  The possible external couplings of an extended non-relativistic classical
system are characterized by gauging its maximal dynamical symmetry group at the
center-of-mass. The Galilean one-time and two-times harmonic oscillators are
exploited as models. The following remarkable results are then obtained: 1) a
peculiar form of interaction of the system as a whole with the external gauge
fields; 2) a modification of the dynamical part of the symmetry
transformations, which is needed to take into account the alteration of the
dynamics itself, induced by the {\it 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
apparently define a sort of Galilean spin connection, while the Yang-Mills
fields associated to the quadrupole momentum and to the internal energy have
the effect of introducing a sort of dynamically induced internal metric in the
relative space.

General Relativity and Quantum Cosmology