There is a resurgence of applications in which the calculus of variations has direct relevance. In addition to application to solid mechanics and dynamics, it is now being applied in a variety of numerical methods, numerical grid generation, modern physics, various optimization settings and fluid dynamics. Many applications, such as nonlinear optimal control theory applied to continuous systems, have only recently become tractable computationally, with the advent of advanced algorithms and large computer systems. This book reflects the strong connection between calculus of variations and the applications for which variational methods form the fundamental foundation. The mathematical fundamentals of calculus of variations (at least those necessary to pursue applications) is rather compact and is contained in a single chapter of the book. The majority of the text consists of applications of variational calculus for a variety of fields.

In a review of the book Mathematics for Physics: A Guided Tour for Graduate Students by Michael Stone and Paul Goldbart (2009), David Khmelnitskii perceptively writes:
Without textbooks, the education of scientists is unthinkable. Textbook authors rearrange, repackage, and present established facts and discoveries – along the way straightening logic, excluding unnecessary details, and, finally, shrinking the volume of preparatory reading for the next generation. Writing them is therefore one of the most important collective tasks of the academic community, and an often underrated one at that. Textbooks are not easy to create, but once they are, the good ones become cornerstones, often advancing and redefining common knowledge.1
There is perhaps no other branch of applied mathematics that is more in need of such a "repackaging" than the calculus of variations.
A glance through the reference list at the end of the book will reveal that there are a number of now-classic texts on the calculus of variations from up through the mid-1960s, such as Weinstock (1952) and Gelfand and Fomin (1963), with a sharp drop subsequent to that period. The classic texts that emphasize applications, such as Morse and Feshback (1953) and Courant and Hilbert (1953), typically focus the majority of their discussion of variational methods on classical mechanics, for example, statics, dynamics, elasticity, and vibrations. Since that time, it has been more common to simply include the necessary elements of variational calculus in books dedicated to specific topics, such as analytical dynamics, dynamical systems, mechanical vibrations, elasticity, finite-element methods, and optimal control theory. More recently, the trend has been to avoid treating these subjects from a variational point of view altogether. Until very recently, therefore, modern stand-alone texts on the calculus of variations have been nearly nonexistent, and a small subset of modern texts in the above-mentioned subject areas include a limited treatment of variational calculus.
Not having had a course on the subject myself, I learned calculus of variations along with my students in a first-year graduate engineering analysis course that I first taught at the Illinois Institute of Technology (IIT) in 2001. The course covers matrices, eigenfunction theory, complex variable theory, and calculus
1 Physics Today, October 2009.
of variations. It required several semesters of teaching the course for me to transition from the – unstated – attitude of "we all need to suffer through the calculus of variations for the sake of those students in solid mechanics, dynamics, and controls" to the – stated – attitude that "calculus of variations is one of the most widely applicable branches of mathematics for scientists and engineers!"
There has been a remarkable resurgence of applications in the last half century in which the calculus of variations has direct relevance. In addition to its traditional application to mechanics of solids and dynamics, it is now being applied in a variety of numerical methods, numerical grid generation, modern physics, various optimization settings, and fluid dynamics. Many of these applications, such as nonlinear optimal control theory applied to continuous systems, have only recently become tractable computationally with the advent of advanced algorithms and large computer systems. In my area of fluid mechanics, which has not traditionally been considered an area ripe for utilization of calculus of variations, there is increasing interest in applying optimal control theory to fluid mechanics and applying variational calculus to numerical methods and grid generation in computational fluid dynamics. With the growing interest in flow control, whereby small actuators are used to excite mechanisms within a flow to produce large changes in the overall flow structure, optimal control theory is making significant headway in providing a formal framework in which to consider various flow control techniques. In addition, numerical grid generation increasingly is being based on formal optimization principles and variational calculus rather than ad hoc techniques, such as elliptic grid generation. Variational calculus also impacts fluid mechanics through its applications to shape optimization and recent advances in hydrodynamic stability theory via transient-growth analysis.
