The key theorem of calculus of variations is the EulerLagrange equation[?]. This corresponds to the stationary condition on a functional. As in the case of finding the maxima and minima of a function, the analysis of small changes round a supposed solution gives a condition, to first order. It cannot tell one directly whether a maximum or minimum has been found.
Variational methods are important in theoretical physics[?]: in Lagrangian mechanics and in application of the principle of stationary action[?] to quantum mechanics. They were also much used in the past in pure mathematics, for example the use of the Dirichlet principle for harmonic functions by Bernhard Riemann.
In modern mathematics the calculus of variations as such is no longer much used. The same material can appear under other headings, such as Hilbert space techniques, Morse theory[?], or symplectic geometry[?]. The term variational is used of all extremal functional questions. The study of geodesics in differential geometry is a field with an obvious variational content. Much work has been done on the minimal surface (soap bubble) problem, known as Plateau's problem[?].
Search Encyclopedia

Featured Article
