Introduction

Abstract

The first two parts of this book were devoted to generalized functions and Hilbert spaces whose operators are primarily of importance for quantum mechanics and quantum field theory. These two physical theories were born and developed in the 20th century. In sharp contrast to this are the variational methods which have a much longer history. In 1744, L. Euler published a first textbook on what soon after was called the calculus of variations, with the title ‘A method for finding curves enjoying certain maximum or minimum properties’. In terms of the calculus which had recently been invented by Leibniz and Newton, optimal curves were determined by Euler. Depending on the case which is under investigation optimal means “maximal” or “minimal”. Though not under the same name the calculus of variations is actually older and closely related to the invention and development of differential calculus, since already in 1684 Leibniz’ first publication on differential calculus appeared under the title Nova methodus pro maximis et minimis itemque tangentibus. This can be considered as the beginning of a mathematical theory which intends to solve problems of “optimization” through methods of analysis and functional analysis. Later in the 20th century methods of topology were also used for this. Here ‘optimal’ can mean a lot of very different things, for instance: shortest distance between two points in space, optimal shapes or forms (of buildings, of plane wings, of natural objects), largest area enclosed by a fence of given length, minimal losses (of a company in difficult circumstances), maximal profits (as a general objective of a company). And in this wider sense of ‘finding optimal solutions’ as part of human nature or as part of human belief that in nature an optimal solution exists and is realized there, the calculus of variations goes back more than 2000 years to ancient Greece. In short, the calculus of variations has a long and fascinating history. However ‘variational methods’ are not a mathematical theory of the past, related to classical physics, but an active area of modern mathematical research as the numerous publications in this field show, with many practical or potential applications in science, engineering and economics. Clearly this means for us that in this short third part we will be able to present only the basic aspects of one direction of the modern developments in the calculus of variations, namely those with close links to the previous parts, mainly to Hilbert space methods.