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File: Calculus Pdf 169307 | Cvtypsetnew
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 ...

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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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...Calculusofvariations and tensorcalculus u h gerlach september beta edition contents fundamentalideas multivariable calculus as a prelude to the of variations some typical problems in methods for solving method finite dierences variants euler lagrange equation variational derivative s dierential solved example integration generalizations functional with several unknown functions extremum problem side conditions heuristic solution via constraint manifold constraints variable end point principle at moment time symmetry generic endpoint general transversality junction many degrees freedom parametrization invariant invariance homogeneous function geodesic motion geodesics their terms curve length physical signicance free oat frame rotating uniformly accelerated equivalence gravitation geometry formulation mechanics hamilton prologue why r k e p dt minimum its conceptual economy physics mathematics jacobi theory dynamical phase momentum hamiltonian single degree description constructive inte...

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