Bead, hoop and spring as a classical spontaneous symmetry breaking problem

Abstract

Through a simple mathematical analysis, we describe a mechanical system that exhibits some features related to more advanced topics in physics and that involves a complex mathematical structure. The system consists of two beads constrained to slide along a hoop and attached to each other through a spring. When the hoop rotates about a fixed axis, the spring–beads system will change its equilibrium position as a function of the angular velocity. This system is shown as an instructive exercise to introduce the Lagrangian method in analytical mechanics, where two different regions of symmetry separated by a critical point analogous to a second-order transition are found, resembling the behaviour of some thermodynamics systems. We even show how the competitive balance between the rotational dynamics and the interaction of the spring behaves in the same way as the balance between temperature and the spin interaction in ferromagnetic systems, which provides some insights about the physical sense of concepts such as spontaneous symmetry breaking, phase transitions, parameter of order and critical exponents. In addition, the gravitational potential act as an external force that causes explicit symmetry breaking where the differences between spontaneous and explicit symmetry breaking is emphasized. We also found a chaotic behaviour near the critical point. Through a demonstrative device we perform some qualitative observations that describe important features of the system.