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Reduction of almost Poisson brackets and Hamiltonization of the Chaplygin sphere. (English) Zbl 1278.70009

Summary: We construct different almost Poisson brackets for nonholonomic systems than those existing in the literature and study their reduction. Such brackets are built by considering non-canonical two-forms on the cotangent bundle of configuration space and then carrying out a projection onto the constraint space that encodes the Lagrange-D’Alembert principle. We justify the need for this type of brackets by working out the reduction of the celebrated Chaplygin sphere rolling problem. Our construction provides a geometric explanation of the Hamiltonization of the problem given by A. V. Borisov and I. S. Mamaev [Sib. Mat. Zh. 48, No. 1, 33–45 (2007); translation in Sib. Math. J. 48, No. 1, 26–36 (2007; Zbl 1164.37342)].

MSC:

70F25 Nonholonomic systems related to the dynamics of a system of particles
37J60 Nonholonomic dynamical systems
70E18 Motion of a rigid body in contact with a solid surface
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics

Citations:

Zbl 1164.37342
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