A pendulum theoremt

Stephenson (1908a, b) showed that it is possible to stabilize a single rigid pendulum in its inverted, or upside-down, equilibrium position by subjecting the pivot to small vertical oscillations of suitably high frequency. He confirmed his theoretical predictions by a practical demonstration of the phenomenon. While this is a well-known curiosity of classical mechanics it does not seem to be generally known that an inverted double, or even triple pendulum can be stabilized in the same way. This was, again, first predicted theoretically by Stephenson, in a comparatively overlooked paper of 1909, though the idea has reappeared in a number of subsequent studies (Lowenstern 1932; Hsu 1961; Kalmus 1970; Otterbein 1982; Leiber & Risken 1988). Here we present a simple but general theorem on the linear stability of an inverted N-pendulum of any kind. The generality of the theorem is achieved by relating the stability question to just two elementary properties of the system as a whole when it is in its non-inverted, or downward-hanging, state.