Is there any example of spontaneous symmetry breaking in simple mechanical systems?

Question

The case of bead in a rotating hoop is a beautiful example of spontaneous symmetry breaking(SSB) in a simple classical mechanical model. I was wondering whether there are other such examples of SSB in a mechanical model which can be treated analytically.

Answer 1

First let me emphasize that in classical mechanics, spontaneous symmetry breaking can occur in finite systems, in contrast to quantum mechanics in which it occurs only in infinite systems. (Please see, for example, Landsman).

Thus, it is easier to find simple examples of spontaneous symmetry breaking in classical mechanics.

One of the strikingly simple examples is the pencil example:

This figure is taken from van Wezel's work. (Please see also another work of his where this example is mentioned).

This example was treated on physics stack exchange here, and also mentioned here. In this answer I am providing additional detail based on van Wezel works.

In this example, the symmetry is the $U(1)$ rotation symmetry about the $z$ axis. The Hamiltonian of the system, which is the gravitational potential $V = mg z$ is invariant under this symmetry. The order parameter is the equilibrium distance $r$ between the pencil's center of mass and the z axis. There are two types of equilibrium position: the symmetric equilibrium position when the pencil is standing $r = 0$ and the broken equilibrium position $r \ne 0$.

In the figure, $b$ is the radius of curvature of the pencil's tip and $\theta$ is the initial inclination angle. When $b$ is very large, the pencil can cope a very large inclination without falling, but when the pencil is very sharp ($b$ very small), the pencil will be very sensitive to the initial inclination. This fact is encoded in the singular limit characteristic to spontaneous symmetry breaking (given in a slightly different notation in equations (2)-(3) in the first van Wezel work. $$\lim_{b \rightarrow 0 }\lim_{\theta \rightarrow 0 } r = 0 \\ \lim_{\theta \rightarrow 0 }\lim_{b \rightarrow 0 } r \ne 0 $$

Please see the following question, where another formulation of this limit is given: