Deriving Lagrangian mechanics

When someone first learns classical mechanics, they're given Newtons equations of motion. But there is another -- at first highly counter-intuitive -- approach to classical mechanics, discovered by Lagrange: you can reformulate classical mechanics as about solving optimization problems! This is very strange (at least for me) to ponder, because Newtonian mechanics explains things in "cause and effect" terms which fit better into my head. The sci-fi writer Ted Chiang wrote a short story (and now feature film!) Arrival which tried to imagine humans meeting an alien species who thought in terms of Lagrangian mechanics instead of Newtonian mechanics. But one thing I had always been upset by with Lagrangian mechanics was how most treatments start by telling you what the Lagrangian is, instead of telling you how Lagrange might have invented it (even Feynman's lectures on physics start by telling you what the Lagrangian is). My friend and I wrote a derivation of the Lagrangian, and of the Euler-Lagrange equation (a fantastic result of the calculus of variations!) at hidden-phenomena.com/articles/lagrangian ; we hope this can be a fun introduction!