  We develop the Floquet-Magnus expansion for a classical equation of motion
under a periodic drive that is applicable to both isolated and open systems.
For classical systems, known approaches based on the Floquet theorem fail due
to the nonlinearity and the stochasticity of their equations of motion (EOMs)
in contrast to quantum ones. Here, employing their master equation, we
successfully extend the Floquet methodology to classical EOMs to obtain their
Floquet-Magnus expansions, thereby overcoming this difficulty. Our method has a
wide range of application from classical to quantum as long as they are
described by differential equations including the Langevin equation, the
Gross-Pitaevskii equation, and the time-dependent Ginzburg-Landau equation. By
analytically evaluating the higher-order terms of the Floquet-Magnus expansion,
we find that it is, at least asymptotically, convergent and well approximates
the relaxation to their prethermal or non-equilibrium steady states. To support
these analytical findings, we numerically analyze two examples: (i) the Kapitza
pendulum with friction and (ii) laser-driven magnets described by the
stochastic Landau-Lifshitz-Gilbert equation. In both cases, the effective EOMs
obtained from their Floquet-Magnus expansions correctly reproduce their exact
time evolution for a long time up to their non-equilibrium steady states. In
the example of driven magnets, we demonstrate the controlled generations of a
macroscopic magnetization and a spin chirality by laser and discuss possible
applications to spintronics.