Unfortunately, the majority of modern texts on applied mathematics do not reflect this revival of interest in variational methods. This is demonstrated by the fact that the genre of modern advanced engineering mathematics textbooks in wide use today typically do not include calculus of variations. See, for example, Kreyszig (2011), which is now in its 10th edition; O'Neil (2012); Zill and Wright (2014); Jeffrey (2002); and Greenberg (1998). Regrettably, calculus of variations has become too closely linked with certain application areas, such as mechanics of solids, analytical dynamics, and control theory. Such connections are reflected in many of the classic books on these subjects. This gives the impression that it is not a subject that has wide applicability in other areas of science and engineering. The paucity of current stand-alone textbooks on the calculus of variations reinforces the impression that the subject has matured and that there is little modern interest in the field.
Owing to the renaissance of applications in which variational methods have direct relevance and development of the computational resources required to treat them, there is need for a modern treatment of the subject from a general point of view that highlights the breadth of applications demonstrating the widespread applicability of variational methods. The present text is my humble attempt at providing such a treatment of variational methods. The objectives are twofold and include assisting the reader in learning and applying variational methods:
1. Learn variational methods: Part I provides a concise treatment of the fundamentals of the calculus of variations that are accessible to the typical student having a background in differential calculus and ordinary differential equations, but none in variational calculus.
2. Apply variational methods: Parts II and III provide a bridge between the fundamental material in Part I that is common to many areas of application and the area-specific applications. They provide an introduction to applications of calculus of variations in various fields and a link to dedicated texts and research literature in these subjects.
The content of the text reflects the strong connection between calculus of variations and the applications for which variational methods form the fundamental foundation. Most readers will be pleased to note that the mathematical fundamentals of calculus of variations (at least those necessary to pursue applications) are rather compact and are contained in a single chapter of the book. Therefore, the majority of the text consists of applications of variational calculus in various fields. The priority of these 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 emphasis is on illustrating the wide range of applications of the calculus of variations, and the reader is referred to dedicated texts for more complete treatments of each topic. The centerpiece of these disparate subjects is Hamilton's principle, which provides a compact form of the dynamical equations of motion (its traditional area of application) and the governing equations for many other physical phenomena as illustrated throughout the text.
Given the emphasis on breadth of applications addressed using variational methods, it is necessary to sacrifice depth of treatment of each topic. This is the case not merely for space considerations but, more importantly, for clarity and unification of presentation. The objective in the application chapters is to provide a clear and concise introduction to the variational underpinnings of each field as the basis for further study and investigation. For example, optimization and control are being applied so broadly, including financial optimization, shape optimization, control of dynamical systems, grid generation, image processing, and so on, that it is instructive to view these topics within a common optimal control theory framework. It is also felt that there is value in readers with a background in a given field to be exposed to other related and complementary topics in order to see potential analogies in approach or opportunities for application of one topic in another.
A unique feature of the present text is the 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 of discrete and continuous systems are unified with the physics of fluids, electromagnetic fields, relativistic mechanics, quantum mechanics, and so on through Hamilton's principle. I hope to impart to the reader some of the joy of discovering the interrelationship between these seemingly disparate physical principles that submit to the variational framework in general and the first law of thermodynamics through Hamilton's principle specifically.
In writing and arranging the text, I have made the following pedagogical assumptions:
