  In order to describe the impact of nonholonomic constraints for the dynamics
of a regular controlled Hamiltonian (RCH) system, in this paper, for an RCH
system with nonholonomic constraint, we first derive its distributional RCH
system, by analyzing carefully the structure of dynamical vector field of the
nonholonomic RCH system. Secondly, we derive precisely the geometric constraint
conditions of the induced distributional two-form for the dynamical vector
field of the distributional RCH system, which are called the Type I and Type II
of Hamilton-Jacobi equations. Thirdly, we generalize the above results for the
nonholonomic reducible RCH system with symmetry, and prove two types of
Hamilton-Jacobi theorems for the nonholonomic reduced distributional RCH
system. Moreover, we consider the nonholonomic reducible RCH system with
momentum map, by combining with the regular point and regular orbit reduction
theory and the analysis of dynamics of RCH system, we give the geometric
formulations of the nonholonomic regular point reduced and orbit reduced
distributional RCH systems, and prove two types of Hamilton-Jacobi theorems for
these reduced distributional RCH systems. These researches reveal the deeply
internal relationships of the nonholonomic constraints, the induced (resp.
reduced) distributional two-forms, the dynamical vector fields and controls of
the nonholonomic RCH system and its (reduced) distributional RCH systems.

Symplectic Geometry;Differential Geometry;Dynamical Systems