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 differential equations, partial differential equations, and integral equations. Although this mathematics is applicable to a wide range physical phenom- ena, the illustrative examples are mostly drawn from classical and quantum mechanics. Classical mechanics is a subject familiar to all physics students and the point being illustrated is immediately understandable without any further specialized knowledge. Similarly all physics students have studied quantum mechanics, and here the matrix/differential-operator analogy lies at the heart of the subject. 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 Examples . 3 1.2.3 First Integral . 9 1.3 Lagrangian Mechanics . 10 1.3.1 One Degree of Freedom . 11 1.3.2 Noether's Theorem . 15 1.3.3 Many Degrees of Freedom . 18 1.3.4 Continuous Systems . 19 1.4 Variable End Points . 28 1.5 Lagrange Multipliers . 34 1.6 Maximum or Minimum? . 38 1.7 Further Problems . 40 2 Function Spaces 47 2.1 Motivation . 47 2.1.1 Functions as Vectors . 48 2.2 Norms and Inner Products . 49 2.2.1 Norms and Convergence . 49 2.2.2 Norms from Integrals . 51 2.2.3 Hilbert Space . 53 2.2.4 Orthogonal Polynomials . 60 2.3 Linear Operators and Distributions . 65 2.3.1 Linear Operators . 65 v vi CONTENTS 2.3.2 Distributions and Test-functions . 68 3 Linear Ordinary Differential Equations 77 3.1 Existence and Uniqueness of Solutions . 77 3.1.1 Flows for First-Order Equations . 77 3.1.2 Linear Independence . 79 3.1.3 The Wronskian . 80 3.2 Normal Form . 85 3.3 Inhomogeneous Equations . 87 3.3.1 Particular Integral and Complementary Function . 87 3.3.2 Variation of Parameters . 88 3.4 Singular Points . 90 3.4.1 Regular Singular Points . 90 4 Linear Differential Operators 93 4.1 Formal vs. Concrete Operators . 93 4.1.1 The Algebra of Formal Operators . 93 4.1.2 Concrete Operators . 95 4.2 The Adjoint Operator . 96 4.2.1 The Formal Adjoint . 96 4.2.2 A Simple Eigenvalue Problem . 100 4.2.3 Adjoint Boundary Conditions . 102 4.2.4 Self-adjoint Boundary Conditions . 104 4.3 Completeness of Eigenfunctions . 111 4.3.1 Discrete Spectrum . 111 4.3.2 Continuous spectrum . 118 5 Green Functions 131 5.1 Inhomogeneous Linear equations . 131 5.1.1 Fredholm Alternative . 131 5.2 Constructing Green Functions . 132 5.2.1 Sturm-Liouville equation . 133 5.2.2 Initial Value Problems . 135 5.2.3 Modified Green Functions . 140 5.3 Applications of Lagrange's Identity . 142 5.3.1 Hermiticity of Green function . 142 5.3.2 Inhomogeneous Boundary Conditions . 143 5.4 Eigenfunction Expansions . 146 CONTENTS vii 5.5 Analytic Properties of Green Functions . 147 5.5.1 Causality Implies Analyticity . 147 5.5.2 Plemelj Formulæ . 152 5.5.3 Resolvent Operator . 154 5.6 Locality and the Gelfand-Dikii equation . 159 6 Partial Differential Equations 163 6.1 Classification of PDE's . 163 6.1.1 Cauchy Data . 165 6.1.2 Characteristics and first-order equations . 167 6.2 Wave Equation . 168 6.2.1 d'Alembert's Solution . 168 6.2.2 Fourier's Solution . 170 6.2.3 Causal Green Function . 171 6.2.4 Odd vs. Even Dimensions . 176 6.3 Heat Equation . 182 6.3.1 Heat Kernel . 182 6.3.2 Causal Green Function . 183 6.3.3 Duhamel's Principle . 185 6.4 Laplace's Equation . 187 6.4.1 Separation of Variables . 189 6.4.2 Eigenfunction Expansions . 198 6.4.3 Green Functions . 200 6.4.4 Kirchhoff vs. Huygens . 205 7 The Mathematics of Real Waves 211 7.1 Dispersive waves . 211 7.1.1 Ocean Waves . 211 7.1.2 Group Velocity . 215 7.1.3 Wakes . 218 7.1.4 Hamilton's Theory of Rays . 221 7.2 Making Waves . 223 7.2.1 Rayleigh's Equation . 223 7.3 Non-linear Waves . 227 7.3.1 Sound in Air . 228 7.3.2 Shocks . 230 7.3.3 Weak Solutions . 236 7.4 Solitons . 237 viii CONTENTS 7.5 Further problems . 243 8 Special Functions I 247 8.1 Curvilinear Co-ordinates . 247 8.1.1 Div, Grad and Curl in Curvilinear Co-ordinates . 250 8.1.2 The Laplacian in Curvilinear Co-ordinates . 253 8.2 Spherical Harmonics . 254 8.2.1 Legendre Polynomials . 254 8.2.2 Axisymmetric potential problems . 257 8.2.3 General spherical harmonics . 261 8.3 Bessel Functions . 264 8.3.1 Cylindrical Bessel Functions . 264 8.3.2 Orthogonality and Completeness . 272 8.3.3 Modified Bessel Functions . 275 8.3.4 Spherical Bessel Functions . 279 8.4 Singular Endpoints . 282 8.4.1 Weyl's Theorem . 283 9 Integral Equations 291 9.1 Illustrations . 291 9.2 Classification of Integral Equations . 292 9.3 Integral Transforms . 294 9.3.1 Fourier Methods . 294 9.3.2 Laplace Transform Methods . 295 9.4 Separable Kernels . 302 9.4.1 Eigenvalue problem . 302 9.4.2 Inhomogeneous problem . 303 9.5 Singular Integral Equations . 305 9.5.1 Solution via Tchebychef Polynomials . 305 9.6 Wiener-Hopf equations . 309 9.7 Some Functional Analysis . 314 9.7.1 Bounded and Compact Operators . 314 9.7.2 Closed Operators . 317 9.8 Series Solutions . 320 9.8.1 Neumann Series . 320 9.8.2 Fredholm Series . 321 CONTENTS ix A Elementary Linear Algebra 325 A.1 Vector Space . 325 A.1.1 Axioms . 325 A.1.2 Bases and Components . 326 A.2 Linear Maps and Matrices . 328 A.2.1 Range-Nullspace Theorem . 329 A.2.2 The Dual Space . 330 A.3 Inner-Product Spaces . 331 A.3.1 Inner Products . 331 A.3.2 Adjoint Operators . 335 A.4 Sums and Differences of Vector Spaces . 336 A.4.1 Projection operators . 338 A.5 Inhomogeneous Linear Equations . 338 A.5.1 Fredholm Alternative . 340 A.6 Determinants . 341 A.6.1 Skew-symmetric n-linear Forms . 341 A.6.2 The Adjugate Matrix . ..