  Classical models of spin systems traditionally retain only the dipole
moments, but a quantum spin state will frequently have additional structure.
Spins of magnitude $S$ have $N=2S+1$ levels. Alternatively, the spin state is
fully characterized by a set of $N^{2}-1$ local physical observables, which we
interpret as generalized spin components. For example, a spin with $S=1$ has
three dipole components and five quadrupole components. These components evolve
under a generalization of the classical Landau-Lifshitz dynamics, which can be
extended with noise and damping terms. In this paper, we reformulate the
dynamical equations of motion as a Langevin dynamics of SU($N$) coherent states
in the Schr\"odinger picture. This viewpoint is especially useful as the basis
for an efficient numerical method to sample spin configurations in thermal
equilibrium and to simulate the relaxation and driven motion of topological
solitons. To illustrate the approach, we simulate a non-equilibrium relaxation
process that creates CP$^{2}$ Skyrmions, which are topological defects with
both dipole and quadrupole character.
