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