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Applied Calculus of Variations for Engineers

By Louis Komzsik

Published October 27th 2008 by CRC Press – 175 pages

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Description

The subject of calculus of variations is to find optimal solutions to engineering problems where the optimum may be a certain quantity, a shape, or a function. Applied Calculus of Variations for Engineers addresses this very important mathematical area applicable to many engineering disciplines. Its unique, application-oriented approach sets it apart from the theoretical treatises of most texts. It is aimed at enhancing the engineer’s understanding of the topic as well as aiding in the application of the concepts in a variety of engineering disciplines.

The first part of the book presents the fundamental variational problem and its solution via the Euler–Lagrange equation. It also discusses variational problems subject to constraints, the inverse problem of variational calculus, and the direct solution techniques of variational problems, such as the Ritz, Galerkin, and Kantorovich methods. With an emphasis on applications, the second part details the geodesic concept of differential geometry and its extensions to higher order spaces. It covers the variational origin of natural splines and the variational formulation of B-splines under various constraints. This section also focuses on analytic and computational mechanics, explaining classical mechanical problems and Lagrange’s equations of motion.

Reviews

"Dr. Komzsik’s latest book offers the readers, especially engineering graduate students, a very valuable, application-oriented, unique, and clear explanation about the topic. As an engineering university professor (and the author of multiple textbooks), I firmly believe that this is a must-read book for engineers and applied mathematicians who wish to learn the important topic of calculus of variations. I have been a loyal reader of all of Dr. Komzsik’s previously published books, due to his consistent and clear presentation style, which is demonstrated in this book as well."

—Professor Duc T. Nguyen, Multidisciplinary Parallel-Vector Computation Center, Old Dominion University, Norfolk, Virginia, USA

Contents

MATHEMATICAL FOUNDATION

The Foundations of Calculus of Variations

The fundamental problem and lemma of calculus of variations

The Legendre test

The Euler–Lagrange differential equation

Application: minimal path problems

Open boundary variational problems

Constrained Variational Problems

Algebraic boundary conditions

Lagrange’s solution

Application: iso-perimetric problems

Closed-loop integrals

Multivariate Functionals

Functionals with several functions

Variational problems in parametric form

Functionals with two independent variables

Application: minimal surfaces

Functionals with three independent variables

Higher Order Derivatives

The Euler–Poisson equation

The Euler–Poisson system of equations

Algebraic constraints on the derivative

Application: linearization of second order problems

The Inverse Problem of the Calculus of Variations

The variational form of Poisson’s equation

The variational form of eigenvalue problems

Application: Sturm–Liouville problems

Direct Methods of Calculus of Variations

Euler’s method

Ritz method

Galerkin’s method

Kantorovich’s method

ENGINEERING APPLICATIONS

Differential Geometry

The geodesic problem

A system of differential equations for geodesic curves

Geodesic curvature

Generalization of the geodesic concept

Computational Geometry

Natural splines

B-spline approximation

B-splines with point constraints

B-splines with tangent constraints

Generalization to higher dimensions

Analytic Mechanics

Hamilton’s principle for mechanical systems

Elastic string vibrations

The elastic membrane

Bending of a beam under its own weight

Computational Mechanics

Three-dimensional elasticity

Lagrange’s equations of motion

Heat conduction

Fluid mechanics

Computational techniques

Closing Remarks

References

Index

Name: Applied Calculus of Variations for Engineers (Hardback) – CRC Press 
Description: By Louis Komzsik. The subject of calculus of variations is to find optimal solutions to engineering problems where the optimum may be a certain quantity, a shape, or a function. Applied Calculus of Variations for Engineers addresses this very important mathematical area...
Categories: Electrical & Electronic Engineering, Applied Mathematics, Mathematics & Statistics for Engineers