Nonholonomic Noether theorem and reduction of symmetries

A necessary and sufficient condition for a function to be a constant of motion of a Lagrangian system with nonlinear nonholonomic constraints is derived in the framework of the Hamiltonian formulation. It is shown that the symmetries generated by constants of motion form a normal subgroup G0 of the symmetry group G of the theory. The reduction of symmetries, taking into account that G0 is generated by constants of motion, decreases the number of the dynamical variables by dim G + dim Gμ, where Gμ ⊂ G0 is the isotropy group of the value μ of the conserved momentum.