  We employ high-resolution N-body simulations of isolated spiral galaxy
models, from low-amplitude, multi-armed galaxies to Milky Way-like disks, to
estimate the vertical action of ensembles of stars in an axisymmetric
potential. In the multi-armed galaxy the low-amplitude arms represent tiny
perturbations of the potential, hence the vertical action for a set of stars is
conserved, although after several orbital periods of revolution the
conservation degrades significantly. For a Milky Way-like galaxy with vigorous
spiral activity and the formation of a bar, our results show that the potential
is far from steady, implying that the action is not a constant of motion.
Furthermore, because of the presence of high-amplitude arms and the bar,
considerable in-plane and vertical heating occurs that forces stars to deviate
from near-circular orbits, reducing the degree at which the actions are
conserved for individual stars, in agreement with previous results, but also
for ensembles of stars. If confirmed, this result has several implications,
including the assertion that the thick disk of our Galaxy forms by radial
migration of stars, under the assumption of the conservation of the action
describing the vertical motion of stars.
