  We consider a system composed of a mobile slab and the electromagnetic field.
We assume that the slab is made of a material that has the following properties
when it is at rest: it is linear, isotropic, non-magnetizable, and ohmic with
zero free charge density. Using instantaneous Lorentz transformations, we
deduce the set of self-consistent equations governing the dynamics of the
system and we obtain approximate equations to first order in the velocity and
the acceleration of the slab. As a consequence of the motion of the slab, the
field must satisfy a wave equation with damping and slowly varying coefficients
plus terms that are small when the time-scale of the evolution of the mirror is
much smaller than that of the field. Also, the motion of the slab and its
interaction with the field introduce two effects in the slab's equation of
motion. The first one is a position- and time-dependent mass related to the
$\textit{effective mass}$ taken in phenomenological treatments of this type of
systems. The second one is a velocity-dependent force that can give rise to
friction and that is related to the much sought $\textit{cooling}$ of
mechanical objects.
