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