  We study the dipole oscillations of strongly correlated 1D bosons, in the
hard-core limit, on a lattice, by an exact numerical approach. We show that far
from the regime where a Mott insulator appears in the system, damping is always
present and increases for larger initial displacements of the trap, causing
dramatic changes in the momentum distribution, $n_k$. When a Mott insulator
sets in the middle of the trap, the center of mass barely moves after an
initial displacement, and $n_k$ remains very similar to the one in the ground
state. We also study changes introduced by the damping in the natural orbital
occupations, and the revival of the center of mass oscillations after long
times.
