3rd Edition
Applied Calculus of Variations for Engineers, Third edition
Calculus of variations 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 needed to do was to state a variational problem, apply the appropriate Euler-Lagrange equation 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 in various fields of engineering are derived from a variational problem. Hence it is an extremely important topic justifying the new edition of this book.
This third edition extends the focus of the book to academia and supports both variational calculus and mathematical modeling classes. The newly added sections, extended explanations, numerous examples and exercises aid the students in learning, the professors in teaching, and the engineers in applying variational concepts.
Preface
Acknowledgments
Author
Introduction
I Mathematical foundation
1 The foundations of calculus of variations
1.1 The fundamental problem and lemma of calculus of variations
1.2 The Legendre test
1.3 The Euler-Lagrange differential equation
1.4 Minimal path problems
1.5 Open boundary variational problems
1.6 Exercises
2 Constrained variational problems
2.1 Algebraic boundary conditions
2.2 Lagrange’s solution
2.3 Isoperimetric problems
2.4 Closed-loop integrals
2.5 Exercises
3 Multivariate functionals
3.1 Functionals with several functions
3.2 Variational problems in parametric form
3.3 Functionals with two independent variables
3.4 Minimal surfaces
3.5 Functionals with three independent variables
3.6 Exercises
4 Higher order derivatives
4.1 The Euler-Poisson equation
4.2 The Euler-Poisson system of equations
4.3 Algebraic constraints on the derivative
4.4 Linearization of second order problems
4.5 Exercises
5 The inverse problem
5.1 Linear differential operators
5.2 The variational form of Poisson’s equation
5.3 The variational form of eigenvalue problems
5.4 Sturm-Liouville problems
5.5 Exercises
6 Analytic solutions
6.1 Laplace transform solution
6.2 d’Alembert’s solution
6.3 Complete integral solutions
6.4 Poisson’s integral formula
6.5 Method of gradients
6.6 Exercises
7 Approximate methods
7.1 Euler’s method
7.2 Ritz method
7.3 Galerkin’s method
7.4 Approximate solutions of Poisson’s equation
7.5 Kantorovich’s method
7.6 Boundary integral method
7.7 Finite element method
7.8 Exercises
II Modeling applications
8 Differential geometry
8.1 The geodesic problem
8.2 A system of differential equations for geodesic curves
8.3 Geodesic curvature
8.4 Generalization of the geodesic concept
9 Computational geometry
9.1 Natural splines
9.2 B-spline approximation
9.3 B-splines with point constraints
9.4 B-splines with tangent constraints
9.5 Generalization to higher dimensions
9.6 Weighting and nonuniform parametrization
9.7 Industrial applications
10 Variational equations of motion
10.1 Legendre’s dual transformation
10.2 Hamilton’s principle
10.3 Hamilton’s canonical equations
10.4 Lagrange’s equations of motion
10.5 Orbital motion
10.6 Variational foundation of fluid motion
11 Analytic mechanics
11.1 Elastic string vibrations
11.2 The elastic membrane
11.3 Bending of a beam under its own weight
11.4 Buckling of a beam under axial load
11.4.1 Axial vibration of a beam
11.5 Simultaneous axial and transversal loading of beam
11.6 Heat diffusion in a beam
12 Computational mechanics
12.1 The finite element technique
12.2 Three-dimensional elasticity
12.3 Mechanical system analysis
12.4 Heat conduction
12.5 Fluid mechanics
Solutions
Notations
List of Tables
List of Figures
References
Index
Biography
Dr. Louis Komzsik worked in the industry as an engineering mathematician for 42 years and during those years also lectured as a Visiting Professor at various southern California colleges and universities. Since his retirement he is lecturing in the Mathematics Department of the University of California at Irvine.
