  In this paper, we consider the controlled rigid spacecraft with an internal
rotor as a regular point reducible regular controlled Hamiltonian (RCH) system.
In the cases of coincident and non-coincident centers of buoyancy and gravity,
we first give the regular point reduction and the dynamical vector field of the
reduced controlled rigid spacecraft-rotor system, respectively. Then, we derive
precisely the geometric constraint conditions of the reduced symplectic form
for the dynamical vector field of the regular point reducible controlled
spacecraft-rotor system, that is, the two types of Hamilton-Jacobi equation for
the reduced controlled spacecraft-rotor system by calculation in detail. These
researches reveal the deeply internal relationships of the geometrical
structures of phase spaces, the dynamical vector fields and controls of the
system.
