The notion of gauge momenta is a generalization of the momentum map which is relevant for nonholonomic systems with symmetry. Weakly Noetherian functions are functions which are constants of motion of all `natural' nonholonomic systems with a given kinetic energy and any G-invariant potential energy. We show that, when the action of the symmetry group on the conguration manifold is free and proper, a function which is linear in the velocities is weakly-Noetherian if anf only if it is a gauge momenta which has a horizontal generator.
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