So far we have dealt with boundary conditions of the form y(a) = A, y(b) = B or y(a) = A, y'(b) = B. For some problems the natural boundary conditions are expressed using an integral. The standard example is Dido's problem^{3}: if you have a piece of rope with a fixed length, what shape should you make with it in order to enclose the largest possible area? Here we are trying to choose a function y to maximise an integral I(y) giving the area enclosed by y, but the fixed length constraint is also expressed in terms of an integral involving y. This kind of problem, where we seek an extremal of some function subject to `ordinary' boundary conditions and also an integral constraint, is called an isoperimetric problem.
A typical isoperimetric problem is to find an extremum of
Consider the functions of two variables
It follows from the theory of Lagrange multipliers that a necessary condition for a function I[ε, δ] of two variables subject to a constraint J[ε, δ] = L to take an extreme value at (0, 0) is that there is a constant λ (called the Lagrange multiplier) such that
+ λ | = 0 | |
+ λ | = 0 |
[ε, δ] + λ[ε, δ] | = | F(x, Y + εη + δζ, Y' + εη' + δζ') + λG(x, Y + εη + δζ, Y' + εη' + δζ') dx | |
= | ηF + λG + η'F + λG dx (chain rule) | ||
= | ηF + λG - F + λG dx (integration by parts) | ||
= | 0 |