There is an ongoing resurgence of applications in which the calculus of variations has direct relevance. Variational Methods with Applications in Science and Engineering reflects the strong connection between calculus of variations and the applications for which variational methods form the fundamental foundation. The material is presented in a manner that promotes development of an intuition about the concepts and methods with an emphasis on applications, and the priority of the application chapters is to provide a brief introduction to a variety of physical phenomena and optimization principles from a unified variational point of view. The first part of the book provides a modern treatment of the calculus of variations suitable for advanced undergraduate students and graduate students in applied mathematics, physical sciences, and engineering. The second part gives an account of several physical applications from a variational point of view, such as classical mechanics, optics and electromagnetics, modern physics, and fluid mechanics. A unique feature of this part of the text is derivation of the ubiquitous Hamilton's principle directly from the first law of thermodynamics, which enforces conservation of total energy, and the subsequent derivation of the governing equations of many discrete and continuous phenomena from Hamilton's principle. In this way, the reader will see how the traditional variational treatments of statics and dynamics are unified with the physics of fluids, electromagnetic fields, relativistic mechanics, and quantum mechanics through Hamilton's principle. The third part covers applications of variational methods to optimization and control of discrete and continuous systems, including image and data processing as well as numerical grid generation. The application chapters in parts two and three are largely independent of each other so that the instructor or reader can choose a path through the topics that aligns with their interests.
Author(s): Kevin W. Cassel
Publisher: Cambridge University Press
Year: 2013
Language: English
Pages: 434
Tags: Физика;Матметоды и моделирование в физике;
CONTENTS......Page 9
Preface......Page 15
PART I VARIATIONAL METHODS......Page 21
1 Preliminaries......Page 23
1.1 A Bit of History......Page 24
1.2 Introduction......Page 27
1.3 Motivation......Page 28
1.4 Extrema of Functions......Page 34
1.5 Constrained Extrema and Lagrange Multipliers......Page 37
1.6 Integration by Parts......Page 40
1.7 Fundamental Lemma of the Calculus of
Variations......Page 41
1.8 Adjoint and Self-Adjoint Differential Operators......Page 42
Exercises......Page 46
2 Calculus of Variations......Page 48
2.1 Functionals of One Independent Variable......Page 49
2.2 Natural Boundary Conditions......Page 64
2.3 Variable End Points......Page 73
2.5 Functionals of Two Independent Variables......Page 76
2.6 Functionals of Two Dependent Variables......Page 84
2.7 Constrained Functionals......Page 86
2.8 Summary of Euler Equations......Page 100
Exercises......Page 101
3 Rayleigh-Ritz, Galerkin, and Finite-Element Methods......Page 110
3.1 Rayleigh-Ritz Method......Page 111
3.2 Galerkin Method......Page 120
3.3 Finite-Element Methods......Page 123
Exercises......Page 130
PART II PHYSICAL APPLICATIONS......Page 135
4 Hamilton’s Principle......Page 137
4.1 Hamilton’s Principle for Discrete Systems......Page 138
4.2 Hamilton’s Principle for Continuous Systems......Page 148
4.3 Euler-Lagrange Equations......Page 151
4.4 Invariance of the Euler-Lagrange Equations......Page 156
4.5 Derivation of Hamilton’s Principle from the First Law of
Thermodynamics......Page 157
4.6 Conservation of Mechanical Energy and the Hamiltonian......Page 161
4.7 Noether’s Theorem – Connection Between
Conservation Laws and Symmetries in Hamilton’s
Principle......Page 163
4.8 Summary......Page 166
4.9 Brief Remarks on the Philosophy of Science......Page 168
Exercises......Page 172
5 Classical Mechanics......Page 180
5.1 Dynamics of Nondeformable Bodies......Page 181
5.2 Statics of Nondeformable Bodies......Page 198
5.3 Statics of Deformable Bodies......Page 204
5.4 Dynamics of Deformable Bodies......Page 217
6.1 Introduction......Page 222
6.2 Simple Pendulum......Page 223
6.3 Linear, Second-Order, Autonomous Systems......Page 227
6.4 Nonautonomous Systems – Forced Pendulum......Page 232
6.5 Non-Normal Systems – Transient Growth......Page 235
6.6 Continuous Systems – Beam-Column Buckling......Page 242
7.1 Optics......Page 245
7.2 Maxwell’s Equations of Electromagnetics......Page 249
7.3 Electromagnetic Wave Equations......Page 252
7.4 Discrete Charged Particles in an Electromagnetic Field......Page 253
7.5 Continuous Charges in an Electromagnetic Field......Page 257
8 Modern Physics......Page 260
8.1 Relativistic Mechanics......Page 261
8.2 Quantum Mechanics......Page 271
9 Fluid Mechanics......Page 279
9.1 Introduction......Page 280
9.2 Inviscid Flow......Page 282
9.3 Viscous Flow – Navier-Stokes Equations......Page 289
9.4 Multiphase and Multicomponent Flows......Page 295
9.5 Hydrodynamic Stability Analysis......Page 301
9.6 Flow Control......Page 317
PART III OPTIMIZATION......Page 321
10 Optimization and Control......Page 323
10.1 Optimization and Control Examples......Page 325
10.2 Shape Optimization......Page 326
10.3 Financial Optimization......Page 330
10.4 Optimal Control of Discrete Systems......Page 332
10.5 Optimal Control of Continuous Systems......Page 362
10.6 Control of Real Systems......Page 371
10.7 Postscript......Page 376
Exercises......Page 377
11 Image Processing and Data Analysis......Page 381
11.1 Variational Image Processing......Page 382
11.2 Curve and Surface Optimization Using Splines......Page 391
11.3 Proper-Orthogonal Decomposition......Page 394
12.1 Fundamentals......Page 399
12.2 Algebraic Grid Generation......Page 401
12.3 Elliptic Grid Generation......Page 405
12.4 Variational Grid Adaptation......Page 409
Bibliography......Page 423
Index......Page 429
