  A semiclassical method is used to study Landau damping of transverse
pseudo-spin waves in harmonically trapped ultracold gases in the collisionless
Boltzmann limit. In this approach, the time evolution of a spin is calculated
numerically as it travels in a classical orbit through a spatially dependent
mean field. This method reproduces the Landau damping results for spin-waves in
unbounded systems obtained with a dielectric formalism. In trapped systems, the
simulations indicate that Landau damping occurs for a given spin-wave mode
because of resonant phase space trajectories in which spins are "kicked out" of
the mode (in spin space). A perturbative analysis of the resonant and nearly
resonant trajectories gives the Landau damping rate, which is calculated for
the dipole and quadrupole modes as a function of the interaction strength. The
results are compared to a numerical solution of the kinetic equation by Nikuni
et al.
