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

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.

43 pages. To appear in Archive for Rational Mechanics and Analysis

References in corpus (3)

• Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces
• The Rolling Motion of a Ball on a Surface. New Integrals and Hierarchy of Dynamics
• Poisson structures for reduced non-holonomic systems

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• Hamiltonian Structure and Nonlinear Stability of Steady Solutions of the Generalized Hasegawa-Mima Equation for Drift Wave Turbulence in Curved Magnetic Fields
• Guiding Center Derivation of the Generalized Hasegawa-Mima Equation for Drift Wave Turbulence in Curved Magnetic Fields
• Poissonization of Three Dimensional Nonholonomic Dynamics with the Method of Extension
• Reduction of hybrid Hamiltonian systems with non-equivariant momentum maps
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