The Kapitza equation for the inverted pendulum

It is well known that a simple pendulum can be made to perform finite amplitude oscillations about the ‘up’ equilibrium position by subjecting the pivot to small amplitude high frequency oscillations. In this article, we show that the autonomous nonlinear ordinary differential equation, first derived by Kapitza in 1951 using the method of averaging and historically used to describe this phenomenon, is not uniformly valid in time and hence deviates appreciably from the exact solution. We explore why this is so and show that we can provide significantly improved temporal accuracy by way of a simple modification to the original Kapitza equation.