Variational Methods with Applications in Science and Engineering

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 vii viii Contents 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 P A R T I I P H Y S I C A L A P P L I C A T I O N S 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 Hamil