The Kapitsa pendulum is an inverted pendulum on a vibrating foundation. If the frequency and amplitude of the fast vibrations of the foundation satisfy the criterion A*Omega > sqr(2*g*L), the inverted (upside down) position of the pendulum becomes stable (here L is the length of the pendulum). This can be proven mathematically by two different methods: either by the Mathieu equations stability analysis or by the so called method separation of motions. The latter implies averaging of small fast vibrations and substituting them by an effective fictitious force (or energy potential term). (See a nice demonstration at https://www.youtube.com/watch?v=rwGAzy0noU0).
Russian mechanician Ilya Blekhman showed that the method is separation of motions is very powerful and applicable to wide range of mechanical problems, in particular, the granular media and fluid mechanics. The small fast vibrations literally change the laws of mechanics. An unstable equilibrium (such as with the upside-down inverted pendulum) can become stable. A soft rope can become a stiff rod. Granular media can flows like a liquid through a pipe without frictional jamming when shaken. In all of these problems hysteresis is present. One can build a pump using fast vibrations, vibrations can be used for propulsion in liquids or fluids and for many other purposes.
We are applying the same principle to wetting and phase transition, which can be affected by fast small vibrations (temporal pattern) or spatial micro-patterns. This is because the equations of mechanics and those equations which govern physico-chemical processes are similar in structure. By using fast vibrations (vibro-levitation) one can stabilize levitation droplets. There is also a new field of surface pattern-induced phase control.