An Analytical Solution of the Inverted Pendulum

A rigorous solution of the inverted pendulum requires several mathematical techniques which are not in the repetoire of the typical student of classical mechanics at the senior level. However, the problem is tractable by Lagrange's equations and the physics of the situation is sufficiently simple so that it is an excellent example to use to extend the student's analytical abilities. Two unexpected dividends are derived from this problem: (1) the vertically driven case is an example of a time-dependent potential energy function, a rare situation in mechanics; (2) it is a vivid example of a classical system undergoing strong focusing and therefore can be used to introduce a discussion of the strong-focusing synchrotron.