  In this paper, for a variety of nonholonomic (reducible) Hamiltonian systems,
we first give to various distributional Hamiltonian systems, by analyzing
carefully the dynamics and structures of the nonholonomic Hamiltonian systems.
Secondly, we derive precisely the geometric constraint conditions of the
induced distributional two-form for the nonholonomic dynamical vector field,
which are called the Type I and Type II of Hamilton-Jacobi equations. Thirdly,
we generalize the above results for the nonholonomic reducible Hamiltonian
systems with symmetries, as well as with momentum maps, and prove two types of
Hamilton-Jacobi theorems for various nonholonomic reduced distributional
Hamiltonian systems. Finally, as an application, we give two examples to
illustrate the theoretical results. These researches reveal the deeply internal
relationships of the nonholonomic constraints, the induced (resp. reduced)
distributional two-forms and the dynamical vector fields of the nonholonomic
Hamiltonian system and its various distributional Hamiltonian systems.
