Applied Calculus of Variations for Engineers: 2nd Edition (Hardback) book cover

Applied Calculus of Variations for Engineers

2nd Edition

By Louis Komzsik

CRC Press

233 pages | 24 B/W Illus.

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pub: 2014-06-06
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The purpose of the calculus of variations is to find optimal solutions to engineering problems whose optimum may be a certain quantity, shape, or function. Applied Calculus of Variations for Engineers addresses this important mathematical area applicable to many engineering disciplines. Its unique, application-oriented approach sets it apart from the theoretical treatises of most texts, as it is aimed at enhancing the engineer’s understanding of the topic.

This Second Edition text:

  • Contains new chapters discussing analytic solutions of variational problems and Lagrange-Hamilton equations of motion in depth
  • Provides new sections detailing the boundary integral and finite element methods and their calculation techniques
  • Includes enlightening new examples, such as the compression of a beam, the optimal cross section of beam under bending force, the solution of Laplace’s equation, and Poisson’s equation with various methods

Applied Calculus of Variations for Engineers, Second Edition extends the collection of techniques aiding the engineer in the application of the concepts of the calculus of variations.


"There is definitely a need for engineers and scientists alike to master a plethora of tools and techniques in their careers. The calculus of variations has long been viewed as esoteric and theoretical, hence explaining its absence from most universities’ engineering curricula. But mentalities need to be changed as products developed today are becoming more and more sophisticated. Hence there is a need for more books in this field that are targeted to the engineering profession, and I expect that this second edition of Dr. Komzsik’s book will gain widespread popularity. … All scientific and non-scientific fields (such as financial engineering) can benefit from the concept of calculus of variations. This book, with its rather high level of math, will appeal most to those in engineering and the natural sciences."

—Dr. Yogeshwarsing Calleecharan, Department of Engineering Sciences and Mathematics, Lulea University of Technology, Sweden

"The author has explained a very difficult subject in a manner which can be understood, even by those with limited backgrounds. … The book’s subject is one of the basic building blocks for deeper understanding of the finite element method. … This topic has been written about by many mathematicians. However, a complete discussion of this topic, clearly explained by a practical engineer, is definitely a plus for both the engineering and mathematical communities."

—Prof. Duc T. Nguyen Old Dominion University, Norfolk, Virginia, USA

Table of Contents

Preface to the Second Edition

Preface to the First Edition


About the Author

List of Notations

I. 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

Shortest Curve between Two Points

The Brachistochrone Problem

Fermat’s Principle

Particle Moving in the Gravitational Field

Open Boundary Variational Problems

Constrained Variational Problems

Algebraic Boundary Conditions

Lagrange’s Solution

Application: Iso-Perimetric Problems

Maximal Area under Curve with Given Length

Optimal Shape of Curve of Given Length under Gravity

Closed-Loop Integrals

Multivariate Functionals

Functionals with Several Functions

Variational Problems in Parametric Form

Functionals with Two Independent Variables

Application: Minimal Surfaces

Minimal Surfaces of Revolution

Functionals with Three Independent Variables

Higher Order Derivatives

The Euler-Poisson Equation

The Euler-Poisson System of Equations

Algebraic Constraints on the Derivative

Linearization of Second Order Problems

The Inverse Problem of Calculus of Variations

The Variational Form of Poisson’s Equation

The Variational Form of Eigenvalue Problems

Orthogonal Eigensolutions

Sturm-Liouville Problems

Legendre’s Equation and Polynomials

Analytic Solutions of Variational Problems

Laplace Transform Solution

Separation of Variables

Complete Integral Solutions

Poisson’s Integral Formula

Method of Gradients

Numerical Methods of Calculus of Variations

Euler’s Method

Ritz Method

Application: Solution of Poisson’s Equation

Galerkin’s Method

Kantorovich’s Method

Boundary Integral Method

II. Engineering Applications

Differential Geometry

The Geodesic Problem

Geodesics of a Sphere

A System of Differential Equations for Geodesic Curves

Geodesics of Surfaces of Revolution

Geodesic Curvature

Geodesic Curvature of Helix

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

Variational Equations of Motion

Legendre’s Dual Transformation

Hamilton’s Principle for Mechanical Systems

Newton’s Law of Motion

Lagrange’s Equations of Motion

Hamilton’s Canonical Equations

Conservation of Energy

Orbital Motion

Variational Foundation of Fluid Motion

Analytic Mechanics

Elastic String Vibrations

The Elastic Membrane

Circular Membrane Vibrations

Non-Zero Boundary Conditions

Bending of a Beam under Its Own Weight

Computational Mechanics

Three-Dimensional Elasticity

Lagrangian Formulation

Heat Conduction

Fluid Mechanics

The Finite Element Method

Finite Element Meshing

Shape Functions

Element Matrix Generation

Element Matrix Assembly and Solution

Closing Remarks



List of Figures

List of Tables

About the Author


Dr. Louis Komzsik is a graduate of the Technical University of Budapest, Hungary and the Eötvös Loránd University, Budapest, Hungary. He has been working in the industry for more than 40 years, and is currently the chief numerical analyst in the Office of Architecture and Technology at Siemens PLM Software, Cypress, California, USA. Dr. Komzsik is the author of the NASTRAN Numerical Methods Handbook, first published by MSC in 1987. His book, The Lanczos Method, published by SIAM, has also been translated into Japanese, Korean, and Hungarian. His book, Computational Techniques of Finite Element Analysis, published by CRC Press, is in its second print, and his Approximation Techniques for Engineers was published by Taylor and Francis in 2006. He is also the coauthor of the book Computational Techniques of Rotor Dynamics with the Finite Element Method, published by Taylor and Francis in 2012.

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