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.
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
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.
Referenced on Wolfram|Alpha:
Calculus of Variations
CITE THIS AS:
Weisstein, Eric W. "Calculus of Variations."
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