Kapitza's pendulum as a classical prelude to Floquet–Magnus theory

We present a pedagogical introduction to Floquet–Magnus theory through the classical example of Kapitza's pendulum—a simple system exhibiting nontrivial dynamical stabilization under rapid periodic driving. By deriving the equations of motion and analyzing the system using Floquet theory and the Magnus expansion, we obtain analytical stability conditions and effective evolution equations. While grounded in classical mechanics, the techniques are directly applicable to periodically driven quantum systems as well. The approach is fully analytical, using only tools from theoretical mechanics, linear algebra, and ordinary differential equations, and is suitable for instruction at the advanced undergraduate or graduate level.