  We consider the dynamics of nonlinear Schr\"odinger equations with strong
constant magnetic fields. In an asymptotic scaling limit the system exhibits a
purely magnetic confinement, based on the spectral properties of the Landau
Hamiltonian. Using an averaging technique we derive an associated effective
description via an averaged model of nonlinear Schr\"odinger type. In a special
case this also yields a derivation of the LLL equation.
