دانلود کتاب Applied calculus of variations for engineers

فهرست مطالب :

Preface to the Second Edition Preface to the First Edition Acknowledgments 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 References Index List of Figures List of Tables