Lagrange multiplier for functionals

In ordinary calculus, to optimize a function while a constraint holds, you introduce a Lagrange multiplier — one extra number — and optimize an augmented function. The same trick lifts to functionals. When you must make one functional stationary while a second functional (a side-condition like fixed length, fixed area, or fixed total) is held constant, you bring in a multiplier lambda and form a combined functional that bundles both. This is the rule behind every isoperimetric problem.

Concretely, to make J[y] = integral of L dx stationary subject to K[y] = integral of G dx being equal to a fixed value, you write the combined integrand L - lambda G and apply the ordinary Euler-Lagrange equation to it as if it were unconstrained. The constant lambda rides along as an unknown; you solve the Euler-Lagrange equation for the family of extremals, then use the constraint K[y] = fixed value to pin down lambda (and any integration constants). The multiplier is not just bookkeeping — it carries physical meaning, often a force, a pressure, or a tension that enforces the constraint, exactly as the finite-dimensional multiplier measures the sensitivity of the optimum to loosening the constraint.

This is how the hanging chain (minimize energy at fixed length), the soap-bubble shape (minimize area at fixed volume, giving the sphere with lambda playing the role of pressure), and the maximum-area enclosure (the circle) are all solved. An honest caveat carried over from the finite-dimensional case: the rule needs the constraint to be 'non-degenerate' — the constraint functional's own variation must not vanish at the extremal (otherwise you may be at an abnormal point where the simple multiplier rule fails). In practice, for the classic problems this condition holds, and the multiplier method is the clean, reliable route through constrained variation.