  We construct a path-integral representation of the generating functional for
the dissipative dynamics of a classical magnetic moment as described by the
stochastic generalization of the Landau-Lifshitz-Gilbert equation proposed by
Brown, with the possible addition of spin-torque terms. In the process of
constructing this functional in the Cartesian coordinate system, we critically
revisit this stochastic equation. We present it in a form that accommodates for
any discretization scheme thanks to the inclusion of a drift term. The
generalized equation ensures the conservation of the magnetization modulus and
the approach to the Gibbs-Boltzmann equilibrium in the absence of non-potential
and time-dependent forces. The drift term vanishes only if the mid-point
Stratonovich prescription is used. We next reset the problem in the more
natural spherical coordinate system. We show that the noise transforms
non-trivially to spherical coordinates acquiring a non-vanishing mean value in
this coordinate system, a fact that has been often overlooked in the
literature. We next construct the generating functional formalism in this
system of coordinates for any discretization prescription. The functional
formalism in Cartesian or spherical coordinates should serve as a starting
point to study different aspects of the out-of-equilibrium dynamics of magnets.
Extensions to colored noise, micro-magnetism and disordered problems are
straightforward.
