ISOPERIMETRIC PROBLEM

in mathematics, the determination of the shape of the closed plane curve having a given length and enclosing the maximum area (the curve is a circle). The calculus of variations evolved from attempts to solve this problem and the brachistochrone (q.v.) problem. Isoperimetrics was made the subject of an investigation by 17th- and 18th-century Swiss mathematicians, the Bernoullis, who found and classified many curves having maximum or minimum properties. A major step in generalization was taken by the Swiss mathematician Leonhard Euler, who published the rule (1744) later known as Euler's differential equation, useful in the determination of a minimizing arc between two points on a curve having continuous second derivatives and second partial derivatives. His work was soon supplemented by that of Joseph-Louis Lagrange, Adrien-Marie Legendre, and others. Techniques of the calculus of variations are frequently applied in seeking a particular arc from some given class for which some parameter (length or other quantity dependent upon the entire arc) is minimal or maximal. Surfaces or functions of several variables may be involved. A problem in three-dimensional Euclidean space (that of finding a surface of minimal area having a given boundary) has received much attention and is called Plateau's problem. As a physical example, consider the shapes of soap bubbles and raindrops, which are determined by the surface tension and cohesive forces tending to maintain the fixed volume while decreasing the area to a minimum. Other examples may be found in mechanics, electricity, relativity, and thermodynamics.