The ordinary differential equation for the motion of an inverted pendulum whose support is subjected to a vertical harmonic excitation with high frequency and small amplitude is studied. A mathematical insight is provided as asymptotic methods are employed to derive the limiting equation when the excitation frequency tends to infinity. These show that as the frequency becomes high and certain parametric conditions are satisfied, the inverted pendulum theoretically can be stabilized. A new and rigorous proof using weak convergence methods is given, to confirm the conclusions of the formal asymptotics. Then the analysis is generalized to a single inverted pendulum on an oscillatory base driven at other angles, a single inverted pendulum on a flywheel and a double inverted pendulum on a vertically oscillatory base and, in each case, if the pendulum or pendula can be stabilized, parametric conditions for stability are given.

Then we also use this method to shed light on the `Paul Trap' which is a method of levitating charged particles via an oscillating electric field. Last, PDE analogues are studied in the same manner.