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In this paper, we consider the hamiltonian formulation of nonholonomicsystems with symmetries and study several aspects of the geometry of theirreduced almost Poisson brackets, including the integrability of theircharacteristic distributions. Our starting point is establishing globalformulas for the nonholonomic Jacobiators, before and after reduction, whichare used to clarify the relationship between reduced nonholonomic brackets andtwisted Poisson structures. For certain types of symmetries (generalizing theChaplygin case), we obtain genuine Poisson structures on the reduced spaces andanalyze situations in which the reduced nonholonomic brackets arise by applyinga gauge transformation to these Poisson structures. We illustrate our resultswith mechanical examples, and in particular show how to recover severalwell-known facts in the special case of Chaplygin symmetries.

The Jacobiator of nonholonomic systems and the geometry of reduced nonholonomic brackets