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.Contents :PrefaceVariational MethodsPreliminaries A Bit of History Introduction Motivation 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 Principle Summary Brief Remarks on the Philosophy of Science ExercisesClassical Mechanics Dynamics of Nondeformable Bodies Statics of Nondeformable Bodies Statics of Deformable Bodies Dynamics of Deformable BodiesIntroduction Simple 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 Mechanics Introduction Inviscid 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 AdaptationBibliography Index
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