Preface to the ﬁrst edition
The topic of this book has a long history. Its fundamentals were laid down
by icons of mathematics like Euler and Lagrange. It was once heralded as
the panacea for all engineering optimization problems by suggesting that all
one needs to do was to apply the Euler-Lagrange equation form and solve the
resulting diﬀerential equation.
This, as most all encompassing solutions, turned out to be not always true
and the resulting diﬀerential equations are not necessarily easy to solve. On
the other hand, many of the diﬀerential equations commonly used by engi-
neers today are derived from a variational problem. Hence, it is important
and useful for engineers to delve into this topic.
The book is organized into two parts: theoretical foundation and engineer-
ing applications. The ﬁrst part starts with the statement of the fundamental
variational problem and its solution via the Euler-Lagrange equation. This
is followed by the gradual extension to variational problems subject to con-
straints, containing functions of multiple variables and functionals with higher
order derivatives. It continues with the inverse problem of variational calcu-
lus, when the origin is in the diﬀerential equation form and the corresponding
variational problem is sought. The ﬁrst part concludes with the direct so-
lution techniques of variational problems, such as the Ritz, Galerkin, and
Kantorovich methods.
With the emphasis on applications, the second part starts with a detailed
discussion of the geodesic concept of diﬀerential geometry and its extensions
to higher order spaces. The computational geometry chapter covers the vari-
ational origin of natural splines and the variational formulation of B-splines
under various constraints.
The ﬁnal two chapters focus on analytic and computational mechanics. Top-
ics of the ﬁrst include the variational form and subsequent solution of several
classical mechanical problems using Hamilton’s principle. The last chapter
discusses generalized coordinates and Lagrange’s equations of motion. Some
fundamental applications of elasticity, heat conduction, and ﬂuid mechanics
as well as their computational technology conclude the book.
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