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

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

This paper aims at answering the simple question, “What is spontaneous symmetry breaking (SSB) in classical systems?” I attempt to do this by analyzing from a philosophical perspective a simple classical model which exhibits some of the main features of SSB Related questions include: What does it mean to say that a symmetry is spontaneously broken? Is it broken without any causes, or is the symmetry not broken but merely hidden? Is the principle, “no asymmetry in, no asymmetry out,” violated by SSB? What really distinguishes SSB from the usual types of symmetry breaking?

https://www.jstor.org/stable/10.1086/377402

Classical Spontaneous Symmetry Breaking

A toy problem of a charged particle in a uniform magnetic field, constrained to move on a vertical rotating loop, is analyzed. After a short review of the model, we move on to its quantum mechanics. We first obtain some exact solutions of the corresponding Schrodinger equation. Thereafter, introducing the notion of instantons and bounces, we explicitly obtain such solutions in this problem. Finally, we delineate briefly the connection of this toy model with some models in field theory and statistical mechanics and mention possible extensions.

Classical and quantum mechanics of a particle on a rotating loop

Some very simple mechanical systems exhibiting instabilities are discussed. They can be used to introduce basic concepts such as first- and second-order phase transitions, critical and triple points and lines, tricritical points, spontaneous symmetry breaking, soft modes and Goldstone modes. Quelques systemes mecaniques tres elementaires presentant des instabilites sont utilises pour introduire des concepts elabores tels que les transitions de phase du premier et du second ordre, les points et lignes critiques et triples, les points tricritiques, les brisures spontanees de symetrie, les modes mous et les modes de Goldstone.

https://iopscience.iop.org/article/10.1088/0143-0807/18/5/013/pdf

Simple mechanical systems exhibiting instabilities

Macroscopic modeling of ferroelectric properties refers usually to Landau-Ginzburg-Devonshire theory. This paper questions the meaningfulness of this term, discussing contributions of the three aut...

Landau, Ginzburg, Devonshire and others

An exactly solvable model is presented that exhibits a phase transition and that can be exactly described by Landau's phase transition theory. The model is used to elucidate the constructs and assumptions that go into the Landau theory, and also to illustrate the role of fluctuations near an instability in a finite system.

Richard Alben

Papers

An Exactly Solvable Model Exhibiting a Landau Phase Transition