New Directions in Non-Relativistic and Relativistic Rotational and Multipole Kinematics for N-Body and Continuous Systems

Abstract

In non-relativistic mechanics the center of mass of an isolated system is easily separated out from the relative variables. For a N-body system these latter are usually described by a set of Jacobi normal coordinates, based on the clustering of the centers of mass of sub-clusters. The Jacobi variables are then the starting point for separating {\it orientational} variables, connected with the angular momentum constants of motion, from {\it shape} (or {\it vibrational}) variables. Jacobi variables, however, cannot be extended to special relativity. We show by group-theoretical methods that two new sets of relative variables can be defined in terms of a {\it clustering of the angular momenta of sub-clusters} and directly related to the so-called {\it dynamical body frames} and {\it canonical spin bases}. The underlying group-theoretical structure allows a direct extension of such notions from a non-relativistic to a special- relativistic context if one exploits the {\it rest-frame instant form of dynamics}. The various known definitions of relativistic center of mass are recovered. The separation of suitable relative variables from the so-called {\it canonical internal} center of mass leads to the correct kinematical framework for the relativistic theory of the orbits for a N-body system with action -at-a-distance interactions. The rest-frame instant form is also shown to be the correct kinematical framework for introducing the Dixon multi-poles for closed and open N-body systems, as well as for continuous systems, exemplified here by the configurations of the Klein-Gordon field that are compatible with the previous notions of center of mass.