A new perspective on nonholonomic brackets and Hamilton–Jacobi theory

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Apr 2024Embargo end date: 01 Jan 2023 Spain, Spain English Publisher:Elsevier BVJournal:Journal of Geometry and Physics, volume 198, page 105,116 (issn: 0393-0440,

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Authors: Manuel de León; Manuel Lainz; Asier López-Gordón; Juan Carlos Marrero;

doi: 10.1016/j.geomphys.2024.105116 , 10.48550/arxiv.2307.06049

arXiv: 2307.06049

handle: 10016/47341 , 10261/380818

A new perspective on nonholonomic brackets and Hamilton–Jacobi theory

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Abstract

The nonholonomic dynamics can be described by the so-called nonholonomic bracket on the constrained submanifold, which is a non-integrable modification of the Poisson bracket of the ambient space, in this case, of the canonical bracket on the cotangent bundle of the configuration manifold. On the other hand, another bracket, also called nonholonomic bracket, was defined using the description of the problem in terms of skew-symmetric algebroids. Recently, reviewing two older papers by R. J. Eden, we have defined a new bracket which we call Eden bracket. In the present paper, we prove that these three brackets coincide. Moreover, the description of the nonholonomic bracket à la Eden has allowed us to make important advances in the study of Hamilton-Jacobi theory and the quantization of nonholonomic systems.

36 pages, no figures. Final version appearing the journal

Countries

Spain, Spain

Related Organizations

Universidad de La Laguna (ULL)
Spain
Institute of Mathematical Sciences
Spain
University of La Laguna
Spain
Spanish National Research Council
Spain

Keywords

nonholonomic mechanics, Mathematics - Differential Geometry, Matemáticas, skew-symmetric algebroids, primary: 37J60, 70F25, 70H20, secondary: 53D17, 53Z05, 70G45, FOS: Physical sciences, Dynamical Systems (math.DS), Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.), Hamilton-Jacobi equations in mechanics, Almost Poisson brackets, Skew-symmetric algebroids, Hamilton–Jacobi equation, FOS: Mathematics, Mathematics - Dynamical Systems, Mathematical Physics, almost Poisson brackets, Nonholonomic systems related to the dynamics of a system of particles, Ingeniería mecánica, Mathematical Physics (math-ph), Hamilton-Jacobi equation, Poisson manifolds; Poisson groupoids and algebroids, Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics, Differential Geometry (math.DG), Nonholonomic mechanics, Nonholonomic dynamical systems, Almost poisson brackets

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