CALCULUS OF VARIATIONS

branch of mathematics concerned with finding maximum or minimum conditions for a relationship between two or more variables that depends not only on the variables themselves, as in the ordinary calculus, but also on an additional arbitrary relation, or constraint, between them. For example, the problem of finding the closed plane curve of given length that will enclose the greatest area is a type of isoperimetric (equal-perimeter) problem that can be treated by the methods of the variational calculus; the solution to this special case is the circle. Another famous problem is the brachistochrone problem, that of finding the curve along which an object will slide to a point not directly below it in the shortest time; the solution is a cycloid curve (a curve traced out by a fixed point on the circumference of a circle as the circle rolls along a straight line). In general, problems in the calculus of variations involve solving the definite integral (single or multiple) of a function of one or more independent variables, x₁, x₂,…, one or more dependent variables, y₁, y₂,…, and derivatives of these, the object being to determine the dependent variables as functions of the independent variables such that the integral will be a maximum or minimum. The calculus of variations was founded at the end of the 17th cent. and was developed by Jakob and Johann Bernoulli, Isaac Newton, G. W. Leibniz, Leonhard Euler, J. L. Lagrange, and others.

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