The Euler–Lagrange Machine

The viewer can (1) say what generalized coordinates and the Lagrangian are, (2) explain a partial derivative as "turn one slot, hold the rest", and why q and q̇ are separate slots, (3) run the three-step recipe on a mass–spring and recover mẍ = −kx, (4) run it on L = ½a(x)ẋ², getting the product-rule and chain-rule terms, and explain why speed changes with no force.