  Nuclear physics is ideal to test and develop techniques to describe the
microscopic dynamics of quantum many-body systems. At low energy, nuclear
dynamics is described with non-relativistic approaches based on the mean-field
approximation and its extensions. Variational principles based on the
stationarity of the action are introduced to build theoretical models with
different levels of approximation. In particular, the time-dependent
Hartree-Fock (TDHF) equation for mean-field dynamics and its linear
approximation, also known as the Random Phase Approximation (RPA), are derived.
Predictions of vibrational spectra at the RPA level are presented as an
application. The inclusion of beyond TDHF correlations and fluctuations are
then discussed. In particular, pairing correlations are treated at the BCS and
Bogoliubov levels. The Balian-Veneroni variational principle is finally
introduced. In addition to provide some insight into mean-field limitations, it
offers a possibility to incorporate quantum fluctuations of one-body
observables with the time-dependent RPA formalism.
