# 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.