Foreword
A foreword is essentially an introductory note penned by an invited writer, scholar, or public figure. As a new textbook does represent a pedagogical experiment, a foreword can serve to illuminate the author's intentions and provide a bit of insight regarding the potential impact of the book.
Alfred James Lotka – the famous chemist, demographer, ecologist, and mathematician – once stated that "The preface is that part of a book which is written last, placed first, and read least." Although the following paragraphs do satisfy Lotka's first two conditions, I hope they will not satisfy the third. For here we have a legitimate chance to adopt the sort of philosophical viewpoint so often avoided in modern scientific treatises. This is partly because the present authors, Lebedev and Cloud, have accepted the challenge of unifying three fundamental subjects that were all rooted in a philosophically-oriented century, and partly because the variational method itself has been the focus of controversy over its philosophical interpretation. The mathematical and philosophical value of the method is anchored in its coordinate-free formulation and easy transformation of parameters. In mechanics it greatly facilitates both the formulation and solution of the differential equations of motion. It also serves as a rigorous foundation for modern numerical approaches such as the finite element method. Through some portion of its history, the calculus of variations was regarded as a simple collection of recipes capable of yielding necessary conditions of minimum for interesting yet very particular functionals. But simple application of such formulas will not suffice for reliable solution of modern engineering problems | we must also understand various convergence-related issues for the popular numerical methods used, say, in elasticity. The basis for this understanding is functional analysis: a relatively young branch of mathematics pioneered by Hilbert, Wiener, von Neumann, Riesz, and many others. It is worth noting that Stefan Banach, who introduced what we might regard as the core of modern functional analysis, lectured extensively on theoretical mechanics; it is therefore not surprising that he knew exactly what sort of mathematics was most needed by engineers.

For a number of years I have delivered lecture courses on system dynamics and control to students and researchers interested in Mechatronics at Johannes Kepler University of Linz, the Technical University of Vienna, and the Technical University of Graz. Mechatronics is an emerging discipline, frequently described as a mixture of mechanics, electronics, and computing; its principal applications are to controlled mechanical devices. Some engineers hold the mistaken view that mechatronics contains nothing new, since both automatic control and computing have existed for a long time. But I believe that mechatronics is a philosophy which happens to overlap portions of the above-mentioned fields without belonging to any of them exclusively. Mechanics, of course, rests heavily on the calculus of variations, and has a long history dating from the works of Bernoulli, Leibniz, Euler, Lagrange, Fermat, Gauss, Hamilton, Routh, and the other pioneers. The remaining disciplines | electronics and computing | are relatively young. Optimal control theory has become involved in mechatronics for obvious reasons: it extends the idea of optimization embodied in the calculus of variations. This involves a significant extension of the class of problems to which optimization can be applied. It also involves an extension of traditional "smooth" analysis tools to the kinds of "non-smooth" tools needed for high-powered computer applications. So again we see how the tools of modern mathematics come into contact with those of computing, and are therefore of concern to mechatronics.

Teaching a combination of the calculus of variations and functional analysis to students in engineering and applied mathematics is a real challenge. These subjects require time, dedication, and creativity from an instructor. They also take special care if the audience wishes to understand the rigorous mathematics used at the frontier of contemporary research. A principal hindrance has been the lack of a suitable textbook covering all necessary topics in a unified and sensible fashion. The present book by Professors Lebedev and Cloud is therefore a welcome addition to the literature. It is lucid, well-connected, and concise. The material has been carefully chosen. Throughout the book, the authors lay stress on central ideas as they present one powerful mathematical tool after another. The reader is thus prepared not only to apply the material to his or her own work, but also to delve further into the literature if desired.

An interesting feature of the book is that optimal control theory arises as a natural extension of the calculus of variations, having a more extensive set of problems and different methods for their solution. Functional analysis, of course, is the basis for justifying the methods of both the calculus of variations and optimal control theory; it also permits us to qualitatively describe the properties of complete physical problems. Optimization and extreme principles run through the entire book as a unifying thread.

The book could function as both (i) an attractive textbook for a course on engineering mathematics at the graduate level, and (ii) a useful reference for researchers in mechanics, electrical engineering, computer science, mechatronics, or related fields such as mechanical, civil, or aerospace engineering, physics, etc. It may also appeal to those mathematicians who lean toward applications in their work. The presence of homework problems at the end of each chapter will facilitate its use as a textbook.

As Poincare once said, mathematicians do not destroy the obstacles with which their science is spiked, but simply push them toward its boundary. I hope that some particular obstacles in the unification of these three branches of science (the calculus of variations, optimal control, and functional analysis) and technology (mechanics, control, and computing) will continue to be pushed out as far as possible. Professors Lebedev and Cloud have made a significant contribution to this process by writing the present book.
Ardeshir Guran
Wien, Austria
March, 2003

Hosted by www.Geocities.ws

1