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| In this talk, we first generalize the geometric Hamilton-Jacobi theorem for Hamiltonian system to the non-holonomic context, and obtain a Hamilton-Jacobi theorem for non-holonomic Hamiltonian system on the cotangent bundle of a configuration manifold, by using its distributional Hamiltonian system under a weaker condition. Then we generalize the above result to non-holonomic reducible Hamiltonian system with symmetry, as well as with momentum map, and obtain the Hamilton-Jacobi theorems for non-holonomic reduced Hamiltonian systems, by using the non-holonomic reduced distributional Hamiltonian systems. As an application of the theoretical results, we consider the motions of the constrained particle in space $\mathbb{R}^3$ and the vertical rolling disk, and give the Hamilton-Jacobi equations of the two systems as non-holonomic reduced distributional Hamiltonian systems.
Keywords: \; Hamilton-Jacobi theorem, \; non-holonomic Hamiltonian system, \; distributional Hamiltonian system, \; nonholonomic reduction, \; momentum map.
AMS Classification: 70H20, \; 70F25,\; 53D20. |
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