Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. Such functionals can for example be formed as integrals involving an unknown function and its derivatives. The interest is in extremal functions: those making the functional attain a maximum or minimum value. Perhaps the simplest example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is obviously a straight line, but if the curve is constrained to lie on a surface in space, then it is less obvious. The solutions of the latter problem are called geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, where the optical length depends upon the material of the medium. A corresponding idea in mechanics is the principle of stationary action. The theory of optimal control concerns a specific kind of problem in the calculus of variations.

The preceding examples have all involved unknown functions of a single variable, which may be identified with a time variable. Other important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet principle: they minimize the potential energy of a membrane. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: the solution or solutions may be found by dipping a wire frame in a solution of soap suds. Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

The Euler-Lagrange equation

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Under ideal conditions, the maxima and minima of a given function may be located by finding the points where its derivative vanishes. In analogy, solutions of smooth variational problems may be obtained by solving the associated Euler-Lagrange equation. In order to illustrate this process, consider the problem of finding the shortest curve in the plane that connects two points $(x\_1,\; y\_1)$ and $(x\_2,\; y\_2)$. The arc length is given by

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