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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

I=int_b^af(y,y^.,x)dx.
(1)

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

(partialf)/(partialy)-d/(dx)((partialf)/(partialy^.))=0.
(2)

The fundamental lemma of calculus of variations states that, if

int_a^bM(x)h(x)dx=0
(3)

for all h(x) with continuous second partial derivatives, then

M(x)=0
(4)

on (a,b).

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.

See also

Beltrami Identity, Bolza Problem, Brachistochrone Problem, Catenary, Envelope Theorem, Euler-Lagrange Differential Equation, Isoperimetric Problem, Isovolume Problem, Lindelof's Theorem, Morse Theory, Plateau's Problem, Line Line Picking, 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.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.

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Calculus of Variations

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Weisstein, Eric W. "Calculus of Variations." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CalculusofVariations.html

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