  We study the time-evolution of initially trapped Bose-Einstein condensates in
the Gross-Pitaevskii regime. Under a physically motivated assumption on the
energy of the initial data, we show that condensation is preserved by the
many-body evolution and that the dynamics of the condensate wave function can
be described by the time-dependent Gross-Pitaevskii equation. With respect to
previous works, we provide optimal bounds on the rate of condensation (i.e. on
the number of excitations of the Bose-Einstein condensate). To reach this goal,
we combine the method of \cite{LNS}, where fluctuations around the Hartree
dynamics for $N$-particle initial data in the mean-field regime have been
analyzed, with ideas from \cite{BDS}, where the evolution of Fock-space initial
data in the Gross-Pitaevskii regime has been considered.
