Kapitza pendulum

What happens

Shake the pivot of a pendulum up and down fast enough and the pendulum stands on its head - not balanced by any controller, but held by the shaking itself. Kapitza explained it in 1951 by averaging: over one fast vibration the pole feels, besides gravity, a push that depends on where it points, and averaged over the cycle that push adds a term ¼ m a² ω² sin² φ to the potential. Once a ω exceeds √(2 g l) the top of the potential turns into a well, and the pendulum oscillates slowly about the upright with a frequency the average predicts. The bench does not trust the average. It integrates the pendulum's equation through every vibration, and it decides stability the exact way: integrate the linearised equation over one drive period, read the trace of the monodromy matrix, and the equilibrium holds when that trace is between −2 and 2. The map on the board behind the shaker is this test over all amplitudes and frequencies - it depends only on a/l and ω/ω₀, so one map serves every bench - with Kapitza's line chalked across it. Far up the map, where the vibration is fast, the line and the Floquet boundary agree to a fraction of a percent. Lower down the map shows what the average cannot: dark tongues where even the HANGING position is unstable, parametric resonance near twice the natural frequency - the trick of pumping a swing by standing up and crouching in rhythm. The stroboscope draws the pendulum once per vibration, so the slow motion appears without the blur; the effective potential on the panel carries the slow angle riding in its well.