In this paper we study the problem of Hamiltonization of nonholonomic systems from a geometric point of view. We use gauge transformations by 2-forms (in the sense of Ševera and Weinstein in Progr Theoret Phys Suppl 144:145 154 2001) to construct different almost Poisson structures describing the same nonholonomic system. In the presence of symmetries, we observe that these almost Poisson structures, although gauge related, may have fundamentally different properties after reduction, and that brackets that Hamiltonize the problem may be found within this family. We illustrate this framework with the example of rigid bodies with generalized rolling constraints, including the Chaplygin sphere rolling problem. We also see through these examples how twisted Poisson brackets appear naturally in nonholonomic mechanics. © 2012 Springer-Verlag.
Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems
Garcia-Naranjo L. C.
2012
Abstract
In this paper we study the problem of Hamiltonization of nonholonomic systems from a geometric point of view. We use gauge transformations by 2-forms (in the sense of Ševera and Weinstein in Progr Theoret Phys Suppl 144:145 154 2001) to construct different almost Poisson structures describing the same nonholonomic system. In the presence of symmetries, we observe that these almost Poisson structures, although gauge related, may have fundamentally different properties after reduction, and that brackets that Hamiltonize the problem may be found within this family. We illustrate this framework with the example of rigid bodies with generalized rolling constraints, including the Chaplygin sphere rolling problem. We also see through these examples how twisted Poisson brackets appear naturally in nonholonomic mechanics. © 2012 Springer-Verlag.
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