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.
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.