1. Most readers are handicapped in learning a difficult topic unless they are persuaded of its relevance to their academic and/or professional development. That is, engineers and scientists must be convinced of the need for a subject or topic before learning it. It is insufficient to say (or imply), "Trust me; you need to know this." Although this should be a primary concern of instructors on the front lines of pedagogy, there is much that can be done in selecting and arranging the content of a textbook to assist both the instructor and the reader. For example, to motivate the engineering, scientific, and/or mathematical need for a subject or topic, one could include both broad and specific applications early in the development of each topic that students are, or will become, familiar with and/or provide historical motivations for development of a topic.
2. Engineers and scientists learn details better when they have an overall intuition or understanding on which to hang the details. This is why we typically feel like we have learned more about a topic when encountering it a second (or third, ...) time as compared to the first. Therefore, one must seek to provide and develop a physical and/or mathematical intuition that will inform theoretical developments, derivations, or proofs.
3. It is better to err on the side of too much detail in derivations and worked examples than too little. It is easier for the reader to skip a familiar step than to fill in an unfamiliar one. This is also done in order to clearly illustrate the concepts and methods developed in the text and to promote good problem-solving techniques with a minimum of "shortcuts."
This treatment is aimed at the advanced undergraduate or graduate engineering, physical sciences, or applied mathematics student. In addition, it could serve as a reference for researchers and practitioners in fields that are based on, or make use of, variational methods. The text assumes that readers are proficient with differential calculus, including vector calculus, and ordinary differential equations. Familiarity with basic matrix operations and partial differential equations is helpful, but not essential. Moreover, it is not necessary to have any background in the application areas treated in Chapters 4–12; however, such a background would certainly provide a useful perspective.
The book provides sufficient material to form the basis for a one-semester course on calculus of variations or a portion of an applied mathematics, mathematical physics, or engineering analysis course that includes such topics. The latter is the case at IIT, where this material is part of the engineering analysis course taken by many first-year graduate students in mechanical, aerospace, materials, chemical, civil, structural, electrical, and biomedical engineering. In addition, it could serve as a supplement or reference for courses in analytical dynamics, elasticity, mechanical vibrations, modern physics, fluid mechanics, optimal control theory, image processing, and so on that are taught from a variational point of view. Exercises are provided at the end of selected "fundamentals," or nonapplication, chapters, including each chapter in Part I and the first chapter in each of Parts II and III. Solutions are available to instructors at the book's website.
Because of the intended audience of the book, there is little emphasis on mathematical proofs. Instead, the material is presented in a manner that promotes development of an intuition about the concepts and methods with an emphasis on applications. Although primarily couched in terms of optimization theory, Luenberger (1969) does an excellent job of putting calculus of variations in the context of the underlying branches of mathematics known as linear vector spaces and functional analysis and is replete with useful geometric interpretations. Therefore, it provides an excellent complement to the present text for those with a more mathematical orientation than the present text typifies.
I would like to acknowledge certain individuals who have contributed to this book project. Professors Sudhakar Nair and Xiaoping Qian of IIT provided helpful comments and insights throughout development of the book. In particular, Professor Qian directed me to several useful references, and Professor Nair laid the foundation for the course from which this book had its start. Moreover, Professor Nair is largely responsible for keeping the mechanics tradition alive at IIT. I would also like to thank the reviewers who provided a host of useful comments that helped shape the emphasis and content of the book. The numerous students over the years whose questions have challenged me to continually improve the core content of the book are owed a special thank you; you are the ones that I had pictured in my mind as I penned each word and equation. In particular, I would like to acknowledge Mr. Jiacheng Wu for his insightful comments, probing questions, and thorough reading of the text. Finally, Mr. Peter Gordon of Cambridge University Press is to be thanked for his encouragement, advice, and editorial assistance throughout this book project. His "old-school" approach is refreshing in an age of increasingly detached publishers.
I would welcome any comments on the text from instructors and readers. I can be reached at cassel@iit.edu.

