The Euler–Lagrange Equation

Why a whole curve can be the answer to a question: the quickest way down, the equation behind it, least action, symmetry and conservation, and the laws of physics written as one function

The Euler–Lagrange Equation — the brachistochrone race and the cycloid, the derivation from varied paths, the tautochrone, Euler's polygon method, minimum and saddle of the action, a chaotic double pendulum, Noether's theorem, soap-film catenoids, mirage rays, variational integrators, field Lagrangians and Weierstrass's missing minimum