lecture notes by Michael Stone Mathematics for Physics I

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Mathematics for Physics I A set of lecture notes by Michael Stone PIMANDER-CASAUBON Alexandria • Florence • London ii M. Stone. All righ ... ts 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 . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . 29 1.5 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . 34 1.6 Maximum or Minimum? . . . . . . . . . . . . . . . . . . . . . 38 1.7 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . 40 2 Function Spaces 49 2.1 Motivation 49 2.1.1 Functions as vectors 50 2.2 Norms and Inner Products 51 2.2.1 Norms and convergence 51 2.2.2 Norms from integrals 53 2.2.3 Hilbert space 55 2.2.4 Orthogonal polynomials 63 2.3 Linear Operators and Distributions 68 v vi CONTENTS 2.3.1 Linear operators 68 2.3.2 Distributions and test-functions 71 2.4 Exercises and Problems 78 3 Linear Ordinary Differential Equations 89 3.1 Existence and Uniqueness of Solutions 89 3.1.1 Flows for first-order equations 89 3.1.2 Linear independence 91 3.1.3 The Wronskian 92 3.2 Normal Form 96 3.3 Inhomogeneous Equations 97 3.3.1 Particular integral and complementary function 97 3.3.2 Variation of parameters 98 3.4 Singular Points 100 3.4.1 Regular singular points 100 3.5 Exercises and Problems 101 4 Linear Differential Operators 105 4.1 Formal vs. Concrete Operators 105 4.1.1 The algebra of formal operators 105 4.1.2 Concrete operators 107 4.2 The Adjoint Operator 108 4.2.1 The formal adjoint 108 4.2.2 A simple eigenvalue problem 113 4.2.3 Adjoint boundary conditions 115 4.2.4 Self-adjoint boundary conditions 116 4.3 Completeness of Eigenfunctions 122 4.3.1 Discrete spectrum 123 4.3.2 Continuous spectrum 129 4.4 Exercises and Problems 139 5 Green Functions 145 5.1 Inhomogeneous Linear equations 145 5.1.1 Fredholm alternative 145 5.2 Constructing Green Functions 146 5.2.1 Sturm-Liouville equation 147 5.2.2 Initial-value problems 149 5.2.3 Modified Green function 154 5.3 Applications of Lagrange’s Identity 156 5.3.1 Hermiticity of Green function 156 5.3.2 Inhomogeneous boundary conditions 158 5.4 Eigenfunction Expansions 160 5.5 Analytic Properties of Green Functions 161 5.5.1 Causality implies analyticity 161 5.5.2 Plemelj formulæ 166 5.5.3 Resolvent operator 168 5.6 Locality and the Gelfand-Dikii equation 173 5.7 Exercises and problems 175 6 Partial Differential Equations 181 6.1 Classification of PDE’s . . . . . . . . . . . . . . . . . . . . . . 181 6.2 Cauchy Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.2.1 Characteristics and first-order equations . . . . . . . . 185 6.2.2 Second-order hyperbolic equations . . . . . . . . . . . . 186 6.3 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.3.1 d’Alembert’s Solution . . . . . . . . . . . . . . . . . . . 188 6.3.2 Fourier’s Solution . . . . . . . . . . . . . . . . . . . . . 193 6.3.3 Causal Green Function . . . . . . . . . . . . . . . . . . 194 6.3.4 Odd vs. Even Dimensions . . . . . . . . . . . . . . . . 198 6.4 Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6.4.1 Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . 204 6.4.2 Causal Green Function . . . . . . . . . . . . . . . . . . 206 6.4.3 Duhamel’s Principle . . . . . . . . . . . . . . . . . . . 207 6.5 Laplace’s Equation . . . . . . . . . . . . . . . . . . . . . . . . 209 6.5.1 Separation of Variables . . . . . . . . . . . . . . . . . . 212 6.5.2 Eigenfunction Expansions . . . . . . . . . . . . . . . . 221 6.5.3 Green Functions . . . . . . . . . . . . . . . . . . . . . 223 6.5.4 Boundary-value problems . . . . . . . . . . . . . . . . 225 6.5.5 Kirchhoff vs. Huygens . . . . . . . . . . . . . . . . . . 229 6.6 Exercises and problems . . . . . . . . . . . . . . . . . . . . . . 233 7 The Mathematics of Real Waves 241 7.1 Dispersive waves . . . . . . . . . . . . . . . . . . . . . . . . . 241 7.1.1 Ocean Waves . . . . . . . . . . . . . . . . . . . . . . . 241 7.1.2 Group Velocity . . . . . . . . . . . . . . . . . . . . . . 245 7.1.3 Wakes . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 7.1.4 Hamilton’s Theory of Rays 251 7.2 Making Waves 253 7.2.1 Rayleigh’s Equation 253 7.3 Non-linear Waves 257 7.3.1 Sound in Air 258 7.3.2 Shocks 260 7.3.3 Weak Solutions 266 7.4 Solitons 267 7.5 Exercises and Problems 273 8 Special Functions I 277 8.1 Curvilinear Co-ordinates 277 8.1.1 Div, Grad and Curl in Curvilinear Co-ordinates 280 8.1.2 The Laplacian in Curvilinear Co-ordinates 283 8.2 Spherical Harmonics 284 8.2.1 Legendre Polynomials 284 8.2.2 Axisymmetric potential problems 287 8.2.3 General spherical harmonics 291 8.3 Bessel Functions 294 8.3.1 Cylindrical Bessel Functions 294 8.3.2 Orthogonality and Completeness 302 8.3.3 Modified Bessel Functions 306 8.3.4 Spherical Bessel Functions 