Calculus of Variations

A branch of mathematics which 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

$$I = \int_b^a f(y,\dot y,x)\,dx.$$ (1)

$I$ has an extremum only if the Euler-Lagrange Differential Equation is satisfied, i.e., if

$${\partial f\over\partial y}-{d\over dx}\left({\partial f\over\partial\dot y}\right)= 0.$$ (2)

The Fundamental Lemma of Calculus of Variations states that, if

$$\int^b_a M(x)h(x)\,dx = 0$$ (3)

for all $h(x)$ with Continuous second Partial Derivatives, then

$$M(x) = 0$$ (4)

on $(a,b)$.

See also Beltrami Identity, Bolza Problem, Brachistochrone Problem, Catenary, Envelope Theorem, Euler-Lagrange Differential Equation, Isoperimetric Problem, Isovolume Problem, Lindelof's Theorem, Plateau's Problem, Point-Point Distance--2-D, Point-Point Distance--3-D, Roulette, Skew Quadrilateral, Sphere with Tunnel, Unduloid, Weierstraß-Erdman Corner Condition

References

Calculus of Variations

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.

Menger, K. ``What is the Calculus of Variations and What are Its Applications?'' In The World of Mathematics (Ed. K. Newman). Redmond, WA: Microsoft Press, pp. 886-890, 1988.

Sagan, H. Introduction to the Calculus of Variations. New York: Dover, 1992.

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.