Re: [Phys-L] Spontaneous Symmetry Breaking in Unexpected Places: A Looped Double Catenary & A Block/Plank Balanced on a Cylinder (very long)

[John Denker <jsd@av8n.com>]

On 6/28/22 8:17 PM, David Bowman wrote:

> I recently came across an example of spontaneous symmetry breaking in
> a context that was new to me and a priori, unexpected.

Interesting

> [1] is the situation of a loop of a perfectly flexible,
> frictionless, narrow, and inextensible string/cable of uniform weight
> density looped around and across two horizontally separated narrow 
> pegs/nails/pins,

Very cute.

> [2] Another mundane situation involving a possibly un expected
> spontaneous symmetry breaking is the case of a uniform rectangular
> plank or block straddling and balanced on a sideways circular
> cylinder

[3] Here's an even more mundane version of that, namely a
balance beam:

Preparation: Orient a yardstick with the longest dimension
and the shortest dimension both horizontal. Drill a tiny
horizontal hole just above the absolute center, and insert
a needle.

Operation: Support it by the needle, so that the needle rolls
on two horizontal surfaces (one for each end of the needle).
I use two salt-shakers or two spice-bottles.

You can then use this as a balance, a home-made instrument
for weighing things. If the needle is above the midpoint, the
beam has positive stability. The stability goes to zero as
the needle approaches the midpoint. If you flip it over so
the needle is below the midpoint, the beam is unstable.

This is more convenient than the beam-on-a-barrel setup,
since you don't need to assume a rocking motion with no
slippage. Slippage doesn't matter here, since the needle
is supported on horizontal surfaces.

[4] Imagine a right circular cylinder, standing upright,
very tall compared to its diameter. It is metastable.
By rounding off the bottom end, you can reduce the
stability to zero.

====================

System [1] is particularly interesting, if I understand
it rightly, because it is a second-order phase transition.
There is a nontrivial relationship between Δ and σ in the
slightly-broken regime. It might be amusing to plot this
and to work out the critical exponent.

Systems [2], [3], and [4] exhibit a first-order phase
transition. That is, as soon as the symmetry is broken,
it is grossly broken. There is step change in the order
parameter.

=====

Systems [2] and [3] can be generalized to higher dimensions.
The case of d=2 is easy to construct.
 [2]: A large circular slab resting on a spherical support.

 [3]: A large circular slab dangling from a fiber.

For d>2 I don't know how to build it mechanically, but the
equations can be generalized easily enough.

System [4] is d=2 as described. The d=1 version is
straightforward.

I have no clue how to generalize the loops in system [1].

=====

System [2] can be reversible or irreversible. If the
deflection is allowed to get large enough, the moving
part will slip.

System [4] can be engineered to control the onset of
irreversibility.

I do not believe everything Robert K. G. Temple says,
but he claimed there was a primitive Chinese seismometer
that worked on this principle. A stick was precariously
balanced. If disturbed, it would fall to one side,
indicating the approximate direction to or from the
site of the earthquake.

A group of such instruments with different sensitivities
would give an indication of the magnitude. A network of
them at various locations would give even more information.

=====

Systems [2] and [3] merge together if you let the radius
of the support be moderately small (but bigger than a
needle) and do something to lower the center of mass of
the moving part, relative to where it would be for a
uniform beam ... perhaps a shallow concave-down bowl,
or perhaps something like the classic gambrel shape:
  https://cdn11.bigcommerce.com/s-yi9ctewjl6/images/stencil/1280x1280/products/1327/7466/20210118_105144__98796.1611082523.jpg