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In this paper we study the reduction of a nonholonomic system by a group ofsymmetries in two steps. Using the so-called 'vertical-symmetry' condition, wefirst perform a 'compression' of the nonholonomic system leading to an almostsymplectic manifold. Second, we perform an (almost) symplectic reduction,relying on the existence of a momentum map. In this case, we verify that theresulting (almost) symplectic reduced spaces are the leaves of thecharacteristic foliation of the reduced nonholonomic bracket. On each leaf westudy the (Lagrangian) equations of motion, obtaining a nonholonomic version ofthe Lagrange-Routh equations, and we analyze the existence of a conformalfactor for the reduced bracket in terms of the existence of conformal factorsfor the almost symplectic leaves. We also relate the conditions for theexistence of a momentum map for the compressed system with gaugetransformations by 2-forms. The results are illustrated in three differentexamples.

Reduction of nonholonomic systems in two stages and Hamiltonization