  Galilean invariance is usually violated in self-consistent mean-field
calculations that employ effective density-dependent nuclear forces. We present
a novel approach, based on variational quantum Monte Carlo techniques, suitable
to preserve this symmetry and assess the effect of its violation, seldom
attempted in the past. To this aim, we generalize the linear optimization
method to encompass the density-dependence of effective Hamiltonians, and study
$^4$He, $^{16}$O, and $^{40}$Ca ground-state properties employing the Gogny
interaction.
