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.
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
