Great post! But don't discount the importance of the simple pendulum or other instances of the simple harmonic oscillator. It may not be exciting compared to principle of least action, but the toy problems start to build intuition that pays a dividend when you encounter complex problems. Sometimes you can exploit symmetries to reduce part(s) of a more complex problem into toy problems.
And sometimes the toy problems are useful in experimental physics. For example, one of the ways of detecting superfluid helium is measuring change in the rotational period of a torsional oscillator as the temperature is lowered to the transition temperature. At the transition, the superfluid begins to flow without viscosity, so it stops rotating as the torsion rod turns because there is no more friction between the helium atoms and the walls inside the torsion bob. This lowers the mass that is rotating, which changes the moment of inertia, which changes the period of the rotation.
Ken, sorry for my delay here! This is a beautiful comment; I'm filing it away to investigate later, and hopefully animate, now that I have my Mathematica-lite implementation coming creakily to life in the browser.
Yes, of course you're right that the harmonic oscillator deserves all of the attention it gets. My "lowly pendulum" was tongue in cheek - I meant more that, yes, this is a totally foundational thing, deeply important, and very visualizable... and wouldn't it be wonderful if every textbook and learning resource could summon a shared "pendulum" model? And, further, if that model could accumulate "references" to all of the other models folks have built that decompose into the pendulum. I'd love to have a whole menu of brilliant examples like yours presented to me right at the beginning.
I have the same feeling about toy-problem-like things in Clojure or Lisp. If you really have your head wrapped around a concept like "Semigroup" or "Monoid", you start to see them everywhere. The harmonic oscillator is certainly a "portal abstraction", to lean on my own lingo from here: https://corecursive.com/050-sam-ritchie-portal-abstractions/
Thanks again for the comment and the lovely description!
Great post! But don't discount the importance of the simple pendulum or other instances of the simple harmonic oscillator. It may not be exciting compared to principle of least action, but the toy problems start to build intuition that pays a dividend when you encounter complex problems. Sometimes you can exploit symmetries to reduce part(s) of a more complex problem into toy problems.
And sometimes the toy problems are useful in experimental physics. For example, one of the ways of detecting superfluid helium is measuring change in the rotational period of a torsional oscillator as the temperature is lowered to the transition temperature. At the transition, the superfluid begins to flow without viscosity, so it stops rotating as the torsion rod turns because there is no more friction between the helium atoms and the walls inside the torsion bob. This lowers the mass that is rotating, which changes the moment of inertia, which changes the period of the rotation.
Ken, sorry for my delay here! This is a beautiful comment; I'm filing it away to investigate later, and hopefully animate, now that I have my Mathematica-lite implementation coming creakily to life in the browser.
Yes, of course you're right that the harmonic oscillator deserves all of the attention it gets. My "lowly pendulum" was tongue in cheek - I meant more that, yes, this is a totally foundational thing, deeply important, and very visualizable... and wouldn't it be wonderful if every textbook and learning resource could summon a shared "pendulum" model? And, further, if that model could accumulate "references" to all of the other models folks have built that decompose into the pendulum. I'd love to have a whole menu of brilliant examples like yours presented to me right at the beginning.
I have the same feeling about toy-problem-like things in Clojure or Lisp. If you really have your head wrapped around a concept like "Semigroup" or "Monoid", you start to see them everywhere. The harmonic oscillator is certainly a "portal abstraction", to lean on my own lingo from here: https://corecursive.com/050-sam-ritchie-portal-abstractions/
Thanks again for the comment and the lovely description!