  We derive the classical counterpart of a previously obtained quantum
mechanical covariant ``continuitylike'' equation for the spin density, and
present an intuitive picture for elucidating the non-conservation of the spin
current. This reveals the equations of motion for a particle with spin under
the Yang-Mills field (in certain semiconductors) as well as the Maxwell field,
from which the condition for an infinite spin relaxation time is drawn out
directly. As a concrete example, we discuss the procession of the spin
orientation in spin Hall effect with the so called ReD field, which undergoes a
circle with the frequency dependent on both the strength of the spin-orbit
coupling and the initial velocity. The anti-commutation of the Pauli matrices
is found to be crucial in simplifying the equations of motion in the view of
quantum mechanism of the same topics.
