Constrained Variation

Real problems rarely let you flex a curve freely. The classic shape questions come with a constraint: extremise one functional while a second functional is held fixed. Make the area as large as possible — but with the perimeter pinned. Drop the potential energy as low as possible — but with the length of the chain fixed. These are the isoperimetric problems:

\text{extremise } \int_a^b L\,dx \quad \text{subject to} \quad \int_a^b G\,dx = C.

Lift the multiplier trick to functions

In finite dimensions the way to optimise under a constraint is Lagrange multipliers: to extremise f subject to g = c you introduce a number \lambda and extremise f - \lambda g freely. The calculus of variations inherits the trick wholesale — the only change is that the "variables" are now the values of a function at its continuum of points.

Step 1 — combine into one Lagrangian. Fold the constraint into the integrand with a single multiplier \lambda:

L^\ast = L + \lambda G.

Step 2 — vary freely. Apply the ordinary Euler–Lagrange equation to the combined Lagrangian L^\ast, as if there were no constraint at all:

\frac{\partial L^\ast}{\partial y} - \frac{d}{dx}\frac{\partial L^\ast}{\partial y'} = 0.

Step 3 — pin \lambda with the constraint. The solution carries the unknown \lambda; the requirement \int G\,dx = C is the extra equation that fixes its value.

To extremise \int L\,dx subject to \int G\,dx = C:

introduce a Lagrange multiplier \lambda and extremise the augmented functional \int (L + \lambda G)\,dx freely;
the extremal solves the Euler–Lagrange equation for L^\ast = L + \lambda G;
the constraint \int G\,dx = C then determines \lambda — exactly the finite-dimensional recipe, lifted to functions.

Two famous answers

Dido's problem. Of all closed curves of a fixed perimeter, which encloses the greatest area? Here L builds the area and G builds the perimeter; extremising L + \lambda G forces the boundary to have constant curvature. The answer is the circle — the same shape that, legend says, Queen Dido enclosed with an oxhide to found Carthage.

The hanging chain. A flexible chain of fixed length hangs between two posts and settles into the shape that minimises its gravitational potential energy \int y\,ds, subject to a fixed length \int ds = \ell. Both integrands are functions of y and y' only, so after folding in \lambda the Beltrami identity applies, and the curve that comes out is the catenary:

y = a\cosh\!\left( \frac{x}{a} \right).

It looks like a parabola but isn't — it is the hyperbolic cosine, and the constant a (set by the chain's length and the multiplier) controls how sharply it sags.

Galileo guessed a hanging chain was a parabola; he was close, but wrong. The true curve, named the catenary from the Latin catena ("chain"), was found in 1691 by Huygens, Leibniz and Johann Bernoulli, responding to a challenge from Jacob Bernoulli. The difference matters: turn a catenary upside down and it becomes the ideal self-supporting arch, carrying its own weight in pure compression — which is exactly why Gaudí hung chains as model "arches" and why the St. Louis Gateway Arch is a flattened catenary, not a parabola.

Sag the chain

Turn the slider to change a in the catenary y = a\cosh(x/a). Small a gives a deep, sharply sagging chain; large a pulls it nearly straight. Each value of a is the energy-minimising shape for a chain of a particular fixed length — the constraint hiding inside the multiplier.