The Formulations of Classical Mechanics with Foucault’s Pendulum

Nicolas Boulanger,Fabien Buisseret

doi:10.3390/physics2040030

Nicolas Boulanger, Fabien Buisseret

Open Access

https://doi.org/10.3390/physics2040030

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Abstract

Since the pioneering works of Newton (1643–1727), mechanics has been constantly reinventing itself: reformulated in particular by Lagrange (1736–1813) then Hamilton (1805–1865), it now offers powerful conceptual and mathematical tools for the exploration of dynamical systems, essentially via the action-angle variables formulation and more generally through the theory of canonical transformations. We propose to the (graduate) reader an overview of these different formulations through the well-known example of Foucault’s pendulum, a device created by Foucault (1819–1868) and first installed in the Panthéon (Paris, France) in 1851 to display the Earth’s rotation. The apparent simplicity of Foucault’s pendulum is indeed an open door to the most contemporary ramifications of classical mechanics. We stress that adopting the formalism of action-angle variables is not necessary to understand the dynamics of Foucault’s pendulum. The latter is simply taken as well-known and simple dynamical system used to exemplify and illustrate modern concepts that are crucial in order to understand more complicated dynamical systems. The Foucault’s pendulum first installed in 2005 in the collegiate church of Sainte-Waudru (Mons, Belgium) will allow us to numerically estimate the different quantities introduced.

Highlights

Sainte-Waudru’s PendulumThe simple pendulum consists of a bob of mass m attached at one end of a rigid cable of length l whose mass is negligible compared to m
We stress that adopting the formalism of action-angle variables is not necessary to understand the dynamics of Foucault’s pendulum
A schematic representation of a simple pendulum is given in Figure 1, particularized to the pendulum installed in the collegiate church of Sainte-Waudru (Mons, Belgium)

Summary

Introduction

The simple pendulum consists of a bob of mass m attached at one end of a rigid cable of length l whose mass is negligible compared to m. With respect to Si the normal to the instantaneous plane of oscillation of the FP defines an inertial direction: as Newton’s mechanics shows, it is a consequence of the fact that the force undergone by the bob of mass m is always directed towards the center of the Earth. The trajectory in the horizontal plane is a hypocycloid (Figure 3) It can be computed from (1) that the FP never turns back to its equilibrium position (r = 0): there is a minimal radius rmin = Ω ω r0 , whose origin is the Earth rotation. During one period [8], that is 0.0322◦ in our example This remains very small, but just wait 10 min in Saint-Waudru’s nave and the angle of deviation will be 1.93◦ , which corresponds to a perfectly observable displacement of 5.73 cm on a circle of 1.7 m radius.

Parallel Transport along a Sphere

Lagrange

From Configuration Space to Phase Space

Motion on a Torus

Conclusions