Calculus of variations

The calculus of variations (or variational calculus) is a field of <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a> that uses variations, which are small changes in <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a>
and <a href="/facts/Functional_(mathematics)/3Cp3PHBR">functionals</a>, to find maxima and minima of functionals: <a href="/facts/Map_(mathematics)/FABF1l2f">mappings</a> from a set of <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> to the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>. Functionals are often expressed as <a href="/facts/Definite_integral/V7lyV997">definite integrals</a> involving functions and their <a href="/facts/Derivative/7Ito70Dh">derivatives</a>. Functions that maximize or minimize functionals may be found using the <a href="/facts/Euler%25E2%2580%2593Lagrange_equation/Ymq3r3Dh">Euler–Lagrange equation</a> 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 <a href="/facts/Straight_line/gJqsr4Gj">straight line</a> 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 <a href="/facts/Geodesic/fbWR6l80">geodesics</a>. A related problem is posed by <a href="/facts/Fermat%2527s_principle/UL9XumdW">Fermat's principle</a>: light follows the path of shortest <a href="/facts/Optical_length/qpXIMzqz">optical length</a> connecting two points, which depends upon the material of the medium. One corresponding concept in <a href="/facts/Mechanics/A3yHLQA5">mechanics</a> is the <a href="/facts/Principle_of_least_action/QaBTpoKz">principle of least/stationary action</a>.
Many important problems involve functions of several variables. Solutions of <a href="/facts/Boundary_value_problem/QEfmUqmP">boundary value problems</a> for the <a href="/facts/Laplace_equation/8n1YkmFS">Laplace equation</a> satisfy the <a href="/facts/Dirichlet%2527s_principle/2PUtzTL8">Dirichlet's principle</a>. <a href="/facts/Plateau%2527s_problem/UyRuUh2c">Plateau's problem</a> 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 <a href="/facts/Topology/2EqW47cU">topology</a>.

Calculus of variations open-in-new

Calculus of variations