This is the equation of motion of the magnetization. It describes a Larmor precession of the magnetization around the effective field, with an additional damping term arising from the coupling of the magnetic system to the environment. The equation can be written in the so-called Gilbert form (or implicit form) as:
where γ is the electron gyromagnetic ratio and α the Gilbert damping constant.
It can be shown that this is mathematically equivalent to the following Landau-Lifshitz (or explicit) form:
Where  is the Gilbert Damping constant, characterizing how quickly the damping term takes away energy from the system (  = 0, no damping, permanent precession).