Preface page xiii

PART I VARIATIONAL METHODS 1

1 Preliminaries 3

1.1 A Bit of History 4

1.2 Introduction 7

1.3 Motivation 8

      1.3.1 Optics 8

      1.3.2 Shape of a Liquid Drop 10

      1.3.3 Optimization of a River-Crossing Trajectory 12

      1.3.4 Summary 14

1.4 Extrema of Functions 14

1.5 Constrained Extrema and Lagrange Multipliers 17

1.6 Integration by Parts 20

1.7 Fundamental Lemma of the Calculus of Variations 21

1.8 Adjoint and Self-Adjoint Differential Operators 22

Exercises 26

2 Calculus of Variations 28

2.1 Functionals of One Independent Variable 29

      2.1.1 Functional Derivative 30

      2.1.2 Derivation of Euler's Equation 31

      2.1.3 Variational Notation 33

      2.1.4 Special Cases of Euler's Equation 37

2.2 Natural Boundary Conditions 44

2.3 Variable End Points 53

2.4 Higher-Order Derivatives 56

2.5 Functionals of Two Independent Variables 56

      2.5.1 Euler's Equation 57

      2.5.2 Minimal Surfaces 61

      2.5.3 Dirichlet Problem 62

2.6 Functionals of Two Dependent Variables 64

2.7 Constrained Functionals 66

      2.7.1 Integral Constraints 66

      2.7.2 Sturm-Liouville Problems 74

      2.7.3 Algebraic and Differential Constraints 76

2.8 Summary of Euler Equations 80

Exercises 81

3 Rayleigh-Ritz, Galerkin, and Finite-Element Methods 90

3.1 Rayleigh-Ritz Method 91

      3.1.1 Basic Procedure 91

      3.1.2 Self-Adjoint Differential Operators 94

      3.1.3 Estimating Eigenvalues of Differential Operators 96

3.2 Galerkin Method 100

3.3 Finite-Element Methods 103

      3.3.1 Rayleigh-Ritz-Based Finite-Element Method 104

      3.3.2 Finite-Element Methods in Multidimensions 109

Exercises 110

PART II PHYSICAL APPLICATIONS 115

4 Hamilton's Principle 117

4.1 Hamilton's Principle for Discrete Systems 118

4.2 Hamilton's Principle for Continuous Systems 128

4.3 Euler-Lagrange Equations 131

4.4 Invariance of the Euler-Lagrange Equations 136

4.5 Derivation of Hamilton's Principle from the First Law of Thermodynamics 137

4.6 Conservation of Mechanical Energy and the Hamiltonian 141

4.7 Noether's Theorem - Connection Between Conservation Laws and Symmetries in Hamilton's Principle 143

4.8 Summary 146

4.9 Brief Remarks on the Philosophy of Science 148

Exercises 152

5 Classical Mechanics 160

5.1 Dynamics of Nondeformable Bodies 161

      5.1.1 Applications of Hamilton's Principle 161

      5.1.2 Dynamics Problems with Constraints 174

5.2 Statics of Nondeformable Bodies 178

      5.2.1 Vectorial Approach 179

      5.2.2 Virtual Work Approach 181

5.3 Statics of Deformable Bodies 184

      5.3.1 Static Deflection of an Elastic Membrane - Poisson Equation 184

      5.3.2 Static Deflection of a Beam 185

      5.3.3 Governing Equations of Elasticity 189

      5.3.4 Principle of Virtual Work 196

5.4 Dynamics of Deformable Bodies 197

      5.4.1 Longitudinal Vibration of a Rod - Wave Equation 197

      5.4.2 Lateral Vibration of a String - Wave Equation 199

6 Stability of Dynamical Systems 202

6.1 Introduction 202

6.2 Simple Pendulum 203

6.3 Linear, Second-Order, Autonomous Systems 207

6.4 Nonautonomous Systems - Forced Pendulum 212

6.5 Non-Normal Systems - Transient Growth 215

6.6 Continuous Systems - Beam-Column Buckling 222

7 Optics and Electromagnetics 225

7.1 Optics 225

7.2 Maxwell's Equations of Electromagnetics 229

7.3 Electromagnetic Wave Equations 232

7.4 Discrete Charged Particles in an Electromagnetic Field 233

7.5 Continuous Charges in an Electromagnetic Field 237

8 Modern Physics 240

8.1 Relativistic Mechanics 241

      8.1.1 Special Relativity 241

      8.1.2 General Relativity 247

