The calculus of variations studies extrema and stationary values of functionals. A functional assigns a number to a function, curve, or surface, such as its arc length or surface area. Johann Bernoulli's 1696 brachistochrone problem, which asks for a path of least descent time, was an early example (Gelfand and Fomin 1963, p. 3). Euler developed a general method for such problems, and Lagrange obtained the equation now named for them by varying the unknown function (Kline 1990, pp. 579 and 582-584).

A basic problem is to find a function with specified endpoint values for which the functional

is stationary, where and the integrand is called the Lagrangian function. Varying to for a continuously differentiable with gives the first variation

Here subscripts denote partial derivatives of evaluated along . For and twice continuously differentiable, integration by parts moves the derivative from to . The endpoint term vanishes, leaving the integral of . The fundamental lemma of calculus of variations says that a continuous factor whose integral against every smooth of compact support in vanishes must itself vanish there. Stationarity therefore gives the Euler-Lagrange differential equation

This equation determines candidates for a minimum or maximum. If the endpoint values are free, the first variation also gives natural boundary conditions, which can include derivatives of a cost at the endpoints in the Bolza problem.

The brachistochrone problem and Fermat's principle both use a travel time functional. A geodesic makes the arc length stationary, and the search for a minimal surface of revolution varies the surface area. In mechanics, a variational symmetry of the action integral can yield a conservation law through Noether's symmetry theorem.

The calculus of variations in the large uses the topology of the space of admissible curves to study the existence and multiplicity of extremals.

For variational problems specified by a Lagrangian function, a future version of the Wolfram Language will provide EulerLagrangeD and EulerLagrangeEquation to form the Euler-Lagrange derivative and the Euler-Lagrange differential equation, and EulerLagrangeSolve and NEulerLagrangeSolve to solve variational problems.

See also

Beltrami Identity, Bolza Problem, Brachistochrone Problem, Catenary, Envelope Theorem, Euler-Lagrange Differential Equation, Fermat's Principle, Functional Derivative, Fundamental Lemma of Calculus of Variations, Isoperimetric Problem, Isovolume Problem, Lindeloef's Theorem, Line Line Picking, Morse Theory, Noether's Symmetry Theorem, Plateau's Problem, Roulette, Skew Quadrilateral, Sphere with Tunnel, Surface of Revolution, Unduloid, Weierstrass-Erdman Corner Condition Explore this topic in the MathWorld classroom

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References

Arfken, G. "Calculus of Variations." Ch. 17 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 925-962, 1985.Bliss, G. A. Calculus of Variations. Chicago, IL: Open Court, 1925.Forsyth, A. R. Calculus of Variations. New York: Dover, 1960.Fox, C. An Introduction to the Calculus of Variations. New York: Dover, 1988.Gelfand, I. M. and Fomin, S. V. Calculus of Variations. Englewood Cliffs, NJ: Prentice-Hall, 1963.Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, 1992.Jeffreys, H. and Jeffreys, B. S. "Calculus of Variations." Ch. 10 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 314-332, 1988.Kline, M. "The Calculus of Variations in the Eighteenth Century." Ch. 24 in Mathematical Thought from Ancient to Modern Times, Vol. 2. Oxford, England: Oxford University Press, 1990.Menger, K. "What Is the Calculus of Variations and What Are Its Applications?" Part V, Ch. 8 in The World of Mathematics, Vol. 2 (Ed. K. Newman). New York: Dover, pp. 886-890, 2000.Sagan, H. Introduction to the Calculus of Variations. New York: Dover, 1992.Smith, D. R. Variational Methods in Optimization. New York: Dover, 1998.Todhunter, I. History of the Calculus of Variations during the Nineteenth Century. New York: Chelsea, 1962.Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, 1974.Weisstein, E. W. "Books about Calculus of Variations." http://www.ericweisstein.com/encyclopedias/books/CalculusofVariations.html.

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