  We present a first-principles theory of dynamical spin excitations in the
presence of spin-orbit coupling. The broken global spin rotational invariance
leads to a new sum rule. We explore the competition between the magnetic
anisotropy energy and the external magnetic field, as well as the role of
electron-hole excitations, through calculations for 3$d$-metal adatoms on the
Cu(111) surface. The spin excitation resonance energy and lifetime display
non-trivial behavior, establishing the strong impact of relativistic effects.
We legitimate the use of the Landau-Lifshitz-Gilbert equation down to the
atomic limit, but with parameters that differ from a stationary theory.
