In this paper we study the problem of Hamiltonization of nonholonomic systemsfrom a geometric point of view. We use gauge transformations by 2-forms (in thesense of Severa and Weinstein [29]) to construct different almost Poissonstructures describing the same nonholonomic system. In the presence ofsymmetries, we observe that these almost Poisson structures, although gaugerelated, may have fundamentally different properties after reduction, and thatbrackets that Hamiltonize the problem may be found within this family. Weillustrate this framework with the example of rigid bodies with generalizedrolling constraints, including the Chaplygin sphere rolling problem. We alsosee how twisted Poisson brackets appear naturally in nonholonomic mechanicsthrough these examples.
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