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

[Brian Whatcott <betwys1@sbcglobal.net>]

 John M praises David B's models as almost intuitive.
I go further and suggest that David's teetering beam problem is intuitive, if a hand-waving approach suffices, as follows. 
A uniform beam whose horizontal axis of rotation is above its center of mass is thereby stable in horizontal repose.A uniform beam is unstable when its mass center is above its axis and falls one side down in typical teeter-totter way.(This is John D's discussion)
However, David's interesting observations occur when the pivot point at the beam's lower surface moves further than the center of mass, if the beams position is disturbed.This occurs when the pivot point bears on a convex surface which is sufficiently shallow so that in the limit, if the beam rests on a flat surface at its pivot point, it is immobile in the horizontal position.This situation is similar to balancing a narrow triangular  prism on its apex: inherently unstable unless some dynamic means of moving the pivot point to stay under the mass center -  comparable to the juggler's skill, or more recently to the magnetic mass levitated over a dynamically varied magnetic repulsion force.
The catenaries are less transparent. It is evident that a catenary suspension point sees greater tension, the smaller the angle of depreession, while the longer catenary sees less tension at the support, while its weight is increased.    On Thursday, June 30, 2022 at 08:56:07 AM CDT, David Bowman <david_bowman@georgetowncollege.edu> wrote:  
 
 Regarding John Mallinkrodt's kind comment & encouraging suggestion:

> Just a quick note to say that, while I haven’t gone through
> David’s work in ANY detail, its simple beauty is immediately
> obvious.  Moreover, the results are not only reasonable, but, I
> might suggest, almost intuitively “obvious.”
>
> I REALLY hope he considers writing it up for publication in AJP
> because it is exactly the kind of paper there should be a LOT
> more of—readily understood and accessible scenarios that nicely
> illustrate advanced concepts and methods.

It looks like I may need to come clean here and confess that when in the first sentence of my initial post on the topic I said, "I recently came across an example of spontaneous symmetry breaking in a context that was new to me and a priori, unexpected." I had come across the example *because* I had read about it in a published Mathematics Magazine, article entitled 'The Curious Case of the Double Catenary' (DOI: 10.1080/0025570X.2022.2055335, https://doi.org/10.1080/0025570X.2022.2055335 ).  The author, Subharanil De, is on the physics faculty at Indiana University Southeast.  That article already covers many of the points I mentioned in my phys-l post, albeit with mostly quite different notation and somewhat different emphasis and in less depth.  Although the spontaneous symmetry breaking phenomenon for the looped double catenary system is prominent in the article, for some reason, that phrase never actually appears in the article to describe it.  Perhaps because the audience is mostly mathematicians and not physicists they may not be likely to know just what is meant by the phrase.

I'm not in the habit of reading Mathematics Magazine and don't subscribe to it.  The only reason I came across it is because the chairman of the Math, Physics & Comp. Sci. at the College I worked at before recently retiring had specifically brought it to my attention because he thought I may appreciate it.  And the reason he thought I may appreciate it is because of one of the challenge problems I had earlier put up on a hallway whiteboard on the MPC wing of the campus science building.  For the last few years I have been in the habit of putting up on the hallway whiteboard various math & physics challenge problems (a la Good Will Hunting) for a few weeks each for anyone passing by to attempt to try to solve.  Usually I don't get any takers.  Occasionally a student or other faculty person puts up a solution or attempts to, wrongly.  But usually, I just answer my own problem a few weeks after I first put it up.  One of the challenge problems I had put up a couple of years ago concerned a (more realistic) generalization of the usual standard catenary statics problem.  The generalization is to relax the constraint that the cable is inextensible, and allow it to have a finite Young's modulus so it stiffly obeys Hooke's law and stretches somewhat when hanging in equilibrium between the supports.  The problem's challenge was to then relate the length, the sag, the unstretched length, the stiffness, the weight density, the  tensions, and distance between the supports all to each other.

It appears Mathematics Magazine plays a role in the mathematics community quite similar to the one AJP plays for the physics community.  According to their web site, "Mathematics Magazine publishes expository articles meant to appeal to a broad mathematical audience that includes strong undergraduate students. Our editorial philosophy is that if a topic is in any way related to mathematics, then it is potentially of interest to the magazine."  The level of presentation in the above article seemed to be maybe just a little lower than what is seen in a typical AJP article.

Nevertheless, it appears this topic has mostly already seen print, and I have been sufficiently scooped on it to not feel right about also publishing on it in AJP.

David Bowman
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