,the author studies the kinematic geometry of n-body systems. The main results are (i) theintroduction of a canonical coordinate svstem which reveals the total amount of kinematicsymmetry by an SO(3) x O(n-1) action in such a canonical coordinate representationt (ii) anin depth analysis of the above kinematic system both in the setting of classical invariant theoryand by the technique of equivariant Riemannian geometry; (iii) a remarkably simple formulafor the potential function in such a canonical coordinate system which reveals the well-fittingbetween the kinematic symmetry and the Potential energy.