# Calculus of variations

**Calculus of variations** is a field of mathematics which 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. Some classical problems on curves were posed in this form: one example is the brachistochrone, the path along which a particle would descend under gravity in the shortest time from a given point A to a point B not directly beneath it. Amongst the curves from A to B one has to minimise the expression representing the time of descent.

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 theoretical physics: in Lagrangian mechanics 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 problems. 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

- Isoperimetric inequality
- Variational principle
- Fermat's principle
- Principle of least action
- Infinite-dimensional optimization
- Functional analysis
- Perturbation methods

## External links

- Chapter III: Introduction to the calculus of variations by Johan Byström, Lars-Erik Persson, and Fredrik Strömberg
- PlanetMath.org: Calculus of variations
- Wolfram Research's MathWorld: Calculus of Variations

## Reference books

- Fomin, S.V. and Gelfand, I.M.: Calculus of Variations, Dover Publ., 2000
- Lebedev, L.P. and Cloud, M.J.: The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics, World Scientific, 2003, pages 1-98
- Charles Fox: An Introduction to the Calculus of Variations, Dover Publ., 1987de:Variationsrechnung

es:Cálculo de variaciones ja:変分法 pl:Rachunek wariacyjny ru:Вариационное исчисление sl:Variacijski račun