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

摘要

In this paper, some of formulations of Hamilton-Jacobi equations forHamiltonian system and regular reduced Hamiltonian systems are given. At first,an important lemma is proved, and it is a modification for the correspondingresult of Abraham and Marsden in [1], such that we can prove two types ofgeometric Hamilton-Jacobi theorem for a Hamiltonian system on the cotangentbundle of a configuration manifold, by using the symplectic structure anddynamical vector field. Then these results are generalized to the regularreducible Hamiltonian system with symmetry and momentum map, by using thereduced symplectic structure and the reduced dynamical vector field. TheHamilton-Jacobi theorems are proved and two types of Hamilton-Jacobi equations,for the regular point reduced Hamiltonian system and the regular orbit reducedHamiltonian system, are obtained. As an application of the theoretical results,the regular point reducible Hamiltonian system on a Lie group is considered,and two types of Lie-Poisson Hamilton-Jacobi equation for the regular pointreduced system are given. In particular, the Type I and Type II of Lie-PoissonHamilton-Jacobi equations for the regular point reduced rigid body and heavytop systems are shown, respectively.