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      Hamilton-Jacobi Theorems for Nonholonomic Reducible Hamiltonian Systems on a Cotangent Bundle

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          Abstract

          Hamilton-Jacobi theorem reveals the deeply internal relationship between the generating function and the dynamical vector field of a Hamiltonian system. Because of the restriction given by constraints, in general, the dynamical vector field of nonholonomic Hamiltonian system is not Hamiltonian, however, it can be described by the dynamical vector field of a distributional Hamiltonian system. In this paper, we give two types of Hamilton-Jacobi equation for a distributional Hamiltonian system, by the calculation in detail, and we give also an exactly restricted condition between the solution of the Type II of Hamilton-Jacobi equation, for the distributional Hamiltonian system, and the solution of the Type II of Hamilton-Jacobi equation, for the associated unconstrained Hamiltonian system. Moreover, we generalize the above results to nonholonomic reducible Hamiltonian system with symmetry, as well as with momentum map, and obtain two types of Hamilton-Jacobi equations for the nonholonomic reduced distributional Hamiltonian systems. As an application, we give two examples to illustrate the theoretical results.

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          Journal
          1508.07548

          Differential equations & Dynamical systems,Geometry & Topology
          Differential equations & Dynamical systems, Geometry & Topology

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