  In order to describe the impact of different geometric structures and
constraints for the dynamics of a Hamiltonian system, in this paper, for a
magnetic Hamiltonian system defined by a magnetic symplectic form, we first
drive precisely the geometric constraint conditions of magnetic symplectic form
for the magnetic Hamiltonian vector field.which are called the Type I and Type
II of Hamilton-Jacobi equation. Secondly, for the magnetic Hamiltonian system
with nonholonomic constraint, we first define a distributional magnetic
Hamiltonian system, then derive its two types of Hamilton-Jacobi equation.
Moreover, we generalize the above results to nonholonomic reducible magnetic
Hamiltonian system with symmetry. We define a nonholonomic reduced
distributional magnetic Hamiltonian system, and prove two types of
Hamilton-Jacobi theorem. These research work reveal the deeply internal
relationships of the magnetic symplectic structure, nonholonomic constraint,
the distributional two-form, and the dynamical vector field of the nonholonomic
magnetic Hamiltonian system.
