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Pacifico Printed in the Federal Republic of Germany Printed on acid-free paper ISBN: 978-3-527-40683-8 www.com V Contents Preface XV Introduction 1 Bibliography 9 1 Review of Classical Mechanics and String Field Theory 11 1.1 Preview and Rationale 11 1.2 Review of Lagrangians and Hamiltonians 13 1.1 Hamilton’s Equations in Multiple Dimensions 14 1.3 Derivation of the Lagrange Equation from Hamilton’s Principle 16 1.4 Linear, Multiparticle Systems 18 1.1 The Laplace Transform Method 23 1.2 Damped and Driven Simple Harmonic Motion 24 1.3 Conservation of Momentum and Energy 26 1.5 Effective Potential and the Kepler Problem 26 1.7 Longitudinal Oscillation of a Beaded String 32 1.2 The Continuum Limit 34 1.8 Field Theoretical Treatment and Lagrangian Density 36 1.9 Hamiltonian Density for Transverse String Motion 39 1.10 String Motion Expressed as Propagating and Reflecting Waves 40 1.11 Problems 42 Bibliography 44 2 Geometry of Mechanics, I, Linear 45 2.1 Pairs of Planes as Covariant Vectors 47 2.2 Calculus of Differential Forms 57 2.3 Familiar Physics Equations Expressed Using Differential Forms 61 www.com VI Contents 2.1 Vectors and Their Duals 66 2.2 Transformation of Coordinates 68 2.3 Transformation of Distributions 72 2.4 Multi-index Tensors and their Contraction 73 2.5 Representation of a Vector as a Differential Operator 76 2.4 (Possibly Complex) Cartesian Vectors in Metric Geometry 79 2.2 Skew Coordinate Frames 81 2.3 Reduction of a Quadratic Form to a Sum or Difference of Squares 81 2.4 Introduction of Covariant Components 83 2.5 The Reciprocal Basis 84 Bibliography 86 3 Geometry of Mechanics, II, Curvilinear 89 3.1 (Real) Curvilinear Coordinates in n-Dimensions 90 3.1 The Metric Tensor 90 3.2 Relating Coordinate Systems at Different Points in Space 92 3.3 The Covariant (or Absolute) Differential 97 3.2 Derivation of the Lagrange Equations from the Absolute Differential 102 3.1 Practical Evaluation of the Christoffel Symbols 108 3.3 Intrinsic Derivatives and the Bilinear Covariant 109 3.4 The Lie Derivative – Coordinate Approach 111 3.1 Lie-Dragged Coordinate Systems 111 3.2 Lie Derivatives of Scalars and Vectors 115 3.5 The Lie Derivative – Lie Algebraic Approach 120 3.1 Exponential Representation of Parameterized Curves 120 3.6 Identification of Vector Fields with Differential Operators 121 3.8 Lie-Dragged Congruences and the Lie Derivative 125 3.9 Commutators of Quasi-Basis-Vectors 130 Bibliography 132 4 Geometry of Mechanics, III, Multilinear 133 4.1 Generalized Euclidean Rotations and Reflections 133 4.2 Expressing a Rotation as a Product of Reflections 135 4.3 The Lie Group of Rotations 136 4.com Contents VII 4.1 Volume Determined by 3- and by n-Vectors 138 4.3 Multivectors and Generalization to Higher Dimensionality 141 4.4 Local Radius of Curvature of a Particle Orbit 143 4.6 Sums of p-Vectors 145 4.7 Bivectors and Infinitesimal Rotations 145 4.3 Curvilinear Coordinates in Euclidean Geometry (Continued) 148 4.1 Repeated Exterior Derivatives 148 4.2 The Gradient Formula of Vector Analysis 149 4.3 Vector Calculus Expressed by Differential Forms 151 4.4 Derivation of Vector Integral Formulas 154 4.5 Generalized Divergence and Gauss’s Theorem 157 4.6 Metric-Free Definition of the “Divergence” of a Vector 159 4.4 Spinors in Three-Dimensional Space 161 4.1 