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The Jacobiator of Nonholonomic Systems and the Geometry of Reduced Nonholonomic Brackets

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Abstract

In this paper, we consider the Hamiltonian formulation of nonholonomic systems with symmetries and study several aspects of the geometry of their reduced almost Poisson brackets, including the integrability of their characteristic distributions. Our starting point is establishing global formulas for the nonholonomic Jacobiators, before and after reduction, which are used to clarify the relationship between reduced nonholonomic brackets and twisted Poisson structures. For certain types of symmetries (generalizing the Chaplygin case), we obtain genuine Poisson structures on the reduced spaces and analyze situations in which the reduced nonholonomic brackets arise by applying a gauge transformation to these Poisson structures. We illustrate our results with mechanical examples, and in particular show how to recover several well-known facts in the special case of Chaplygin symmetries.

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  1. Instituto de Matemática, Universidade Federal Fluminense, Rua Mario Santos Braga S/N, 24020-140, Niterói, Rio de Janeiro, Brazil

    Paula Balseiro

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  1. Paula Balseiro
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Correspondence to Paula Balseiro.

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Communicated by M. Ortiz

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Balseiro, P. The Jacobiator of Nonholonomic Systems and the Geometry of Reduced Nonholonomic Brackets. Arch Rational Mech Anal 214, 453–501 (2014). https://doi.org/10.1007/s00205-014-0759-4

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