The Foucault Pendulum (with a Twist)

Abstract

A Foucault pendulum is supposed to precess in a direction opposite to the earth's rotation, but nonlinear terms in the equations of motion can also produce precession. The goal of this paper is to study the motion of a nonlinear, spherical pendulum on a rotating planet. It turns out that the problem on a fixed energy level reduces to the study of a monotone twist map of an annulus. For certain values of the parameters, this leads to existence proofs for orbits which do not precess or else precess in the wrong direction. In fact, there will be nonprecessing periodic solutions which return to their initial state after swinging back and forth just once. For pendulums of modest size, these nonprecessing periodic solutions can be very nearly planar.