309 8.4 Singular Endpoints 313 8.4.1 Weyl’s Theorem 314 8.5 Exercises and Problems 321 9 Integral Equations 327 9.1 Illustrations 327 9.2 Classification of Integral Equations 328 9.3 Integral Transforms 330 9.3.1 Fourier Methods 330 9.3.2 Laplace Transform Methods 332 9.4 Separable Kernels 338 9.4.1 Eigenvalue problem 338 9.4.2 Inhomogeneous problem 339 9.5 Singular Integral Equations 341 9.5.1 Solution via Tchebychef Polynomials 341 9.6 Wiener-Hopf equations 345 9.7 Some Functional Analysis 350 9.7.1 Bounded and Compact Operators 351 9.7.2 Closed Operators 354 9.8 Series Solutions 357 9.8.1 Neumann Series 357 9.8.2 Fredholm Series 358 A Linear Algebra Review 363 A.1 Vector Space 363 A.1.1 Axioms 363 A.1.2 Bases and components 364 A.2 Linear Maps 366 A.2.1 Matrices 366 A.2.2 Range-nullspace theorem 367 A.2.3 The dual space 368 A.3 Inner-Product Spaces 369 A.3.1 Inner products 369 A.3.2 Euclidean vectors 371 A.3.3 Bra and ket vectors 371 A.3.4 Adjoint operator 373 A.4 Sums and Differences of Vector Spaces 374 A.4.1 Direct sums 374 A.4.2 Quotient spaces 375 A.4.3 Projection-operator decompositions 376 A.5 Inhomogeneous Linear Equations 376 A.5.1 Rank and index 377 A.5.2 Fredholm alternative 379 A.6 Determinants 379 A.6.1 Skew-symmetric n-linear Forms 379 A.6.2 The adjugate matrix 382 A.6.3 Differentiating determinants 384 A.7 Diagonalization and Canonical Forms 385 A.7.1 Diagonalizing linear maps 385 A.7.2 Diagonalizing quadratic forms 391 A.7.3 Block-diagonalizing symplectic forms 394 B Fourier Series and Integrals. 399 B.1 Fourier Series 399 B.1.1 Finite Fourier series 399 B.1.2 Continuum limit 401 B.2 Fourier Integral Transforms 404 B.2.1 Inversion formula 404 B.2.2 The Riemann-Lebesgue lemma 406 B.3 Convolution 407 B.3.1 The convolution theorem 408 B.3.2 Apodization and Gibbs’ phenomenon 408 B.4 The Poisson Summation Formula 414 C Bibliography 417 Chapter 1 Calculus of Variations We begin our tour of useful mathematics with what is called the calculus of variations. Many physics problems can be formulated in the language of this calculus, and once they are there are useful tools to hand. In the text and associated exercises we will meet some of the equations whose solution will occupy us for much of our journey. 1.1 What is it good for? The classical problems that motivated the creators of the calculus of varia- tions include: i) Dido’s problem: In Virgil’s Aeneid we read how Queen Dido of Carthage must find largest area that can be enclosed by a curve (a strip of bull’s hide) of fixed length. ii) Plateau’s problem: Find the surface of minimum area for a given set of bounding curves. A soap film on a wire frame will adopt this minimal- area configuration. iii) Johann Bernoulli’s Brachistochrone: A bead slides down a curve with fixed ends. Assuming that the total energy 1 mv2 + V (x) is constant, find the curve that gives the most rapid descent. iv) Catenary: Find the form of a hanging heavy chain of fixed length by minimizing its potential energy. These problems all involve finding maxima or minima, and hence equating some sort of derivative to zero. In the next section we will define this deriva- tive, and show how to compute it. 1 1.2 Functionals In variational problems we are provided with an expression J[y] that “eats” whole functions y(x) and returns a single number. Such objects are often called functionals to distinguish them from ordinary functions. An ordinary function is a map f : R R. A functional J is a map J : C∞(R) R where C∞(R) is the space of smooth (having derivatives of all orders) functions. To find the function y(x) that maximizes or minimizes a given functional J[y] we need to define, and evaluate, its functional derivative. 1.2.1 The functional derivative We will restrict ourselves to expressions of the form J[y] = x2 f (x, y, yJ, y x1 JJ, · · · y(n)) dx, (1.1) where f depends on the value of y(x) and only finitely many of its derivatives. Such functionals are said to be local in x. Consider first a functional J = f dx in which f depends only x, y and yJ, and suppose that we vary y(x) y(x) + εη(x) where ε is a (small) x-independent constant. The resultant change in J is then ∫ x2 x2 ∂f = εη x1 ∂y dη ∂f + ε dx ∂yJ + O(ε2) dx = εη ∂f x2 x2 + (εη(x)) ∂f − d ∂f dx + O(ε2). ∂yJ x1 x1 ∂y dx ∂yJ For the moment we assume that η(x1) = η(x2) = 0. The change δy(x) εη(x) that we have made in y(x) is therefore a “fixed-endpoint variation.” For such variations the integrated-out part [. . .]x2 to be the O(ε) part of J[y + εη] − J[y], we have ......................................................................................continued................................................................................................................ [Show More]

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