What is "calculus of variations"

Variation \Va`ri*a"tion\, n. [OE. variatioun, F. variation, L. variatio. See Vary.]

1. The act of varying; a partial change in the form, position, state, or qualities of a thing; modification; alternation; mutation; diversity; deviation; as, a variation of color in different lights; a variation in size; variation of language.

   The essences of things are conceived not capable of any such variation.
   --Locke.

2. Extent to which a thing varies; amount of departure from a position or state; amount or rate of change.

3. (Gram.) Change of termination of words, as in declension, conjugation, derivation, etc.

4. (Mus.) Repetition of a theme or melody with fanciful embellishments or modifications, in time, tune, or harmony, or sometimes change of key; the presentation of a musical thought in new and varied aspects, yet so that the essential features of the original shall still preserve their identity.

5. (Alg.) One of the different arrangements which can be made of any number of quantities taking a certain number of them together.

   Annual variation (Astron.), the yearly change in the right ascension or declination of a star, produced by the combined effects of the precession of the equinoxes and the proper motion of the star.

   Calculus of variations. See under Calculus.

   Variation compass. See under Compass.

   Variation of the moon (Astron.), an inequality of the moon's motion, depending on the angular distance of the moon from the sun. It is greater at the octants, and zero at the quadratures.

   Variation of the needle (Geog. & Naut.), the angle included between the true and magnetic meridians of a place; the deviation of the direction of a magnetic needle from the true north and south line; -- called also declination of the needle.

   Syn: Change; vicissitude; variety; deviation.

Calculus \Cal"cu*lus\, n.; pl. Calculi. [L, calculus. See Calculate, and Calcule.]

1. (Med.) Any solid concretion, formed in any part of the body, but most frequent in the organs that act as reservoirs, and in the passages connected with them; as, biliary calculi; urinary calculi, etc.

2. (Math.) A method of computation; any process of reasoning by the use of symbols; any branch of mathematics that may involve calculation.

   Barycentric calculus, a method of treating geometry by defining a point as the center of gravity of certain other points to which co["e]fficients or weights are ascribed.

   Calculus of functions, that branch of mathematics which treats of the forms of functions that shall satisfy given conditions.

   Calculus of operations, that branch of mathematical logic that treats of all operations that satisfy given conditions.

   Calculus of probabilities, the science that treats of the computation of the probabilities of events, or the application of numbers to chance.

   Calculus of variations, a branch of mathematics in which the laws of dependence which bind the variable quantities together are themselves subject to change.

   Differential calculus, a method of investigating mathematical questions by using the ratio of certain indefinitely small quantities called differentials. The problems are primarily of this form: to find how the change in some variable quantity alters at each instant the value of a quantity dependent upon it.

   Exponential calculus, that part of algebra which treats of exponents.

   Imaginary calculus, a method of investigating the relations of real or imaginary quantities by the use of the imaginary symbols and quantities of algebra.

   Integral calculus, a method which in the reverse of the differential, the primary object of which is to learn from the known ratio of the indefinitely small changes of two or more magnitudes, the relation of the magnitudes themselves, or, in other words, from having the differential of an algebraic expression to find the expression itself.

n. (context calculus English) The form of calculus that deals with the maxima and minima of definite integrals of functions of many variables

n. the calculus of maxima and minima of definite integrals

Calculus of variations is a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. The interest is in extremal functions that make the functional attain a maximum or minimum value – or stationary functions – those where the rate of change of the functional is zero.

A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is obviously a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, where the optical length depends upon the material of the medium. One corresponding concept in mechanics is the principle of least action.

Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in a solution of soap suds. Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.