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)
Euler-Lagrange
differential equation is satisfied, i.e., if
(2)
The fundamental lemma of calculus
of variations states that, if
(3)
for all 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 Explore
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References Arfken, G. "Calculus of Variations." Ch. 17 in Mathematical
Methods for Physicists, 3rd ed. Bliss, G. A. Calculus
of Variations. Forsyth, A. R.
Calculus
of Variations. Fox, C. An
Introduction to the Calculus of Variations. Isenberg,
C. The
Science of Soap Films and Soap Bubbles. Jeffreys,
H. and Jeffreys, B. S. "Calculus of Variations." Ch. 10 in Methods
of Mathematical Physics, 3rd ed. Menger, K. "What is the Calculus
of Variations and What are Its Applications?" Part V, Ch. 8 in The
World of Mathematics, Vol. 2 Sagan, H. Introduction
to the Calculus of Variations. Smith,
D. R. Variational
Methods in Optimization. Todhunter, I.
History
of the Calculus of Variations During the Nineteenth Century. Weinstock, R. Calculus
of Variations, with Applications to Physics and Engineering. 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."
From MathWorld https://mathworld.wolfram.com/CalculusofVariations.html
Subject classifications
has an extremum only if the Euler-Lagrange
differential equation is satisfied, i.e., if
with continuous second partial
derivatives, then
.
