Cassel K.W. Variational Methods with Applications in Science and Engineering

Cambridge: Cambridge University Press, 2013. - 434p.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.Variational MethodsPreliminaries
A Bit of HistoryMotivation
Extrema of Functions
Constrained Extrema and Lagrange Multipliers
Integration by Parts
Fundamental Lemma of the Calculus of
Variations
Adjoint and Self-Adjoint Differential Operators
ExercisesCalculus of Variations
Functionals of One Independent Variable
Natural Boundary Conditions
Variable End Points
Functionals of Two Independent Variables
Functionals of Two Dependent Variables
Constrained Functionals
Summary of Euler Equations
ExercisesRayleigh-Ritz, Galerkin, and Finite-Element Methods
Rayleigh-Ritz Method
Galerkin Method
Finite-Element Methods
ExercisesPhysical ApplicationsHamilton’s Principle
Hamilton’s Principle for Discrete Systems
Hamilton’s Principle for Continuous Systems
Euler-Lagrange Equations
Invariance of the Euler-Lagrange Equations
Derivation of Hamilton’s Principle from the First Law of
Thermodynamics
Conservation of Mechanical Energy and the Hamiltonian
Noether’s Theorem – Connection Between
Conservation Laws and Symmetries in Hamilton’s
PrincipleBrief Remarks on the Philosophy of Science
ExercisesClassical Mechanics
Dynamics of Nondeformable Bodies
Statics of Nondeformable Bodies
Statics of Deformable Bodies
Dynamics of Deformable BodiesSimple Pendulum
Linear, Second-Order, Autonomous Systems
Nonautonomous Systems – Forced Pendulum
Non-Normal Systems – Transient Growth
Continuous Systems – Beam-Column BucklingOptics
Maxwell’s Equations of Electromagnetics
Electromagnetic Wave Equations
Discrete Charged Particles in an Electromagnetic Field
Continuous Charges in an Electromagnetic FieldModern Physics
Relativistic Mechanics
Quantum MechanicsFluid MechanicsInviscid Flow
Viscous Flow – Navier-Stokes Equations
Multiphase and Multicomponent Flows
Hydrodynamic Stability Analysis
Flow ControlOptimizationOptimization and Control
Optimization and Control Examples
Shape Optimization
Financial Optimization
Optimal Control of Discrete Systems
Optimal Control of Continuous Systems
Control of Real Systems
Postscript
ExercisesImage Processing and Data Analysis
Variational Image Processing
Curve and Surface Optimization Using Splines
Proper-Orthogonal Decomposition
Fundamentals
Algebraic Grid Generation
Elliptic Grid Generation
Variational Grid Adaptation