  We show how to derive an effective nonlinear dynamics, described by the
Hartree-Fock equations, for fermionic quantum particles confined to a
two-dimensional box and in presence of an external, uniform magnetic field. The
derivation invokes the Dirac-Frenkel principle. We discuss the validity of this
effective description with respect to the many-body Schr\"odinger dynamics for
small times and for weak interactions, and also in regards to the number of
particles.
