CALCULUS OF VARIATIONS

branch of mathematics concerned with the problem of finding a function for which the value of a certain integral is either the largest or the smallest possible. Many problems of this kind are easy to state, but their solutions commonly involve difficult procedures of the differential calculus and differential equations. The isoperimetric problemthat of finding, among all plane figures of a given perimeter, the one enclosing the greatest areawas known to Greek mathematicians of the 2nd century BC. The term isoperimetric problem has been extended to mean any problem in the calculus of variations in which a function is to be made a maximum or a minimum, subject to an auxiliary condition called the isoperimetric condition, although it may have nothing to do with perimeters. For example, the problem of finding a solid of given volume that has the least surface area is an isoperimetric problem, the given volume being the auxiliary, or isoperimetric, condition. Another example of an isoperimetric problem from the field of aerodynamics is that of finding the shape of a solid having a given volume that will encounter minimum resistance as it travels though the atmosphere at a constant velocity. Modern interest in the calculus of variations began when in 1696 Johan Bernoulli of Switzerland proposed a brachistochrone problem. Suppose that a thin wire in the shape of a curve joins two points at different elevations. Suppose a bead is placed on the wire at the higher point and allowed to slide under gravity, starting from rest and assuming no friction. The question is: What should be the shape of the curve so that the bead will reach the lower point in the least time? The problem was solved independently by Johan Bernoulli, his brother Jakob, and Isaac Newton (16421727). The basic idea was to set up an integral for the total time of fall in terms of the unknown curve and then vary the curve so that a minimum time is obtained. This technique, typical of the calculus of variations, led to a differential equation whose solution is a curve called the cycloid. It is possible to formulate various scientific laws in terms of general principles involving the calculus of variations. These are called variational principles and are usually expressed by stating that some given integral is a maximum or a minimum. One example is Hamilton's principle of least action, of importance in the theory of motion. In this case an integral, called an action integral, is to be minimized. This principle, which leads to Newton's laws of motion as a special case, has also been used as a basis for quantum mechanics. Applications of variational principles also occur in elasticity, electromagnetic theory, aerodynamics, the theory of vibrations, and other areas in engineering and science.