Moving Forward by Traveling in Circles

The purpose of this paper is to introduce the reader to the mathematical construct known as holonomy. Holonomy is a measurement of the change in a certain angle as one travels along a curve. For this paper, we will consider two physical situations which involve “traveling in a circle” and comparing an initial and final measurement of an angle. In the first case we will see how this angular displacement can be used to prove that the Earth rotates! In the second example we explore the workings of a nineteenth century cartographic instrument. In both cases, traveling in circles yields interesting mathematical information. Foucault’s Pendulum In the middle of the nineteenth century Foucault constructed a pendulum by hanging a 5kg bob from the ceiling of a cellar (about 2m high). After setting the pendulum in motion he observed “The oscillating point is continuously displaced..., which indicates that the deviation of the plane of oscillation takes place in the same sense as the apparent motion of the celestial sphere...” [1] Later on at the Pantheon he repeated this experiment on a grander scale using a 28kg bob hanging from a height of 67m. Hopefully you have had the opportunity to observe this phenomenon yourself. If not, find such a pendulum nearby (science museums or large universities may have one) and organize a fieldtrip! Can you explain why he observed the plane of oscillation (the direction of the pendulum swing) rotating as time when on?