Variational calculus

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An approach to solving a class of optimization problems that seek a functional LaTeX: (y) to make some integral function LaTeX: (J) an extreme. Given LaTeX: \textstyle F : \Omega \times \mathbb{R} \times \mathbb{R}^n \rightarrow \mathbb{R} is smooth, then the classical unconstrained problem is to find LaTeX: \textstyle y \in C^1 to minimize (or maximize) the following function:

LaTeX: 
\mbox{J}(y) = \int\limits_{x_0}^{x_1}\mbox{F}(x,y(x),\nabla y(x))\, dx.

An example is a min arc length, where LaTeX: \textstyle \mbox{F} = \sqrt{1+y'^2}. Using the Euler-Lagrange equation, the solution is LaTeX: \textstyle y(x) = ax + b, where LaTeX: \textstyle a and LaTeX: \textstyle b are determined by boundary conditions: LaTeX: \textstyle y(x_0) = y_0 \mbox{ and } y(x_1) = y_1.

If constraints take the form LaTeX: \textstyle \mbox{G}(x, y, y') = 0, this is called the problem of Lagrange; other forms are possible.


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