Gauge conservation laws and the momentum equation in nonholonomic mechanics

The gauge mechanism is a generalization of the momentum map which links conservation laws to symmetry groups of nonholonomic systems. This method has been so far employed to interpret conserved quantities as momenta of vector fields which are sections of the constraint distribution. In order to obtain the largest class of conserved quantities of this type, we extend this method to an over-distribution of the constraint distribution, the so-called reaction-annihilator distribution, which encodes the effects that the nonholonomic reaction force has on the conservation laws. We provide examples showing the effectiveness of this generalization. Furthermore, we discuss the Noetherian properties of these conserved quantities, that is, whether and to which extent they depend only on the group, and not on the system. In this context, we introduce a notion of ‘weak Noetherianity’. Finally, we point out that the gauge mechanism is equivalent to the momentum equation (at least for locally free actions), we generalize the momentum equation to the reaction-annihilator distribution, and we introduce a ‘gauge momentum map’ which embodies both methods. For simplicity, we treat only the case of linear constraints, natural Lagrangians, and lifted actions.