Table Of ContentMathematics for Physics I
A set of lecture notes by
Michael Stone
PIMANDER-CASAUBON
Alexandria Florence London
(cid:15) (cid:15)
ii
Copyright c 2001,2002 M. Stone.
(cid:13)
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 (cid:12)rst 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 (cid:12)nite dimensional spaces. The op-
erator language then provides a uni(cid:12)ed framework for investigating ordinary
and partial di(cid:11)erential 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
di(cid:11)erential 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
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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 Di(cid:11)erential Equations 89
3.1 Existence and Uniqueness of Solutions . . . . . . . . . . . . . 89
3.1.1 Flows for (cid:12)rst-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 Di(cid:11)erential 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 Modi(cid:12)ed Green function . . . . . . . . . . . . . . . . . 154
CONTENTS vii
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 formul(cid:26) . . . . . . . . . . . . . . . . . . . . . . 166
5.5.3 Resolvent operator . . . . . . . . . . . . . . . . . . . . 168
5.6 Locality and the Gelfand-Dikii equation . . . . . . . . . . . . 173
5.7 Exercises and problems . . . . . . . . . . . . . . . . . . . . . . 175
6 Partial Di(cid:11)erential Equations 181
6.1 Classi(cid:12)cation of PDE’s . . . . . . . . . . . . . . . . . . . . . . 181
6.2 Cauchy Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.2.1 Characteristics and (cid:12)rst-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 Kirchho(cid:11) 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
viii CONTENTS
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 Modi(cid:12)ed 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 Classi(cid:12)cation 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
CONTENTS ix
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 Di(cid:11)erences 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 Di(cid:11)erentiating 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
x CONTENTS
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
