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The possible external couplings of an extended non-relativistic classicalsystem are characterized by gauging its maximal dynamical symmetry group at thecenter-of-mass. The Galilean one-time and two-times harmonic oscillators areexploited as models. The following remarkable results are then obtained: 1) apeculiar form of interaction of the system as a whole with the external gaugefields; 2) a modification of the dynamical part of the symmetrytransformations, which is needed to take into account the alteration of thedynamics itself, induced by the {\it gauge} fields. In particular, theYang-Mills fields associated to the internal rotations have the effect ofmodifying the time derivative of the internal variables in a scheme of minimalcoupling (introduction of an internal covariant derivative); 3) given theirdynamical effect, the Yang-Mills fields associated to the internal rotationsapparently define a sort of Galilean spin connection, while the Yang-Millsfields associated to the quadrupole momentum and to the internal energy havethe effect of introducing a sort of dynamically induced internal metric in therelative space.

Gauging kinematical and internal symmetry groups for extended systems: the Galilean one-time and two-times harmonic oscillators