Applied Calculus of Variations for Engineers, Third Edition

ISBN-13: 9780367376093 / Angielski / Twarda / 2019 / 273 str.

Louis Komzsik

Opis

This third edition extends the focus of the book to academia to also support 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.

Spis treści

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

O autorze

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.