  We perform a theoretical investigation into the classical and quantum
dynamics of an optical field in a cavity containing a moving membrane
("membrane-in-the-middle" set-up). Our approach is based on the Maxwell wave
equation, and complements previous studies based on an effective Hamiltonian.
The analysis shows that for slowly moving and weakly reflective membranes the
dynamics can be approximated by unitary, first-order-in-time evolution given by
an effective Schr\"{o}dinger-like equation with a Hamiltonian that does not
depend on the membrane speed. This approximate theory is the one typically
adopted in cavity optomechanics and we develop a criterion for its validity.
However, in more general situations the full second-order wave equation
predicts light dynamics which do not conserve energy, giving rise to parametric
amplification (or reduction) that is forbidden under first order dynamics and
can be considered to be the classical counterpart of the dynamical Casimir
effect. The case of a membrane moving at constant velocity can be mapped onto
the Landau-Zener problem but with additional terms responsible for field
amplification. Furthermore, the nature of the adiabatic regime is rather
different from the ordinary Schr\"odinger case, since mode amplitudes need not
be constant even when there are no transitions between them. The Landau-Zener
problem for a field is therefore richer than in the standard single-particle
case. We use the work-energy theorem applied to the radiation pressure on the
membrane as a self-consistency check for our solutions of the wave equation and
as a tool to gain an intuitive understanding of energy pumped into/out of the
light field by the motion of the membrane.