Definition of Spinors 162 4.2 Demonstration that a Spinor is a Euclidean Tensor 162 4.4 Associating a Matrix with a Trivector (Triple Product) 164 4.5 Representations of Reflections 164 4.6 Representations of Rotations 165 4.7 Operations on Spinors 166 4.8 Real Euclidean Space 167 4.9 Real Pseudo-Euclidean Space 167 Bibliography 167 5 Lagrange–Poincaré Description of Mechanics 169 5.1 The Poincaré Equation 169 5.1 Some Features of the Poincaré Equations 179 5.2 Invariance of the Poincaré Equation 180 5.3 Translation into the Language of Forms and Vector Fields 182 5.4 Example: Free Motion of a Rigid Body with One Point Fixed 183 5.2 Variational Derivation of the Poincaré Equation 186 5.3 Restricting the Poincaré Equation With Group Theory 189 5.1 Continuous Transformation Groups 189 5.2 Use of Infinitesimal Group Parameters as Quasicoordinates 193 5.3 Infinitesimal Group Operators 195 5.4 Commutation Relations and Structure Constants of the Group 199 5.5 Qualitative Aspects of Infinitesimal Generators 201 5.6 The Poincaré Equation in Terms of Group Generators 204 5.7 The Rigid Body Subject to Force and Torque 206 Bibliography 217 www.com VIII Contents 6 Newtonian/Gauge Invariant Mechanics 219 6.1 Vector Description in Curvilinear Coordinates 219 6.2 The Frenet–Serret Formulas 222 6.3 Vector Description in an Accelerating Coordinate Frame 226 6.4 Exploiting the Fictitious Force Description 232 6.2 Single Particle Equations in Gauge Invariant Form 238 6.1 Newton’s Force Equation in Gauge Invariant Form 239 6.2 Active Interpretation of the Transformations 242 6.3 Newton’s Torque Equation 246 6.4 The Plumb Bob 248 6.3 Gauge Invariant Description of Rigid Body Motion 252 6.1 Space and Body Frames of Reference 253 6.2 Review of the Association of 2 × 2 Matrices to Vectors 256 6.3 “Association” of 3 × 3 Matrices to Vectors 258 6.4 Derivation of the Rigid Body Equations 259 6.5 The Euler Equations for a Rigid Body 261 6.4 The Foucault Pendulum 262 6.1 Fictitious Force Solution 263 6.2 Gauge Invariant Solution 265 6.3 “Parallel” Translation of Coordinate Axes 270 6.5 Tumblers and Divers 274 Bibliography 276 7 Hamiltonian Treatment of Geometric Optics 277 7.1 Analogy Between Mechanics and Geometric Optics 278 7.1 Scalar Wave Equation 279 7.2 The Eikonal Equation 281 7.3 Determination of Rays from Wavefronts 282 7.4 The Ray Equation in Geometric Optics 283 7.1 The Lagrange Integral Invariant and Snell’s Law 285 7.2 The Principle of Least Time 287 7.3 Paraxial Optics, Gaussian Optics, Matrix Optics 288 7.4 Huygens’ Principle 292 Bibliography 294 8 Hamilton–Jacobi Theory 295 8.1 Hamilton–Jacobi Theory Derived from Hamilton’s Principle 295 8.1 The Geometric Picture 297 8.2 Trajectory Determination Using the Hamilton–Jacobi Equation 299 www.com Contents IX 8.2 Finding a Complete Integral by Separation of Variables 300 8.3 Hamilton–Jacobi Analysis of Projectile Motion 301 8.4 The Jacobi Method for Exploiting a Complete Integral 302 8.5 Completion of Projectile Example 304 8.6 The Time-Independent Hamilton–Jacobi Equation 305 8.7 Hamilton–Jacobi Treatment of 1D Simple Harmonic Motion 306 8.3 The Kepler Problem 307 8.3 Hamilton–Jacobi Formulation.4 Analogies Between Optics and Quantum Mechanics 314 8.1 Classical Limit of the Schrödinger Equation 314 Bibliography 316 9 Relativistic Mechanics 317 9.2 World Points and Intervals 318 9.4 