Calculus of Variations

A branch of mathematics that is a sort of generalization of calculus. Calculus of variations seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum). Mathematically, this involves finding stationary values of integrals of the form

(1)

 has an extremum only if the Euler-Lagrange differential equation is satisfied, i.e., if

(2)

The fundamental lemma of calculus of variations states that, if

(3)

for all  with continuous second partial derivatives, then

(4)

on .

A generalization of calculus of variations known as Morse theory (and sometimes called "calculus of variations in the large") uses nonlinear techniques to address variational problems.

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.

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.

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.