  We present an approach to studying the Casimir effects by means of the
effective theory. An essential point of our approach is replacing the mirror
separation into the size of space S^1 in the adiabatic approximation. It is
natural to identify the size of space S^1 with the scale factor of the
Robertson-Walker-type metric. This replacement simplifies the construction of a
class of effective models to study the Casimir effects. To check the validity
of this replacement we construct a model for a scalar field coupling to the
two-dimensional gravity and calculate the Casimir effects by the effective
action for the variable scale factor. Our effective action consists of the
classical kinetic term of the mirror separation and the quantum correction
derived by the path-integral method. The quantum correction naturally contains
both the Casimir energy term and the back-reaction term of the dynamical
Casimir effect, the latter of which is expressed by the conformal anomaly. The
resultant effective action describes the dynamical vacuum pressure, i.e., the
dynamical Casimir force. We confirm that the force depends on the relative
velocity of the mirrors, and that it is always attractive and stronger than the
static Casimir force within the adiabatic approximation.