The Lorentz Transformation 321 9.5 Transformation of Velocities 322 9.6 4-Vectors and Tensors 322 9.7 Three-Index Antisymmetric Tensor 325 9.9 The 4-Gradient, 4-Velocity, and 4-Acceleration 326 9.1 The Relativistic Principle of Least Action 327 9.2 Energy and Momentum 328 9.5 Hamilton–Jacobi Formulation 330 9.3 Introduction of Electromagnetic Forces into Relativistic Mechanics 332 9.1 Generalization of the Action 332 9.2 Derivation of the Lorentz Force Law 334 9.3 Gauge Invariance 335 Bibliography 338 10 Conservation Laws and Symmetry 339 10.1 Conservation of Linear Momentum 339 10.2 Rate of Change of Angular Momentum: Poincaré Approach 341 www.3 Conservation of Angular Momentum: Lagrangian Approach 342 10.4 Conservation of Energy 343 10.5 Cyclic Coordinates and Routhian Reduction 344 10.1 Integrability; Generalization of Cyclic Variables 347 10.7 Conservation Laws in Field Theory 352 10.1 Ignorable Coordinates and the Energy Momentum Tensor 352 10.8 Transition From Discrete to Continuous Representation 356 10.1 The 4-Current Density and Charge Conservation 356 10.2 Energy and Momentum Densities 360 10.9 Angular Momentum of a System of Particles 362 10.10 Angular Momentum of a Field 363 Bibliography 364 11 Electromagnetic Theory 365 11.1 The Electromagnetic Field Tensor 367 11.1 The Lorentz Force Equation in Tensor Notation 367 11.2 Lorentz Transformation and Invariants of the Fields 369 11.2 The Electromagnetic Field Equations 370 11.1 The Homogeneous Pair of Maxwell Equations 370 11.2 The Action for the Field, Particle System 370 11.3 The Electromagnetic Wave Equation 372 11.4 The Inhomogeneous Pair of Maxwell Equations 373 11.5 Energy Density, Energy Flux, and the Maxwell Stress Energy Tensor 374 Bibliography 377 12 Relativistic Strings 379 12.1 Is String Theory Appropriate? 379 12.3 Postulating a String Lagrangian 381 12.2 Area Representation in Terms of the Metric 383 12.3 The Lagrangian Density and Action for Strings 384 12.2 Parameterization of String World Surface by σ and τ 385 12.3 The Nambu–Goto Action 385 12.4 String Tension and Mass Density 387 12.4 Equations of Motion, Boundary Conditions, and Unexcited Strings 389 12.5 The Action in Terms of Transverse Velocity 391 12.6 Orthogonal Parameterization by Energy Content 394 www.com Contents XI 12.7 General Motion of a Free Open String 396 12.8 A Rotating Straight String 398 12.9 Conserved Momenta of a String 400 12.1 Angular Momentum of Uniformly Rotating Straight String 401 12.10 Light Cone Coordinates 402 12.11 Oscillation Modes of a Relativistic String 406 Bibliography 408 13 General Relativity 409 13.2 Transformation to Locally Inertial Coordinates 412 13.3 Parallel Transport on a Surface 413 13.4 The Twin Paradox in General Relativity 417 13.5 The Curvature Tensor 422 13.1 Properties of Curvature Tensor, Ricci Tensor, and Scalar Curvature 423 13.6 The Lagrangian of General Relativity and the Energy–Momentum Tensor 425 13.7 “Derivation” of the Einstein Equation 428 13.8 Weak, Nonrelativistic Gravity 430 13.9 The Schwarzschild Metric 433 13.1 Orbit of a Particle Subject to the Schwarzschild Metric 434 13.10 Gravitational Lensing and Red Shifts 437 Bibliography 440 14 Analytic Bases for Approximation 441 14.1 The Action as a Generator of Canonical Transformations 441 14.2 Time-Independent Canonical Transformation 446 14.3 Action-Angle Variables 448 14.