If you don't mind, can you elaborate more on what kind of work you do? Do you run a school for the gifted (assuming such a thing exists)?

My most obnoxiously smart student was this 12 year old who would interrupt my lectures on classical mechanics with tricky questions I couldn't solve. The next year he came back, doing the same thing but for quantum mechanics, and the year after that he got bored of this and enrolled in an elite college. I want to stress that none of this required great wealth or privilege. The cost for his books was a small fraction of what other middle class parents would pay for, say, music lessons or SAT tutoring.

When covering the principle of least action: often, applying it will give infinitely many solutions, or none at all. An example with no solutions would be the harmonic oscillator with x(0) = 0, x(t) = 0, and t not a multiple of half the period. An example with infinitely many solutions would be the principle of least time applied between the two foci of an ellipse with reflecting walls. So what happens in these cases? Does Lagrangian mechanics just not work?

Heck, when I was a kid I learned from just checking out books at random from the local public library, and even that was fine. There's just an avalanche of great free resources out there.

0: https://knzhou.github.io/#writing 1: https://physics.stackexchange.com/questions/12175/book-recom...