Gyroscopic theory of the Foucault pendulum : new Berry phases and sensitivity to syzygies

Abstract

The normal way of exploiting a Foucault pendulum is by considering the total precession angle described during a complete cycle and to cumulate those elementary precession increments in order to yield a macroscopic precession angle. Said precession angle has been shown by Hanney to constitute a geometric phase in the sense described by Berry in 1984. The above precession increments per cycle have also been described by Berry as the result of a mathematical two-form corresponding to a pair of orthogonal pendulum circular oscillation states. In this article, the pendulum is analyzed on a half-cycle basis, as a pair of contra-rotating gyroscopes spinning about a horizontal axis. During two consecutive half-cycles, these two gyroscopes, acting in sequence, describe equal precession angles about the vertical axis, but in opposite directions, when the horizontal axis is forced by gravity to change its orientation in free space as the Earth is revolving. That gives rise, within each complete cycle, to a novel 8-shaped orbit. The cumulative precession angle difference over many complete cycles constitutes a new geometric phase of the Berry type. It can be described by a new two-form corresponding to the spin states of the two contrarotating gyroscopes. Instead of evaluating the effect of the two-form after each complete oscillation cycle, the new two-form is assessed after each half cycle and the difference between the half-cycle effects is cumulated. The new geometric phase is related to the tilt rate of the local vertical in free space. Thanks to an 18-hourduration pendulum experiment, evidence of the novel 8-shaped orbit is given. The Foucault pendulum is no longer considered in its geocentric environment, but in the barycentre frame of different celestial bodies. Sensitivity to syzygies between pendulum and said celestial bodies is discussed