Applied Calculus of Variations for Engineers, Third edition

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Acknowledgments
Author
Introduction
Part 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
			1.4.1 Shortest curve between two points
			1.4.2 The brachistochrone problem
			1.4.3 Fermat’s principle
			1.4.4 Particle moving in a gravitational field
		1.5 Open boundary variational problems
		1.6 Exercises
	2. Constrained variational problems
		2.1 Algebraic boundary conditions
			2.1.1 Transversality condition computation
		2.2 Lagrange’s solution
		2.3 Isoperimetric problems
			2.3.1 Maximal area under curve with given length
			2.3.2 Optimal shape of curve of given length under gravity
		2.4 Closed-loop integrals
		2.5 Exercises
	3. Multivariate functionals
		3.1 Functionals with several functions
			3.1.1 Euler–Lagrange system of equations
		3.2 Variational problems in parametric form
			3.2.1 Maximal area by closed parametric curve
		3.3 Functionals with two independent variables
		3.4 Minimal surfaces
			3.4.1 Minimal surfaces of revolution
		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
			5.3.1 Orthogonal eigensolutions
		5.4 Sturm–Liouville problems
			5.4.1 Legendre’s equation and polynomials
		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’s 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
Part II: Modeling applications
	8. Differential geometry
		8.1 The geodesic problem
			8.1.1 Geodesics of a sphere
			8.1.2 Geodesic polyhedra
		8.2 A system of differential equations for geodesic curves
			8.2.1 Geodesics of surfaces of revolution
		8.3 Geodesic curvature
			8.3.1 Geodesic curvature of helix
		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 non-uniform 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
			10.3.1 Conservation of energy
			10.3.2 Newton’s law of motion
		10.4 Lagrange’s equations of motion
			10.4.1 Mechanical system modeling
			10.4.2 Electro-mechanical analogy
		10.5 Orbital motion
			10.5.1 Conservation of angular momentum
			10.5.2 The 3-body problem
		10.6 Variational foundation of fluid motion
	11. Analytic mechanics
		11.1 Elastic string vibrations
		11.2 The elastic membrane
			11.2.1 Circular membrane vibrations
			11.2.2 Non-zero boundary conditions
		11.3 Bending of a beam under its own weight
			11.3.1 Transverse vibration of beam
		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
			11.6.1 Dimensionless heat equation
	12. Computational mechanics
		12.1 The finite element technique
			12.1.1 Finite element meshing
			12.1.2 Shape functions
			12.1.3 Element matrix generation
			12.1.4 Element matrix assembly and solution
		12.2 Three-dimensional elasticity
		12.3 Mechanical system analysis
		12.4 Heat conduction
		12.5 Fluid mechanics
Solutions to selected exercises
Notations
References
Index