  We consider the motion of the end mirror of a cavity in whose standing wave
mode pattern atoms are trapped. The atoms and the light field strongly couple
to each other because the atoms form a distributed Bragg mirror with a
reflectivity that can be fairly high. We analyze how the dipole potential in
which the atoms move is modified due to this backaction of the atoms. We show
that the position of the atoms can become bistable. These results are of a more
general nature and can be applied to any situation where atoms are trapped in
an optical lattice inside a cavity and where the backaction of the atoms on the
light field cannot be neglected. We analyze the dynamics of the coupled system
in the adiabatic limit where the light field adjusts to the position of the
atoms and the light field instantaneously and where the atoms move much faster
than the mirror. We calculate the side band spectrum of the light transmitted
through the cavity and show that these spectra can be used to detect the
coupled motion of the atoms and the mirror.
