An upside-down pendulum theorem

It has long been known that a single pendulum can be maintained stably in its inverted, or `upside-down' position if its pivot is vibrated up and down at a suitably high frequency. In 1993 I proved a simple theorem which shows how the same `trick' can be performed with any finite number of linked pendulums, all balanced on top of one another.

One rather attractive feature of the theorem is that it relates the stability of the inverted state to just two simple properties of the free oscillations of the system about its downward-hanging state. In this way there is a direct link between the new theorem and a classical investigation of multiple pendulums by Daniel Bermoulli, in 1738.

Proc. Roy. Soc. A Vol 443, pp 239-245, 1993
(See also Chapter 12 of From Calculus to Chaos, or
article in New Scientist, 21 Feb 1998 or
radio interview with Thames Valley FM, 26 Feb 1998)

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