GAUGING KINEMATICAL AND INTERNAL SYMMETRY GROUPS FOR EXTENDED SYSTEMS- THE GALILEAN ONE-TIME AND 2-TIMES HARMONIC-OSCILLATORS

The possible external couplings of an extended non-relativistic classi cal system are characterized by gauging its maximal dynamical symmetry group at the centre-of-mass. The Galilean one-time and two-times harm onic oscillators are exploited as models. The following remarkable res ults are then obtained: 1), a peculiar form of interaction of the syst em as a whole with the external gauge fields; 2), a modification of th e dynamical part of the symmetry transformations, which is needed to t ake into account the alteration of the dynamics itself, induced by the gauge fields. In particular, the Yang-Mills fields associated to the internal rotations have the effect of modifying the time derivative of the internal variables in a scheme of minimal coupling (introduction of an internal covariant derivative); 3), given their dynamical effect , the Yang-Mills fields associated to the internal rotations apparentl y define a sort of Galilean spin connection, while the Yang-Mills fiel ds associated to the quadrupole momentum and to the internal energy ha ve the effect of introducing a sort of dynamically induced internal me tric in the relative space.