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Walter Thirring
A Course in Mathematical Physics
1 and 2
Classical Dynamical Systems and Classical Field Theory
Second Edition
Translated by Evans M. Harrell
With 144 lllustrations
Springer-Verlag New York Wien
Dr. Walter Thirring Institute for Theoretical Physics University of Vienna Austria
Printed on acid-free paper.
© 1992 Springer-Verlag New York Inc.
Dr. Evans M. Harrell The lohns Hopkins University Baltimore, Maryland U.S.A.
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA) except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Volume I © 1978, 1992 by Springer-Vedag/Wien Volume 2 © 1979, 1986 by Springer-VedaglWien
Typeset by Asco Trade Typesetting Ltd., Hong Kong.
9 8 7 6 5 4 3 2 1
ISBN-13: 978-0-387-97609-9 DOl: 10.1007/978-1-4684-0517-0
e-ISBN-13: 978-1-4684-0517-0
Summary of Contents
A Course in Mathematical Physics 1: Classical Dynamical Systems, Second Edition
1 Introduction 2 Analysis on Manifolds 3 Hamiltonian Systems 4 Nonrelativistic Motion 5 Relativistic Motion 6 The Structure of Space and Time
A Course in Mathematical Physics 2: Classical Field Theory, Second Edition
1 Introduction 2 The Electromagnetic Field of a Known Charge Distribution 3 The Field in the Presence of Conductors 4 Gravitation
v
Walter Thirring
A Course in Mathematical Physics
1
Classical Dynamical Systems
Translated by Evans M. Harrell
Preface to the Second Edition
The last decade has seen a considerable renaissance in the realm of classical dynamical systems, and many things that may have appeared mathematically overly sophisticated at the time of the first appearance of this textbook have since become the everyday tools of working physicists. This new edition is intended to take this development into account. I have also tried to make the book more readable and to eradicate errors.
Since the first edition already contained plenty of material for a onesemester course, new material was added only when some of the original could be dropped or simplified. Even so, it was necessary to expand the chapter with the proof of the K-A-M Theorem to make allowances for the current trend in physics. This involved not only the use of more refined mathematical tools, but also a reevaluation of the word "fundamental." What was earlier dismissed as a grubby calculation is now seen as the consequence of a deep principle. Even Kepler's laws, which determine the radii of the planetary orbits, and which used to be passed over in silence as mystical nonsense, seem to point the way to a truth unattainable by superficial observation: The ratios of the radii of Platonic solids to the radii of inscribed Platonic solids are irrational, but satisfy algebraic equations of lower order. These irrational numbers are precisely the ones that are the least well approximated by rationals, and orbits with radii having these ratios are the most robust against each other's perturbations, since they are the least affected by resonance effects. Some surprising results about chaotic dynamics have been discovered recently, but unfortunately their proofs did not fit within the scope of this book and had to be left out.
In this new edition I have benefited from many valuable suggestions of colleagues who have used the book in their courses. In particular, I am deeply grateful to H. Grosse, H.-R. Griimm, H. Narnhofer, H. Urbantke, and above
IX
x Preface to the Second Edition
all M. Breitenecker. Once again the quality of the production has benefited from drawings by R. Bertlmann and J. Ecker and the outstanding wordprocessing of F. Wagner. Unfortunately, the references to the literature have remained sporadic, since any reasonably complete list of citations would have overwhelmed the space allotted.
Vienna, July, 1988 Walter Thirring
Preface to the First Edition
This textbook presents mathematical physics in its chronological order. It originated in a four-semester course I offered to both mathematicians and physicists, who were only required to have taken the conventional introductory courses. In order to be able to cover a suitable amount of advanced material for graduate students, it was necessary to make a careful selection of topics. I decided to cover only those subjects in which one can work from the basic laws to derive physically relevant results with full mathematical rigor. Models which are not based on realistic physical laws can at most serve as illustrations of mathematical theorems, and theories whose predictions are only related to the basic principles through some uncontrollable approximation have been omitted. The complete course comprises the following one-semester lecture series:
I. Classical Dynamical Systems II. Classical Field Theory III. Quantum Mechanics of Atoms and Molecules IV. Quantum Mechanics of Large Systems
Unfortunately, some important branches of physics, such as the relativistic quantum theory, have not yet matured from the stage of rules for calculations to mathematically well understood disciplines, and are therefore not taken up. The above selection does not imply any value judgment, but only attempts to be logically and didactically consistent.
