Editing Calculus of variations (section)

{{Short description|Differential calculus on function spaces}}
{{redirect|Variational method|the use as an approximation method in quantum mechanics|Variational method (quantum mechanics)}}
{{Calculus |specialized}}

The '''calculus of variations''' (or '''variational calculus''') is a field of [[mathematical analysis]] that uses variations, which are small changes in [[Function (mathematics)|functions]]
and [[functional (mathematics)|functionals]], to find maxima and minima of functionals: [[Map (mathematics)|mappings]] from a set of [[Function (mathematics)|functions]] to the [[real number]]s.{{efn|Whereas [[Calculus|elementary calculus]] is about [[infinitesimal]]ly small changes in the values of functions without changes in the function itself, calculus of variations is about infinitesimally small changes in the function itself, which are called variations.<ref name='CourHilb1953P184'>{{harvnb|Courant|Hilbert|1953|p=184}}</ref>}} Functionals are often expressed as [[definite integral]]s involving functions and their [[derivative]]s. Functions that maximize or minimize functionals may be found using the [[Euler–Lagrange equation]] of the calculus of variations.

A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a [[straight line]] between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''[[geodesic]]s''. A related problem is posed by [[Fermat's principle]]: light follows the path of shortest [[optical length]] connecting two points, which depends upon the material of the medium. One corresponding concept in [[mechanics]] is the [[Principle of least action|principle of least/stationary action]].

Many important problems involve functions of several variables. Solutions of [[boundary value problem]]s for the [[Laplace equation]] satisfy the [[Dirichlet's principle]]. [[Plateau's problem]] requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial [[topology]].