  The real-time dynamics of a classical spin in an external magnetic field and
locally exchange coupled to an extended one-dimensional system of
non-interacting conduction electrons is studied numerically. Retardation
effects in the coupled electron-spin dynamics are shown to be the source for
the relaxation of the spin in the magnetic field. Total energy and spin is
conserved in the non-adiabatic process. Approaching the new local ground state
is therefore accompanied by the emission of dispersive wave packets of
excitations carrying energy and spin and propagating through the lattice with
Fermi velocity. While the spin dynamics in the regime of strong exchange
coupling J is rather complex and governed by an emergent new time scale, the
motion of the spin for weak J is regular and qualitatively well described by
the Landau-Lifschitz-Gilbert (LLG) equation. Quantitatively, however, the full
quantum-classical hybrid dynamics differs from the LLG approach. This is
understood as a breakdown of weak-coupling perturbation theory in J in the
course of time. Furthermore, it is shown that the concept of the Gilbert
damping parameter is ill-defined for the case of a one-dimensional system.