8.2 Quantum Mechanics 251

      8.2.1 Schrödinger's Equation 252

      8.2.2 Density-Functional Theory 256

      8.2.3 Feynman Path-Integral Formulation of Quantum Mechanics 257

9 Fluid Mechanics 259

9.1 Introduction 260

9.2 Inviscid Flow 262

      9.2.1 Fluid Particles as Nondeformable Bodies - Bernoulli Equation 262

      9.2.2 Fluid Particles as Deformable Bodies - Euler Equations 263

      9.2.3 Potential Flow 266

9.3 Viscous Flow - Navier-Stokes Equations 269

9.4 Multiphase and Multicomponent Flows 275

      9.4.1 Level-Set Methods 276

      9.4.2 Phase-Field Models 278

9.5 Hydrodynamic Stability Analysis 281

      9.5.1 Introduction 281

      9.5.2 Linear Stability of the Navier-Stokes Equations 282

      9.5.3 Modal Analysis 283

      9.5.4 Nonmodal Transient Growth (Optimal Perturbation) Analysis 289

      9.5.5 Energy Methods 296

9.6 Flow Control 297

PART III OPTIMIZATION 301

10 Optimization and Control 303

10.1 Optimization and Control Examples 305

10.2 Shape Optimization 306

10.3 Financial Optimization 310

10.4 Optimal Control of Discrete Systems 312

      10.4.1 Example: Control of an Undamped Harmonic Oscillator 312

      10.4.2 Riccati Equation for the LQ Problem 320

      10.4.3 Properties of Systems for Control 330

      10.4.4 Pontryagin's Principle 331

      10.4.5 Time-Optimal Control 337

10.5 Optimal Control of Continuous Systems 342

      10.5.1 Variational Approach 342

      10.5.2 Adjoint Approach 346

      10.5.3 Example: Control of Plane-Poiseuille Flow 347

10.6 Control of Real Systems 351

      10.6.1 Open-Loop and Closed-Loop Control 352

      10.6.2 Model Predictive Control 353

      10.6.3 State Estimation and Data Assimilation 354

      10.6.4 Robust Control 356

10.7 Postscript 356

Exercises 357

11 Image Processing and Data Analysis 361

11.1 Variational Image Processing 362

      11.1.1 Denoising 367

      11.1.2 Deblurring 368

      11.1.3 Inpainting 368

      11.1.4 Segmentation 370

11.2 Curve and Surface Optimization Using Splines 371

      11.2.1 B-Splines 372

      11.2.2 Spline Functionals 373

11.3 Proper-Orthogonal Decomposition 374

12 Numerical Grid Generation 379

12.1 Fundamentals 379

12.2 Algebraic Grid Generation 381

12.3 Elliptic Grid Generation 385

12.4 Variational Grid Adaptation 389

      12.4.1 One-Dimensional Case 390

      12.4.2 Two-Dimensional Case 398

Bibliography 403

Index 409

Author of the books, "Matrix, Numerical, and Optimization Methods in Science and Engineering" (Cambridge University Press 2021) and "Variational Methods with Applications in Science and Engineering" (Cambridge University Press 2013), Kevin W. Cassel is Professor of Mechanical and Aerospace Engineering and Professor of Applied Mathematics at the Illinois Institute of Technology (IIT). He is a Fellow of the American Society of Mechanical Engineers and an Associate Fellow of the American Institute of Aeronautics and Astronautics, and his honors include IIT's University Excellence in Teaching Award (2008) and Ralph L. Barnett Excellence in Teaching Award (2007, 2001), the 2002 Alfred Noble Prize, and the Army Research Office Young Investigator Award (1998-2001). Professor Cassel has been a visiting researcher at the University of Manchester and University College London, and is a visiting professor at the University of Palermo, Italy. His research utilizes computational fluid dynamics in conjunction with advanced analytical methods to address problems in bio-fluids, unsteady aerodynamics, multiphase flow, and cryogenic fluid flow and heat transfer.

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