Mathematics for Physics: A guided tour for graduate students

919 Pages·2012·5.27 MB·English

Mathematics for Physics
A guided tour for graduate
students
Michael Stone
and
Paul Goldbart
PIMANDER-CASAUBON
Alexandria Florence London
(cid:15) (cid:15)
ii
Copyright c 2002-2008 M. Stone, P. M. Goldbart
(cid:13)
All rights reserved. No part of this material can be reproduced, stored or
transmitted without the written permission of the authors. For information
contact: Michael Stone or Paul Goldbart, Department of Physics, University
of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois
61801-3080, U.S.A.
Dedication
To the memory of Mike’s mother, Aileen Stone: 9 9 = 81.
(cid:2)
To Paul’s mother and father, Carole and Colin Goldbart.
iii
iv DEDICATION
Acknowledgments
A great many people have encouraged us along the way:
Ourteachers attheUniversity ofCambridge, theUniversity ofCalifornia-Los
Angeles, and Imperial College London.
Our students { your questions and enthusiasm have helped shape our under-
standing and our exposition.
Our colleagues|faculty and sta(cid:11)|at the University of Illinois at Urbana-
Champaign { how fortunate we are to have a community so rich in both
accomplishment and collegiality.
Our friends and family: Kyre and Steve and Ginna; and Jenny, Ollie and
Greta { we hope to be more attentive now that this book is done.
Our editor Simon Capelin at Cambridge University Press { your patience is
appreciated.
The sta(cid:11) of the U.S. National Science Foundation and the U.S. Department
of Energy, who have supported our research over the years.
Our sincere thanks to you all.
v
vi ACKNOWLEDGMENTS
Preface
This book is based on a two-semester sequence of courses taught to incoming
graduate students at the University of Illinois at Urbana-Champaign, pri-
marily physics students but also some from other branches of the physical
sciences. The courses aim to introduce students to some of the mathematical
methods and concepts that they will (cid:12)nd useful in their research. We have
sought to enliven the material by integrating the mathematics with its appli-
cations. Wethereforeprovide illustrative examples andproblems drawnfrom
physics. Some of these illustrations are classical but many are small parts of
contemporary research papers. In the text and at the end of each chapter we
provide a collection of exercises and problems suitable for homework assign-
ments. The former are straightforward applications of material presented
in the text; the latter are intended to be interesting, and take rather more
thought and time.
We devote the (cid:12)rst, and longest, part (Chapters 1 to 9, and the (cid:12)rst
semester in the classroom) to traditional mathematical methods. We explore
the analogy between linear operators acting on function spaces and matrices
acting on (cid:12)nite dimensional spaces, and use the operator language to pro-
vide a uni(cid:12)ed framework for working with ordinary di(cid:11)erential equations,
partial di(cid:11)erential equations, and integral equations. The mathematical pre-
requisites are a sound grasp of undergraduate calculus (including the vector
calculus needed for electricity and magnetism courses), elementary linear al-
gebra, and competence atcomplex arithmetic. Fouriersums 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 for this part of the book.
The second part (Chapters 10 to 14) focuses on modern di(cid:11)erential ge-
ometry and topology, with an eye to its application to physics. The tools of
calculus on manifolds, especially the exterior calculus, are introduced, and
vii
viii PREFACE
used to investigate classical mechanics, electromagnetism, and non-abelian
gauge (cid:12)elds. The language of homology and cohomology is introduced and
is used to investigate the in(cid:13)uence of the global topology of a manifold on
the (cid:12)elds that live in it and on the solutions of di(cid:11)erential equations that
constrain these (cid:12)elds.
Chapters 15 and 16 introduce the theory of group representations and
their applications to quantum mechanics. Both (cid:12)nite groups and Lie groups
are explored.
The lastpart(Chapters 17to19)explores thetheoryofcomplex variables
anditsapplications. Althoughmuch ofthematerialisstandard, we make use
of the exterior calculus, and discuss rather more of the topological aspects of
analytic functions than is customary.
A cursory reading of the Contents of the book will show that there is
more material here than can be comfortably covered in two semesters. When
using the book as the basis for lectures in the classroom, we have found it
useful to tailor the presented material to the interests of our students.
Contents
Dedication iii
Acknowledgments v
Preface vii
1 Calculus of Variations 1
1.1 What is it good for? . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Variable endpoints . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5 Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . 36
1.6 Maximum or minimum? . . . . . . . . . . . . . . . . . . . . . 40
1.7 Further exercises and problems . . . . . . . . . . . . . . . . . 42
2 Function Spaces 55
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2 Norms and inner products . . . . . . . . . . . . . . . . . . . . 57
2.3 Linear operators and distributions . . . . . . . . . . . . . . . . 74
2.4 Further exercises and problems . . . . . . . . . . . . . . . . . 85
3 Linear Ordinary Di(cid:11)erential Equations 95
3.1 Existence and uniqueness of solutions . . . . . . . . . . . . . . 95
3.2 Normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.3 Inhomogeneous equations . . . . . . . . . . . . . . . . . . . . . 103
3.4 Singular points . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.5 Further exercises and problems . . . . . . . . . . . . . . . . . 108
ix
x CONTENTS
4 Linear Di(cid:11)erential Operators 111
4.1 Formal vs. concrete operators . . . . . . . . . . . . . . . . . . 111
4.2 The adjoint operator . . . . . . . . . . . . . . . . . . . . . . . 114
4.3 Completeness of eigenfunctions . . . . . . . . . . . . . . . . . 128
4.4 Further exercises and problems . . . . . . . . . . . . . . . . . 145
5 Green Functions 155
5.1 Inhomogeneous linear equations . . . . . . . . . . . . . . . . . 155
5.2 Constructing Green functions . . . . . . . . . . . . . . . . . . 156
5.3 Applications of Lagrange’s identity . . . . . . . . . . . . . . . 167
5.4 Eigenfunction expansions . . . . . . . . . . . . . . . . . . . . . 170
5.5 Analytic properties of Green functions . . . . . . . . . . . . . 171
5.6 Locality and the Gelfand-Dikii equation . . . . . . . . . . . . 184
5.7 Further exercises and problems . . . . . . . . . . . . . . . . . 185
6 Partial Di(cid:11)erential Equations 193
6.1 Classi(cid:12)cation of PDE’s . . . . . . . . . . . . . . . . . . . . . . 193
6.2 Cauchy data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.3 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.4 Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.5 Potential theory . . . . . . . . . . . . . . . . . . . . . . . . . . 223
6.6 Further exercises and problems . . . . . . . . . . . . . . . . . 249
7 The Mathematics of Real Waves 257
7.1 Dispersive waves . . . . . . . . . . . . . . . . . . . . . . . . . 257
7.2 Making waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
7.3 Non-linear waves . . . . . . . . . . . . . . . . . . . . . . . . . 274
7.4 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
7.5 Further exercises and problems . . . . . . . . . . . . . . . . . 289
8 Special Functions 295
8.1 Curvilinear co-ordinates . . . . . . . . . . . . . . . . . . . . . 295
8.2 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . 302
8.3 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . 311
8.4 Singular endpoints . . . . . . . . . . . . . . . . . . . . . . . . 332
8.5 Further exercises and problems . . . . . . . . . . . . . . . . . 340