  Systems of interest in physics are usually composed by a very large number of
interacting particles. At equilibrium, these systems are described by
stationary states of the many-body Hamiltonian (at zero temperature, by the
ground state). The reaction to perturbations, for example to a change of the
external fields, is governed by the time-dependent many-body Schroedinger
equation. Since it is typically very difficult to extract useful information
from the Schroedinger equation, one of the main goals of non-equilibrium
statistical mechanics is the derivation of effective evolution equations which
can be used to predict the macroscopic behavior of the system. In these notes,
we are going to consider systems of interacting bosons in the so called
Gross-Pitaevskii regime, and we are going to show how coherent states and
Bogoliubov transformations can be used to approximate the many body dynamics.
