Robert Hooke's conical pendulum from the modern viewpoint of amplitude equations and its optical analogues

Abstract

As stated by Lord Kelvin a long time ago, ‘It seems to me that the test of “Do we or do we not understand a particular point in physics?” is, “Can we make a mechanical model of it?”’ What is the relationship between the propagation of a light wave in a Kerr medium in the presence of a magnetic field and the oscillation of a spherical pendulum on a rotating platform? A Kerr medium is one that when submitted to an electric field its refraction index becomes a non-linear function of the latter. It is Robert Hooke who first studied the motion of a spherical pendulum in order to approach the notion of central force. Indeed, he was willing to explore the motion of the planets with this analogous device. As a matter of fact, when a pendulum made up of a heavy mass, representing the Earth, and hanging on a wire is moved away from its equilibrium position vertically from the point of suspension, it undergoes a restoring force which tends to bring it back to the center, similar to the gravitational force exerted by the Sun on the Earth. For small amplitudes the trajectories are ellipses which precess. The ellipses are centered on the axis, in contrast to the case of planets, where the attractive center corresponds to one of the focii of the elliptical path. Thanks to the modern formalism of nonlinear dynamics, we were able to show the close relationships between the equations which describe the motion of the pendulum and the propagation of the light wave in a Kerr medium. In both cases, an elliptical motion is induced. It is interesting to note that the application of a magnetic field to a Kerr medium translates into an angular rotation which induces an additional precession of the pendulum—well known as the Foucault effect.