General mathematical knowledge is assumed, at the level of a beginning graduate student or advanced undergraduate majoring in physics or mathematics. Some terminology of the relevant mathematical background is
xi
xii Preface to the First Edition
collected in the glossary at the beginning. More specialized tools are introduced as they are needed; I have used examples and counterexamples to try to give the motivation for each concept and to show just how far each assertion may be applied. The best and latest mathematical methods to appear on the market have been used whenever possible. In doing this many an old and trusted favorite of the older generation has been forsaken, as I deemed it best not to hand dull and worn-out tools down to the next generation. It might perhaps seem extravagant to use manifolds in a treatment of Newtonian mechanics, but since the language of manifolds becomes unavoidable in general relativity, I felt that a course that used them right from the beginning was more unified.
References are cited in the text in square brackets [ ] and collected at the end of the book. A selection of the more recent literature is also to be found there, although it was not possible to compile a complete bibliography.
I am very grateful to M. Breitenecker, J. Dieudonne, H. Grosse, P. Hertel, J. Moser, H. Narnhofer, and H. Urbantke for valuable suggestions. F. Wagner and R. Bertlmann have made the production of this book very much easier by their greatly appreciated aid with the typing, production and artwork.
Walter Thirring
Note about the Translation
In the English translation we have made several additions and corrections to try to eliminate obscurities and misleading statements in the German text. The growing popularity of the mathematical language used here has caused us to update the bibliography. We are indebted to A. Pflug and G. Siegl for a list of misprints in the original edition. The translator is grateful to the Navajo Nation and to the Institute for Theoretical Physics of the University of Vienna for hospitality while he worked on this book.
Evans M. Harrell
Walter Thirring
Contents of Volume 1
Preface to the Second Edition ix
Preface to the First Edition xi
Glossary xv
Symbols Defined in the Text xix
1 Introduction
1.1 Equations of Motion 1.2 The Mathematical Language 4 1.3 The Physical Interpretation 5
2 Analysis on Manifolds 8
2.1 Manifolds 8 2.2 Tangent Spaces 19 2.3 Flows 32 2.4 Tensors 42 2.5 Differentiation 56 2.6' Integrals 61
73
3 Hamiltonian Systems 84
3.1 Canonical Transformations 84 3.2 Hamilton's Equations 91 3.3 Constants of Motion 100 3.4 The Limit t -+ I ± 00 117 3.5 Perturbation Theory: Preliminaries 141 3.6 Perturbation Theory: The Iteration 153
xiii
xiv
4
5
6
Nonrelativistic Motion
4.1 Free Particles 4.2 The Two-Body Problem 4.3 The Problem of Two Centers of Force 4.4 The Restricted Three-Body Problem 4.5 The N-body Problem
Relativistic Motion
5.1 The Hamiltonian Formulation of the Electrodynamic Equations of Motions
5.2 The Constant Field 5.3 The Coulomb Field 5.4 The Betatron 5.5 The Traveling Plane Disturbance 5.6 Relativistic Motion in a Gravitational Field 5.7 Motion in the Schwarzschild Field 5.8 Motion in a Gravitational Plane Wave
The Structure of Space and Time
6.1 The Homogeneous Universe 6.2 The Isotropic Universe 6.3 Me according to Galileo 6.4 Me as Minkowski Space 6.5 Me as a Pseudo-Riemannian Space
Bibliography
Index
Contents
165
165 169 178 186 201
210
210 216 223 229 234 239 245 255
262
262 264 266 268 275
279
283
Glossary
Logical Symbols
V for every :I there exist(s) ~ there does not exist :I! there exists a unique a => b if a then b iff if and only if
Sets
aEA a¢A AuB AnB CA A\B A~B
o C0 AxB
a is an element of A a is not an element of A union of A and B intersection A and B complement of A (In a larger set B: {a: a E B, a¢ A}) {a:aEA,a¢B} symmetric difference of A and B: (A\B) u (B\A) empty set universal set Cartesian product of A and B: the set of all pairs (a, b), a E A, bE B
Important Families of Sets
open sets contains 0 and the universal set and some other specified sets, such that the open sets are closed under union and finite intersection
xv
xvi
closed sets measurable sets
Glossary
the complements of open sets contains 0 and some other specified sets, and
closed under complementation and countable intersection
Borel-measurable sets the smallest family of measurable sets which contains the open sets
null sets, or sets of measure zero the sets whose measure is zero. "Almost everywhere" means" except on a set of measure zero."
An equivalence relation is a covering of a set with a non-intersecting family of subsets. a ~ b means that a and b are in the same subset. An equivalence relation has the properties: i) a ~ a for all a: ii) a ~ b => b ~ a. iii) a ~ b, b ~ c => a ~ c.
