Minimally driven Kapitza oscillator: a pedagogical perspective from Newtonian mechanics and geometry

Abstract

The unstable top-equilibrium point of a simple pendulum turns stable when its pivot point is given a fast and strong enough vertical vibration. Known as the Kapitza oscillator, it has four symmetrically spaced points of equilibrium in absence of gravity, out of which two are stable and two are unstable. This article, completely based on a geometric argument and an elementary intuition in Newtonian mechanics, is a visual and pedagogical exposition of (a) why the oscillator has four symmetrically spaced equilibrium points in absence of gravity, (b) which of them are stable or unstable, (c) why they are so and (d) how the stability, position and number of the equilibrium points change when gravity is turned on gradually along the line of vibration of the pivot of the oscillator. A minimal impulsive drive of the pivot is sufficient to illustrate the bare bones of the phenomenon. I propose a construction that can sustain the minimal drive passively in absence of dissipative forces, or actively if all dissipative forces cannot be eliminated. In either of the cases, the discussed arguments apply.