Geometrical aspects of rapid vibrations and rotations

The counterintuitive phenomenon of stabilization of the inverted pendulum by the vertical vibration of its pivot has been known for over a century. This remarkable effect attracted attention of many mathematicians and physicists, including Kapitsa, Feynman, Arnold, Moser and others. The 1989 Nobel Prize in Physics was awarded to W. Paul for his discovery of the particle trap based on this phenomenon. The inverted pendulum is a tip of an 'iceberg' of related phenomena arising in systems with high-frequency time-dependence. This article surveys some related phenomena discovered more recently, some connections with differential geometry and mechanics, and some new geometrical insights.