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CALCULUSOFVARIATIONS and TENSORCALCULUS U. H. Gerlach September 22, 2019 Beta Edition 2 Contents 1 FUNDAMENTALIDEAS 5 1.1 Multivariable Calculus as a Prelude to the Calculus of Variations. . . 5 1.2 Some Typical Problems in the Calculus of Variations. . . . . . . . . . 6 1.3 Methods for Solving Problems in Calculus of Variations. . . . . . . . 10 1.3.1 Method of Finite Differences. . . . . . . . . . . . . . . . . . . 10 1.4 The Method of Variations. . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.1 Variants and Variations . . . . . . . . . . . . . . . . . . . . . 14 1.4.2 The Euler-Lagrange Equation . . . . . . . . . . . . . . . . . . 17 1.4.3 Variational Derivative . . . . . . . . . . . . . . . . . . . . . . 20 1.4.4 Euler’s Differential Equation . . . . . . . . . . . . . . . . . . . 21 1.5 Solved Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.6 Integration of Euler’s Differential Equation. . . . . . . . . . . . . . . 25 2 GENERALIZATIONS 33 2.1 Functional with Several Unknown Functions . . . . . . . . . . . . . . 33 2.2 Extremum Problem with Side Conditions. . . . . . . . . . . . . . . . 38 2.2.1 Heuristic Solution . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2.2 Solution via Constraint Manifold . . . . . . . . . . . . . . . . 42 2.2.3 Variational Problems with Finite Constraints . . . . . . . . . 54 2.3 Variable End Point Problem . . . . . . . . . . . . . . . . . . . . . . . 55 2.3.1 Extremum Principle at a Moment of Time Symmetry . . . . . 57 2.4 Generic Variable Endpoint Problem . . . . . . . . . . . . . . . . . . . 60 2.4.1 General Variations in the Functional . . . . . . . . . . . . . . 62 2.4.2 Transversality Conditions . . . . . . . . . . . . . . . . . . . . 64 2.4.3 Junction Conditions . . . . . . . . . . . . . . . . . . . . . . . 66 2.5 Many Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . 68 2.6 Parametrization Invariant Problem . . . . . . . . . . . . . . . . . . . 70 2.6.1 Parametrization Invariance via Homogeneous Function . . . . 71 2.7 Variational Principle for a Geodesic . . . . . . . . . . . . . . . . . . . 72 2.8 Equation of Geodesic Motion . . . . . . . . . . . . . . . . . . . . . . 76 2.9 Geodesics: Their Parametrization. . . . . . . . . . . . . . . . . . . . . 77 3 4 CONTENTS 2.9.1 Parametrization Invariance. . . . . . . . . . . . . . . . . . . . 77 2.9.2 Parametrization in Terms of Curve Length . . . . . . . . . . . 78 2.10 Physical Significance of the Equation for a Geodesic . . . . . . . . . . 80 2.10.1 Free float frame . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.10.2 Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.10.3 Uniformly Accelerated Frame . . . . . . . . . . . . . . . . . . 84 2.11 The Equivalence Principle and “Gravitation”=“Geometry” . . . . . . . 85 3 Variational Formulation of Mechanics 89 3.1 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.1.1 Prologue: Why R(K:E:−P:E:)dt = minimum? . . . . . 90 3.1.2 Hamilton’s Principle: Its Conceptual Economy in Physics and Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.2 Hamilton-Jacobi Theory . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.3 The Dynamical Phase . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.4 Momentum and the Hamiltonian . . . . . . . . . . . . . . . . . . . . 103 3.5 The Hamilton-Jacobi Equation . . . . . . . . . . . . . . . . . . . . . 109 3.5.1 Single Degree of Freedom . . . . . . . . . . . . . . . . . . . . 109 3.5.2 Several Degrees of Freedom . . . . . . . . . . . . . . . . . . . 113 3.6 Hamilton-Jacobi Description of Motion . . . . . . . . . . . . . . . . . 114 3.7 Constructive Interference . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.8 Spacetime History of a Wave Packet . . . . . . . . . . . . . . . . . . . 119 3.9 Hamilton’s Equations of Motion . . . . . . . . . . . . . . . . . . . . . 123 3.10 The Phase Space of a Hamiltonian System . . . . . . . . . . . . . . . 125 3.11 Consturctive interference ⇒ Hamilton’s Equations . . . . . . . . . . . 127 3.12 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.12.1 H-J Equation Relative to Curvilinear Coordinates . . . . . . . 129 3.12.2 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . 130 3.13 Hamilton’s Principle for the Mechanics of a Continuum . . . . . . . . 142 3.13.1 Variational Principle . . . . . . . . . . . . . . . . . . . . . . . 144 3.13.2 Euler-Lagrange Equation . . . . . . . . . . . . . . . . . . . . . 145 3.13.3 Examples and Applications . . . . . . . . . . . . . . . . . . . 147 4 DIRECT METHODSINTHECALCULUSOFVARIATIONS 151 4.1 Minimizing Sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4.2 Implementation via Finite-Dimensional Approximation . . . . . . . . 153 4.3 Raleigh’s Variational Principle . . . . . . . . . . . . . . . . . . . . . . 154 4.3.1 The Raleigh Quotient . . . . . . . . . . . . . . . . . . . . . . . 155 4.3.2 Raleigh-Ritz Principle . . . . . . . . . . . . . . . . . . . . . . 159 4.3.3 Vibration of a Circular Membrane . . . . . . . . . . . . . . . . 160
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