Inverted Pendulum

 

An upside-down pendulum becomes stable if its pivot is vibrated rapidly. Don’t believe it? Watch the movie below.

Explanation 1 via the topology of SL(2,R): see this paper: inverted.pendulum.pdf

Explanation 2, via the pursuit curve [sic] is given in [43]. This elementary explanation shows ``what is going on”. See the movie below for the actual demonstration. A similar idea is used also in this paper: Kapitsa.potentials.pdf

These two explanations apply to different regimes: low frequency, i.e. short length (explanation 1) and high frequency (explanation 2).