Kapitza’s Pendulum: A Physically Transparent Simple Treatment

The phenomenon of dynamic stabilization of the inverted rigid planar pendulum whose pivot is forced to oscillate at a high frequency in the vertical direction is revisited. This intriguing nonlinear physical system is analyzed in the paper using time-scale separation and averaging. On this basis, a simple and clear physically meaningful explanation of the phenomenon is presented, followed by the derivation of an approximate quantitative criterion of stability. The advantages and limitations of this approach are discussed. The conventional criterion is compared with the boundaries of stability region on the Ince– Strutt diagram obtained with the help of the linearized differential equation of the system (Mathieu equation). An accompanying computer program simulating the physical system is designed to illustrate and aid the analytical investigation of the phenomenon.