  We construct analytical phase-space solutions for perturbations of flat disks
by performing a power series expansion for the radius and the velocity
coordinates. We show that this approach translates into an elegant mathematical
formulation which is easy to use for a wide variety of distribution functions,
for as far as resonances do not play a role, such as is the case for potentials
which are close to quadratic. As a testcase, the method is applied on the
Kalnajs disks. The results obtained are in full agreement with the analytical
solutions of the mode analysis. The strongest advantages of this method are its
independence of the mathematical complexity of the unperturbed distribution,
the degree of detail with which the solutions can be calculated and its
computational straightforwardness. On the contrary, power series solutions are
not suitable for describing regions where resonant orbits occur, which we
therefore exclude in this paper. We used the technique to analyse perturbations
in the central regions of a galaxy, tracking the dynamical consequences of a
Galactic bar on the kinematics of the solar neighbourhood (Hipparcos). We
showed how the orientation and strength of the bar is related to the properties
of the velocity ellipsoid in our model.
