Isoperimetric Problems and Lagrange Multipliers

The Classical Riddle of Dido

According to legend, the Phoenician princess Dido fled to North Africa and asked the local chief for as much land as could be encompassed by a bull's hide. She cut the hide into thin strips and fenced a plot with them—taking the straight shoreline as one side. What shape did Dido choose? Naturally, a semicircle. It is precisely such problems—to maximize area given a fixed perimeter—that are called isoperimetric. A strict mathematical proof that the circle is the answer required the invention of variational calculus.

Formulation of the Isoperimetric Problem

Classical formulation: among all closed curves of a given length L, find the one that bounds the largest area.

In analytical form: max $J_1[y] = \int_0^a y, dx$ given $J_2[y] = \int_0^a \sqrt{1+y'^2}, dx = L$.

This is a functional with a constraint-functional—an analogue of a constrained optimization problem in finite-dimensional space.

The Method of Lagrange Multipliers for Functionals

Theorem: for the problem $min, J[y] = \int F(x,y,y'), dx$ with constraint $J_2[y] = \int G(x,y,y'), dx = C$, the extremal is an extremal of the auxiliary functional:

$ H[y] = \int \left( F(x,y,y') + \lambda G(x,y,y') \right), dx $

where $\lambda$ is a numerical Lagrange multiplier, i.e., the extremal satisfies the Euler-Lagrange equation for $F + \lambda G$:

$ (F+\lambda G)y - \frac{d}{dx} (F+\lambda G){y'} = 0 $

How to find $\lambda$? From the condition $J_2[y] = C$—substitute the found extremal into the constraint and solve the equation for $\lambda$.

The Problem of the Catenary (Chain Line)

Formulation: a flexible chain of fixed length $L$ hangs between points $A$ and $B$. Under the influence of gravity, it assumes a shape minimizing potential energy. Find the shape of the chain.

Potential energy: $J_1 = \int y\sqrt{1+y'^2}, dx$ (height of the center of mass of an infinitesimal arc element times the length of the element). Length: $J_2 = \int \sqrt{1+y'^2}, dx = L$.

Auxiliary functional with multiplier $\lambda$: $H = \int (y+\lambda)\sqrt{1+y'^2}, dx$.

Euler-Lagrange equation: the functional $H$ does not explicitly depend on $x$ → use the first integral:

$ H - y'H_{y'} = C_1 \rightarrow \frac{y+\lambda}{\sqrt{1+y'^2}} = C_1 $

Separating variables: $dy / \sqrt{(y+\lambda)^2/C_1^2 - 1} = dx$. Integrating: $y + \lambda = C_1 \cosh\left(\frac{x-C_2}{C_1}\right)$.

Answer: $y = a\cosh(x/a + b) - \lambda$, where $a$, $b$ are constants from boundary conditions. This is the chain line (catenary).

Physical interpretation: the shape of the chain is not a parabola (as thought until Huygens and Leibniz)—it is a hyperbolic cosine! The difference is clearly visible for large sags.

Dido's Problem: The Semicircle

Problem: fence the maximum territory along a straight shoreline, using a rope of length $L$.

Boundary conditions: $y(0) = y(a) = 0$. $max,\int_0^a y, dx$ given $\int_0^a \sqrt{1+y'^2}, dx = L$.

Auxiliary functional: $H = \int \left( y + \lambda \sqrt{1+y'^2} \right), dx$.

Euler-Lagrange equation: $1 - \lambda \frac{d}{dx} \left( \frac{y'}{\sqrt{1+y'^2}} \right) = 0$.

$\frac{d}{dx}\left(\frac{y'}{\sqrt{1+y'^2}}\right) = 1/\lambda \rightarrow \frac{y'}{\sqrt{1+y'^2}} = x/\lambda + C$.

From the boundary condition, $y'(0)$ is odd: $C = 0$. Solution: $y = \sqrt{\lambda^2 - x^2}$—a semicircle of radius $\lambda$.

From the length condition: $\pi\lambda = L \rightarrow \lambda = L/\pi$, area $= \pi\lambda^2/2 = L^2/(2\pi)$.

This is the theorem on the isoperimetric inequality: for a curve of length $L$, the area $\leq L^2/(4\pi)$, with equality achieved for the circle.

General Formulation with Several Constraints

$min, J[y_1,...,y_n]$ subject to $K_1[y]=C_1,...,K_m[y]=C_m$

Auxiliary functional: $H = F + \lambda_1 G_1 + ... + \lambda_m G_m$. Euler-Lagrange system for each $y_i$ plus $m$ equations to find the $m$ multipliers $\lambda_j$ from conditions $K_j[y] = C_j$.

Fully Worked Example: Euler's Column Problem

Problem: find the shape of an elastic column supporting the maximum axial load $P$ before loss of stability. The length of the column is fixed.

Energy functional: $J = \int_0^L \frac{EI}{2} \kappa^2(s), ds$, where $\kappa$ is the curvature, $EI$ is stiffness. Constraint: $\int_0^L 1, ds = L$ (column length).

Sturm-Liouville problem: $EI y'' + P y = 0$. Critical loads: $P_n = n^2\pi^2 EI / L^2$ (Euler's formula).

The smallest critical load (buckling): $P_1 = \pi^2 EI / L^2$—Euler's critical load. This is the minimum force at which the straight configuration ceases to be an energy minimum.

Application: in construction, all columns, beams, and thin-walled structures are designed with a safety margin according to Euler's formula. This is a direct application of isoperimetric problems in variational calculus.

Lagrange Principle: General Formulation

The isoperimetric rule generalizes to a wide class of constrained functional optimization problems. The Lagrange principle in variational calculus: for the problem of minimizing $J[y]$ subject to constraints $K_j[y] = c_j$ ($j=1,...,m$), a necessary condition for optimality is the existence of numbers $\lambda_j$ such that $y^$ is an extremal of the auxiliary functional $J[y] + \sum \lambda_j K_j[y]$. Lagrange multipliers have the meaning of "shadow prices": $\partial J^/\partial c_j = -\lambda_j$ (derivative of the optimum with respect to the right-hand side of the constraint).

Second-Order Conditions with Constraints

For an extremal to truly be a minimum, one must check positive definiteness of the second variation on the tangent space to the constraints—the set of variations $\eta$ satisfying the linearized constraints. This is analogous to the reduced Hessian matrix in finite-dimensional optimization.

Modern Isoperimetric Problems

• Spectral optimization: which shape of a membrane has the smallest fundamental eigenvalue of the Laplacian for a given area? The answer is the disk (Faber-Krahn inequality)
• Isoperimetric inequalities in high dimensions: in $\mathbb{R}^n$, the volume of the ball $V(r)$ gives the minimal surface $\Sigma$ for a given volume (Brunn-Minkowski theorem)
• Isoperimetry on manifolds: on the sphere, geodesic balls minimize boundary area (Levy, Gromov)
• Isoperimetry in graph theory (expansion): minimizing the number of edges at the boundary of a subset of vertices for a fixed size—foundation of random walk theory and spectral methods

Applications

In architecture, isoperimetric ideas are used in designing domes and shells: the dome of St. Peter's Basilica in Rome approximates the catenary shape. In biology, the shape of soap bubbles and cell membranes is determined by area minimization for a given volume. In materials science, Plateau's problems (minimal surfaces with a given boundary) describe the shape of soap films and model the structure of crystalline grains.