Reduction of Almost Poisson brackets and Hamiltonization of the Chaplygin Sphere

Abstract

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.

References in corpus (3)

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• Hamiltonization of solids of revolution through reduction
• First Integrals and symmetries of nonholonomic systems
• Generalisation of Chaplygin's Reducing Multiplier Theorem with an application to multi-dimensional nonholonomic dynamics
• A Geometric Characterization of Certain First Integrals for Nonholonomic Systems with Symmetries
• $G$-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball
• Hamiltonian form for general autonomous ODE systems: Low dimensional examples
• Linear connections and shape maps for second order ODEs with and without constraints