Kapitza's pendulum

Periodically driving the fixed endpoint of a pendulum with sufficiently high frequency can lead to new stable configurations or to stable points turning into unstable ones. The most unexpected result is the stabilization of the inverted pendulum.

Kapitza’s calculations show that the periodic force can be associated with an extra potential that changes the shape of the classical sinusoidal one.

Two simple realizations correspond to fixing the endpoint’s oscillation to the vertical and horizontal axis.

• When driving vertically, the extra term is positive and two new hills arise at equal distance from the origin, making a valley around the inverted position, that becomes stable.
• In the horizontal case, the forcing potential is about the same shape, but negative, so it digs to valley, again preserving left-right symmetry, turning the zero angle into a hilltop, thence unstable.

In both cases the driving force doesn’t affect directly the potential energy in the 0 and the inverted configuration, it just makes nearby points more stable or unstable.

As I said, all of this happens when the driving frequency is high enough. It turns out the precise¹ requirement is equivalent to the endpoint’s maximum velocity being greater of equal to the terminal velocity of an object falling from an height equal to the length of the rod.