  Starting from first principle many-body quantum dynamics, we show that the
dynamics of Bose-Einstein condensates can be approximated by the time-dependent
nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the
convergence. Initial data are constructed on the bosonic Fock space applying an
appropriate Bogoliubov transformation on a coherent state with expected number
of particles N. The Bogoliubov transformation plays a crucial role; it produces
the correct microscopic correlations among the particles. Our analysis shows
that, on the level of the one particle reduced density, the form of the initial
data is preserved by the many-body evolution, up to a small error which
vanishes as N^{-1/2} in the limit of large N.