Numbers
N 7L IR IR+(IR-) C sup inf I (a, b) [a,b] (a,b] and [a, b) IR"
natural numbers integers real numbers positive (negative) numbers complex numbers supremum, or lowest upper bound infimum, or greatest lower bound any open interval the open interval from a to b the closed interval from a to b half-open intervals from a to b IRx···xlR '-.--'
N times This is a vector space with the scalar product (Yb ... , YNlx l , ••• , XN) = Li: I YiXi
Maps ( = Mappings, Functions)
f:A -+ B f(A) rl(b)
r l
rl(B) f is injective (one-to-one) f is surjective (onto) f is bijective
1
for every a E A an element f(a) E B is specified image of A, i.e., if f:A -+ B, {f(a) E B:a E A} inverse image of b, i.e. {a E A: f(a) = b} inverse mapping to f. Warning: 1) it is not necessarily a
function 2) distinguish from 11f when B = IR inverse image of B: UbEB f-I(b) al oF a2 => f(ad oF f(a2) f(A) = B f is injective and surjective. Only in this case is f - I a
true function the function defined from A I X A 2 to B I X B 2, so that
(a\> a2) -+ (fl (al),fz(a2» fl composed with f2: if fl : A -+ Band f2: B -+ C, then f2 0 fl:A -+ C so that a -+ f2(fI(a»
identity map, when A = B; i.e., a -+ a. Warning: do not confuse with a --+ 1 when A = B = R
Glossary
fl. fl. f is continuous f is measurable suppf
C' Co XA
Topological Concepts
Topology compact set
connected set
discrete topology trivial topology
simply connected set
(open) neighborhood of a E A
f restricted to a subset V c A e~aluation of the map f at the point a; i.e.,f(a). the inverse image of any open set is open
xvii
the inverse image of any measurable set is measurable support off: the smallest closed set on whose
complement f = 0 the set of r times continuously differentiable functions the set of C' functions of compact (see below) support characteristic function of
A:xia) = 1. ..
any family of open sets, as defined above a set for which any covering with open sets has
a finite subcovering a set for which there are no proper subsets which
are both open and closed the topology for which every set is an open set the topology for which the only open sets are
o andC0 a set in which every path can be continuously
deformed to a point any open subset of A containing a. Usually
denoted by V or V. (open) neighborhood ofB c A p is a point of accumulation
any open subset of A containing B for any neighborhood V containing p,
(= cluster point) B B is dense in A B is nowhere dense in A metric (distance function) for A
separable space homeomorphism product topology on Al x A2
Mathematical Conventions
Of/Oqi
dq(r)
dt
V nB"# {p} or 0 closure of B: the smallest closed set containing B B=A A\B is dense in A a map d: A x A -+ IR such that dCa, a) = 0;
dCa, b) = deb, a) > 0 for b "# a; and dCa, c) :;; dCa, b) + deb, c) for all a, b, c in A. A metric induces a topology on A, in which all sets of the form {b:d(b, a) < II} are open.
a space with a countable dense subset a continuous bijection with a continuous inverse the family of open sets of the form V I X V 2,
where VI is open in Al and V 2 is open in A2 ,
and unions of such sets
q(t)
detlMijl TrM
determinant of the matrix Mij
c5f, c5ij LiMii
I if i = j, otherwise 0
xviii Glossary
the totally antisymmetric tensor of degree m. with values ± 1.
Mt
M* transposed matrix: (Mt)ij = Mji
V· w, (vlw), or (v· w)
v x w or [v 1\ w] Vf
Hermitian conjugate matrix: (M*)ij = (Mji)* scalar (inner, dot) product cross product gradient of f
V x f V ·f
curl off divergence of f
IIvil (in 3 dimensions, Ivl) ds
length of the vector v: Ilv II = (Ii = 1 vl)1/2 = d(O. v) differential line element
dS dmq .1 II 4..
dQ
Mat.(IR) O(x)
differential surface element m-dimensional volume element is perpendicular (orthogonal) to is parallel to angle element of solid angle
the set of real n x n matrices order of x
The summation convention for repeated indices is understood except where it does not make sense. For example. Li1Xl stands for Ik Li1Xl.
