The Foucault Pendulum (with a Twist)

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 pendula of modest size, these nonprecessing periodic solutions can be very nearly planar. The Foucault pendulum is often given as proof of the rotation of the earth. As the pendulum swings back and forth, the positions of maximum amplitude precess in a direction opposite to the earth’s rotation. But a nonlinear spherical pendulum on a nonrotating planet also exhibits precession. So the observed precession of Foucault’s pendulum must be a combination of two effects. The goal of this paper is to see how the two precessions interact to produce the observed motion. In the usual Foucault experiment, the initial conditions are chosen to be such that the pendulum motion is nearly planar. If we consider more general initial conditions, it turns out that some surprising motions are possible. In particular, the direction of precession can be reversed or stopped altogether. Mathematically we find that the problem reduces to a study of a monotone twist map of an annulus. The boundary circles of the annulus are nearly circular periodic orbits of the pendulum, that is, instead of swinging back and forth in a nearly planar motion, the pendulum is sweeping out a circular cone, moving either clockwise or counterclockwise. The points near the boundary of the annulus represent nearly circular elliptical orbits which are slowly precessing. For certain values of the parameters, including those suggested by the original Foucault pendulum, the directions of precession of these orbits are opposite. It follows from the Poincaré-Birkhoff theorem that there are periodic motions of the pendulum which do not precess at all. The precession due to the rotation of the earth is completely canceled by Date: July 8, 2015. 2000 Mathematics Subject Classification. 37E40,37N05,70H08,70K45.