  We consider the Fr\"ohlich Hamiltonian in a mean-field limit where many
bosonic particles weakly couple to the quantized phonon field. For large
particle number and suitably small coupling, we show that the dynamics of the
system is approximately described by the Landau-Pekar equations. These describe
a Bose-Einstein condensate interacting with a classical polarization field,
whose dynamics is effected by the condensate, i.e., the back-reaction of the
phonons that are created by the particles during the time evolution is of
leading order.
