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.
Author(s): Louis Komzsik
Edition: 2nd ed. 2014
Publisher: CRC Press
Year: 2014
Language: English
Pages: xx+214
Preface to the Second Edition
Preface to the First Edition
Acknowledgments
About the Author
List of Notations
I. Mathematical Foundation
1 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
2 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
3 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
4 Higher Order Derivatives
The Euler-Poisson Equation
The Euler-Poisson System of Equations
Algebraic Constraints on the Derivative
Linearization of Second Order Problems
5 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
6 Analytic Solutions of Variational Problems
Laplace Transform Solution
Separation of Variables
Complete Integral Solutions
Poisson’s Integral Formula
Method of Gradients
7 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
8 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
9 Computational Geometry
Natural Splines
B-Spline Approximation
B-Splines with Point Constraints
B-Splines with Tangent Constraints
Generalization to Higher Dimensions
10 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
11 Analytic Mechanics
Elastic String Vibrations
The Elastic Membrane
Circular Membrane Vibrations
Non-Zero Boundary Conditions
Bending of a Beam under Its Own Weight
12 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
References
Index
List of Figures
List of Tables
