Remarks on Calculus of Variations

Statement of the Problem:
Let Ω = { y(x) | y: [a,b]->real number system, where y has a continuous first derivative. } Let F be a functional whose domain is Ω,

F[y] = ∫_[a,b] f(x, y, y') dx.

Suppose the functional F[y] obtains a minimum (or maximum) value. Determine the curve y.

The Euler-Lagrange Equation:

partial f / partial y - d/dx( partial f / partial y' ) = 0.

Example: F[y] = ∫_[a,b] (1 + y'^2)^{1/2}. By Euler-Lagrange equation, 0 = -y''/(1 + y')^{3/2}, or y = Ax+B.

(To prove the Euler-Lagrange equation, we may start a helper function y_ε(x) = y(x) + εh(x), where h: [x,y]->real number system having a continuous second derivative.)

Reference:

• http://online.redwoods.cc.ca.us/instruct/darnold/staffdev/assignments/calcvarb.pdf (Lots of typo, but it's okay.)