  We use high resolution numerical simulations to investigate the adiabatic
contraction of dark matter halos with a Hernquist density profile. We test the
response of the halos to the growth of additional axisymmetric disk potentials
with various central concentrations and the spherically symmetric potential of
a softened point mass. Adding the potentials on timescales that are long
compared to the dynamical time scale of the halo, the contracted halos have
density profiles that are in excellent agreement with analytical predictions
based on the conservation of the adiabatic invariant $M(r)r$. This is
surprising as this quantity is strictly conserved only for particles on
circular orbits and in spherically symmetric potentials. If the same potentials
are added on timescales that are short compared to the dynamical timescale, the
result depends strongly on the adopted potential. The adiabatic approximation
still works for disk potentials. It does, however, fail for the central
potential.
