  In this paper, some of formulations of Hamilton-Jacobi equations for
Hamiltonian system and regular reduced Hamiltonian systems are given. At first,
an important lemma is proved, and it is a modification for the corresponding
result of Abraham and Marsden in [1], such that we can prove two types of
geometric Hamilton-Jacobi theorem for a Hamiltonian system on the cotangent
bundle of a configuration manifold, by using the symplectic structure and
dynamical vector field. Then these results are generalized to the regular
reducible Hamiltonian system with symmetry and momentum map, by using the
reduced symplectic structure and the reduced dynamical vector field. The
Hamilton-Jacobi theorems are proved and two types of Hamilton-Jacobi equations,
for the regular point reduced Hamiltonian system and the regular orbit reduced
Hamiltonian 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 point
reduced system are given. In particular, the Type I and Type II of Lie-Poisson
Hamilton-Jacobi equations for the regular point reduced rigid body and heavy
top systems are shown, respectively.
