  The polaron model of H. Fr\"ohlich describes an electron coupled to the
quantized longitudinal optical modes of a polar crystal. In the strong-coupling
limit one expects that the phonon modes may be treated classically, which leads
to a coupled Schr\"odinger-Poisson system with memory. For the effective
dynamics of the electron this amounts to a nonlinear and non-local
Schr\"odinger equation. We use the Dirac-Frenkel variational principle to
derive the Schr\"odinger-Poisson system from the Fr\"ohlich model and we
present new results on the accuracy of their solutions for describing the
motion of Fr\"ohlich polarons in the strong-coupling limit. Our main result
extends to $N$-polaron systems.
