I love the counterfactual approach, thank you!!!
replyI've done plenty of this in pure math and stats, but this is the first time I've seen it applied to physics, and I love it! Thank you!
If I saw your derivation when I was 18 years old, who knows, maybe I would have caught the physics bug and went that way, this is super cool!
For a very concise treatment of this read the first two chapters of Landau & Lifshitz's Mechanics book. The actual logic behind what can and cannot go into the Lagrangian fits into ~2 pages.
replyIt's essentially the same argument: the Lagrangian can't have a bare a) position or b) velocity vector or it would violate homogeneity or isotropy of space, respectively.