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Mathematics for Physics I Mathematics for Physics I A set of lecture notes by Michael Stone PIMANDER-CASAUBON Alexandria Florence London • • ii Copyright c 2001,2002 M. Stone. All rights reserved. No part of this material can be reproduced, stored or transmitted without the written permission of the author. For information contact: Michael Stone, Loomis Laboratory of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801, USA. Preface These notes were prepared for the first semester of a year-long mathematical methods course for begining graduate students in physics. The emphasis is on linear operators and stresses the analogy between such operators acting on function spaces and matrices acting on finite dimensional spaces. The op- erator language then provides a unified framework for investigating ordinary and partial differential equations, and integral equations. The mathematical prerequisites for the course are a sound grasp of un- dergraduate calculus (including the vector calculus needed for electricity and magnetism courses), linear algebra (the more the better), and competence at complex arithmetic. Fourier sums and integrals, as well as basic ordinary differential equation theory receive a quick review, but it would help if the reader had some prior experience to build on. Contour integration is not required. iii iv PREFACE Contents Preface iii 1 Calculus of Variations 1 1.1 What is it good for? . 1 1.2 Functionals . 2 1.2.1 The functional derivative . 2 1.2.2 The Euler-Lagrange equation . 3 1.2.3 Some applications . 4 1.2.4 First integral . 8 1.3 Lagrangian Mechanics . 9 1.3.1 One degree of freedom . 10 1.3.2 Noether's theorem . 12 1.3.3 Many degrees of freedom . 15 1.3.4 Continuous systems . 16 1.4 Variable End Points . 26 1.5 Lagrange Multipliers . 31 1.6 Maximum or Minimum? . 35 1.7 Exercises and Problems . 37 2 Function Spaces 53 2.1 Motivation . 53 2.1.1 Functions as vectors . 54 2.2 Norms and Inner Products . 55 2.2.1 Norms and convergence . 55 2.2.2 Norms from integrals . 57 2.2.3 Hilbert space . 59 2.2.4 Orthogonal polynomials . 67 2.3 Linear Operators and Distributions . 72 v vi CONTENTS 2.3.1 Linear operators . 72 2.3.2 Distributions and test-functions . 75 2.4 Exercises and Problems . 82 3 Linear Ordinary Differential Equations 93 3.1 Existence and Uniqueness of Solutions . 93 3.1.1 Flows for first-order equations . 93 3.1.2 Linear independence . 95 3.1.3 The Wronskian . 96 3.2 Normal Form . 100 3.3 Inhomogeneous Equations . 101 3.3.1 Particular integral and complementary function . 101 3.3.2 Variation of parameters . 102 3.4 Singular Points . 104 3.4.1 Regular singular points . 104 3.5 Exercises and Problems . 105 4 Linear Differential Operators 109 4.1 Formal vs. Concrete Operators . 109 4.1.1 The algebra of formal operators . 109 4.1.2 Concrete operators . 111 4.2 The Adjoint Operator . 112 4.2.1 The formal adjoint . 112 4.2.2 A simple eigenvalue problem . 117 4.2.3 Adjoint boundary conditions . 119 4.2.4 Self-adjoint boundary conditions . 120 4.3 Completeness of Eigenfunctions . 126 4.3.1 Discrete spectrum . 127 4.3.2 Continuous spectrum . 133 4.4 Exercises and Problems . 143 5 Green Functions 149 5.1 Inhomogeneous Linear equations . 149 5.1.1 Fredholm alternative . 149 5.2 Constructing Green Functions . 150 5.2.1 Sturm-Liouville equation . 151 5.2.2 Initial-value problems . 153 5.2.3 Modified Green function . 158 CONTENTS vii 5.3 Applications of Lagrange's Identity . 159 5.3.1 Hermiticity of Green function . 159 5.3.2 Inhomogeneous boundary conditions . 160 5.4 Eigenfunction Expansions . 162 5.5 Analytic Properties of Green Functions . 164 5.5.1 Causality implies analyticity . 164 5.5.2 Plemelj formulæ . 168 5.5.3 Resolvent operator . 170 5.6 Locality and the Gelfand-Dikii equation . 174 5.7 Further Exercises and problems . 176 6 Partial Differential Equations 187 6.1 Classification of PDE's . 187 6.2 Cauchy Data . 189 6.2.1 Characteristics and first-order equations . 191 6.2.2 Second-order hyperbolic equations . 192 6.3 Wave Equation . 194 6.3.1 d'Alembert's Solution . 194 6.3.2 Fourier's Solution . 199 6.3.3 Causal Green Function . 200 6.3.4 Odd vs. Even Dimensions . 204 6.4 Heat Equation . 210 6.4.1 Heat Kernel . 210 6.4.2 Causal Green Function . 212 6.4.3 Duhamel's Principle . 213 6.5 Laplace's Equation . 215 6.5.1 Separation of Variables . 218 6.5.2 Eigenfunction Expansions . 227 6.5.3 Green Functions . 229 6.5.4 Boundary-value problems . 231 6.5.5 Kirchhoff vs. Huygens . 235 6.6 Exercises and problems . 239 7 The Mathematics of Real Waves 247 7.1 Dispersive waves . 247 7.1.1 Ocean Waves . 247 7.1.2 Group Velocity . 251 7.1.3 Wakes . 254 viii CONTENTS 7.1.4 Hamilton's Theory of Rays . 257 7.2 Making Waves . 259 7.2.1 Rayleigh's Equation . 259 7.3 Non-linear Waves . 263 7.3.1 Sound in Air . 263 7.3.2 Shocks . 266 7.3.3 Weak Solutions . 272 7.4 Solitons . 273 7.5 Exercises and Problems . 279 8 Special Functions I 283 8.1 Curvilinear Co-ordinates . 283 8.1.1 Div, Grad and Curl in Curvilinear Co-ordinates . 286 8.1.2 The Laplacian in Curvilinear Co-ordinates . 289 8.2 Spherical Harmonics . 290 8.2.1 Legendre Polynomials . 290 8.2.2 Axisymmetric potential problems . 293 8.2.3 General spherical harmonics . 297 8.3 Bessel Functions . 300 8.3.1 Cylindrical Bessel Functions . 300 8.3.2 Orthogonality and Completeness . 308 8.3.3 Modified Bessel Functions . 312 8.3.4 Spherical Bessel Functions . 315 8.4 Singular Endpoints . 319 8.4.1 Weyl's Theorem . 320 8.5 Exercises and Problems . 327 9 Integral Equations 333 9.1 Illustrations . 333 9.2 Classification of Integral Equations . 334 9.3 Integral Transforms . 336 9.3.1 Fourier Methods . 336 9.3.2 Laplace Transform Methods . 338 9.4 Separable Kernels . 344 9.4.1 Eigenvalue problem . 344 9.4.2 Inhomogeneous problem . 345 9.5 Singular Integral Equations . 347 9.5.1 Solution via Tchebychef Polynomials . 347 CONTENTS ix 9.6 Wiener-Hopf equations . 351 9.7 Some Functional Analysis . 356 9.7.1 Bounded and Compact Operators . 357 9.7.2 Closed Operators . 360 9.8 Series Solutions . 363 9.8.1 Neumann Series . 363 9.8.2 Fredholm Series . 364 A Linear Algebra Review 369 A.1 Vector Space . 369 A.1.1 Axioms . 369 A.1.2 Bases and components . 370 A.2 Linear Maps . 372 A.2.1 Matrices . 372 A.2.2 Range-nullspace theorem . 373 A.2.3 The dual space . 374 A.3 Inner-Product Spaces . 375 A.3.1.
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