chapter 11, Calculus of Variations

Excerpt

Historical Comments: The earliest problems of elementary calculus that gained wide attention of mathematicians involved the determination of a maximum or minimum of some quantity. This interest stemmed from the fact that, prior to the invention of the calculus by Sir Isaac Newton (1642–1727) and Gottfried Wilhelm von Leibniz (1646–1716), there existed no systematical procedure for solving such extremal problems. Similar types of extremal problems form the central core of the theory of the calculus of variations.

In 1696, John Bernoulli (1667–1748) proposed his now famous brachistochrone problem—that is, finding the path in a vertical plane down which a particle will fall from one point to another in minimal time. The brachistochrone problem gained the attention of many mathematicians like Newton, Leibniz, Guillaume F. A. de L'Hôpital (1661–1704), and the older Bernoulli brother James (1654–1705), all of whom provided a correct solution.

Up until the time of the brachistochrone problem, the problems we associate today with the calculus of variations originated as certain isolated maximum or minimum problems not treatable by techniques of the elementary calculus. People such as Leonhard Euler (1707–1783) continued the development of the subject and introduced the variational notation for which the subject is now named. Today the calculus of variations plays an increasingly important role in the fields of analysis, physics, and engineering.

The purpose of this chapter is to develop an optimization technique known as the calculus of variations. This powerful method can be applied to a variety of mathematical and physical problems to derive the governing differential equation of the problem. Among others, such problems include those that arise from Hamilton's principle and in applying the principle of minimum potential energy to determine the equilibrium configuration of a deformable body. Higher-dimensional problems often lead to some of the classic partial differential equations of mathematical physics.

11.1 Introduction

The calculus of variations deals with the optimization problem of finding an extremal (maxima or minima) of a quantity in the form of an integral. The simplest example of such a problem is to show that the shortest path between two points in space is a straight line. A similar problem formulated from Fermat's principle leads to Snell's law.

From the beginning, the calculus of variations had its development closely interlaced with that of the differential calculus.