| Calculus Of Variations |
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The key theorem of calculus of variations is the Euler-Lagrange 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 (or neither) has been found. Variational methods are important in and in application of the Principle Of Stationary Action to Quantum Mechanics . Variational methods provide the mathematical basis for the Finite Element Method , which is a very powerful tool for solving Boundary Value Problem s. They are also much used for studying material equilibria in Materials Science , and in pure mathematics, for example the use of the '' Dirichlet Principle '' for harmonic functions by Bernhard Riemann . 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 . The theory of Optimal Control is a generalization of the calculus of variations. SEE ALSO
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