Variational Principles — A Survey

Abstract

A physical problem may be formulated as a variational principle rather than as a differential equation with associated conditions. The basic problem of the variational principle is to determine the function from an admissible class of functions such that a certain definite integral involving the function and some of its derivatives takes on a maximum or minimum value in a closed region R. This is a generalisation of the elementary theory of maxima and minima of the calculus which is concerned with the problem of finding a point in a closed region at which a function has a maximum or minimum value compared with neighboring points in the region. The definite integral in the variational principle is referred to as a functional, since it depends on the entire course of a function rather than on a number of variables. The domain of the functional is the space of the admissible functions. The main difficulty with the variational principle approach is that problems which can be meaningfully formulated as variational principles may not have solutions. This is reflected in mathematical terms by the domain of admissible functions of the functional not forming a closed set. Thus the existence of an extremum (maximum or minimum) cannot be assumed for a variational principle. However, in this text we are concerned with approximate solutions of variational principles. These are obtained by considering some closed subset of the space of permissible functions to provide an upper and lower bound for the theoretical solution of the variational principle.