Variational Methods in Theoretical Mechanics

1. Introduction.- 1.1 The Role of Variational Theory in Mechanics.- 1.2 Some Historical Comments.- 1.3 Plan of Study.- 2. Mathematical Foundations of Classical Variational Theory.- 2.1 Introduction.- 2.2 Nonlinear Operators.- 2.3 Differentiation of Operators.- 2.4 Mean Value Theorems.- 2.5 Taylor Formulas.- 2.6 Gradients of Functionals.- 2.7 Minimization of Functionals.- 2.8 Convex Functionals.- 2.9 Potential Operators and the Inverse Problem.- 2.10 Sobolev Spaces.- 3. Mechanics of Continua- A Brief Review.- 3.1 Introduction.- 3.2 Kinematics.- 3.3 Stress and the Mechanical Laws of Balance.- The Principle of Conservation of Mass.- The Principle of Balance of Linear Momentum.- The Principle of Balance of Angular Momentum.- 3.4 Thermodynamic Principles.- The Principle of Conservation of Energy.- The Clausius-Duhem Inequality.- 3.5 Constitutive Theory.- Rules of Constitutive Theory.- Special Forms of Constitutive Equations.- 3.6 Jump Conditions for Discontinuous Fields.- 4. Complementary and Dual Variational Principles in Mechanics.- 4.1 Introduction.- 4.2 Boundary Conditions and Green's Formulas.- 4.3 Examples from Mechanics and Physics.- 4.4 The Fourteen Fundamental Complementary-Dual Principles.- 4.5 Some Complementary-Dual Variational Principles of Mechanics and Physics.- 4.6 Legendre Transformations.- 4.7 Generalized Hamiltonian Theory.- 4.8 Upper and Lower Bounds and Existence Theory.- 4.9 Lagrange Multipliers.- 5. Variational Principles in Continuum Mechanics.- 5.1 Introduction.- 5.2 Some Preliminary Properties and Lemmas.- 5.3 General Variational Principles for Linear Theory of Dynamic Viscoelasticity.- 5.4 Gurtin's Variational Principles for the Linear Theory of Dynamic Viscoelasticity.- 5.5 Variational Principles for Linear Coupled DynamicThermoviscoelasticity.- Linear (Coupled) Thermoelasticity.- 5.6 Variational Principles in Linear Elastodynamics.- 5.7 Variational Principles for Linear Piezoelectric Elastodynamic Problems.- 5.8 Variational Principles for