VARIATIONAL CALCULUS

Variational calculus - An approach to solving a class of optimization problems that seek a functional (y) to make some integral function (J) an extreme. Given F is in C^1, the classical unconstrained problem is to find y in C^1 to minimize (or maximize) the following function:

 _x1 | J(y) = | F(x,y,y') dx. | _|x0 

(Sorry for the sad looking integral, but such is an ASCII world.)

An example is a min arc length, where F = sqrt(1+y '^2). Using the Euler-Lagrange equation , the solution is y(x) = ax + b, where a and b are determined by boundary conditions: y(x0) = y0 and y(x1) = y1. If constraints take the form G(x, y, y') = 0, this is called the problem of Lagrange; other forms are possible.