Three curves the workshop already built — one law.
The hanging chain, the fastest slide, and the minimal soap film are each an extremal of an integral ∫f(y,y′)dx whose integrand carries no explicit x. By the Beltrami identity — the energy theorem of the calculus of variations — every such extremal conserves the same quantity all the way along its arc.
Below: each piece’s own shipped curve, fed into its own Beltrami integrand. The strip under each curve plots H along the arc — it renders dead flat for the true curve. Toggle the impostor and the strip wavers: the law only conserves the genuine extremal.