Groups
GL. group of n x n matrices with nonzero determinant O. group of n x n matrices M with M Mt = 1 (unit matrix) SO. subgroup of O. with determinant 1 E. Euclidean group S. group of permutations of n elements U. group of complex n x n matrices M with M M* = 1 (unit matrix)
SP. group of symplectic n x n matnces
Physical Symbols
mi Xi
t = xO/c S
qi Pi ei K
c Ii = h/21t F; g./I E B
mass of the i-th particle Cartesian coordinates of the i-th particle time proper time generalized coordinates generalized momenta charge of the i-th particle gravitational constant speed of light Planck's constant divided by 21t electromagnetic field tensor gravitational metric tensor (relativistic gravitational potential) electric field strength magnetic field strength in a vacuum is on the order of is much greater than
Symbols Defined in the Text
Df derivative of f: JR" -+ JR"' (2.1.1 ) (V. <II) chart (2.1.3) T" n-dimensional torus (2.1.7; 2) S" n-dimensional sphere (2.1.7; 2) oM boundary of M (2.1.20) 0C<q) mapping of the tangent space into JR"' (2.2.1) 1'q(M) tangent space at the point q (2.2.4) Tq(f) derivative o{ f at the point q (2.2.7) T(M) tangent bundle (2.2.12) n projection onto a basis (2.2.15) T(f) derivative of f: M I -+ M 2 (2.2.17) .rMM) set of vector fields (2.2.19) <II. induced mapping on .r~ (2.2.21) Lx Lie derivative (2.2.25; I), (2.5.7) OJ natural basis on the tangent space (2.2.26) <II'" I flow (2.3.7) r'" I automorphism of a flow (2.3.8) W action (2.3.16) L Lagrangian (2.3.17) H Hamiltonian (2.3.26) T:(M) cotangent space (2.4.1) e'!' , dual basis (2.4.2; I) df differential of a function (2.4.3; I) Tq~(M) space of tensors (2.4.4) ® tensor product (2.4.5) T~(M) tensor bundle (2.4.8) .r~(M) set of tensor fields (2.4.28) tif differential of a function (2.4.13; 1) g pseudo-Riemannian metric (2.4.27) • fiber product (2.4.34) x
xix
xx Symbols Defined in the Text
T*(<1» transposed derivative (2.4.34) (<1>- 1)* pull-back, or inverse image of the covariant tensors (2.4.41) Ep(M) set of p-forms (2.4.27) A wedge (outer, exterior) product (2.4.28) * *-mapping (2.4.18)
ix interior product (2.4.33) d exterior derivative (2.5.1 ) [ ] Lie bracket (2.5.9; 6) 0,w canonical forms (3.1.1 ) Q Liouville measure (3.1.2;3) XH Hamiltonian vector field (3.1.9) b bijection associated with w (3.1.9) { } Poisson brackets (3.1.11)
Me generalized configuration space (3.2.12) Yf Hamiltonian on Me (3.2.12) (I, cp) action-angle variables (3.3.14)
Q± M011er transformations (3.4.4) S scattering matrix (3.4.9) da differential scattering cross-section (3.4.12) L angular momentum (4.1.3) K boost (4.1.9)
'l,p Minkowski space metric (5.1.2)
" I/JI - v21c2 (relativistic dilatation) (5.1.2)
F electromagnetic 2-form (5.1.10; I) A I-form of the potential (5.1.10; I) /\ Lorentz transformation (5.1.12)
ro Schwarzschild radius (5.7.1)
Walter Thirring
A Course in Mathematical Physics
2
Classical Field Theory
Second Edition
Translated by Evans M. Harrell
Preface
In the past decade the language and methods of modern differential geometry have been increasingly used in theoretical physics. What seemed extravagant when this book first appeared 12 years ago, as lecture notes, is now a commonplace. This fact has strengthened my belief that today students of theoretical physics have to learn that language-and the sooner the better. After all, they will be the professors of the twenty-first century and it would be absurd if they were to teach then the mathematics of the nineteenth century. Thus for this new edition I did not change the mathematical language. Apart from correcting some mistakes I have only added a section on gauge theories. In the last decade it has become evident that these theories describe fundamental interactions, and on the classical level their structure is sufficiently clear to qualify them for the minimum amount of knowledge required by a theoretician. It is with much regret that I had to refrain from incorporating the interesting developments in Kaluza-Klein theories and in cosmology, but I felt bound to my promise not to burden the students with theoretical speculations for which there is no experimental evidence.
I am indebted to many people for suggestions concerning this volume. In particular, P. Aichelburg, H. Rumpf and H. Urbantke have contributed generously to corrections and improvements. Finally, I would like to thank Dr. I. Dahl-lensen for redoing some of the figures on the computer.
Vienna December, 1985
W. Thirring
iii
Contents of Volume 2
Symbols Defined in the Text vii
1 Introduction
2
3
4
1.1 Physical Aspects of Field Dynamics 1.2 The Mathematical Formalism 1.3 Maxwell's and Einstein's Equations
The Electromagnetic Field of a Known Charge Distribution
I 10 28
46
2.1 The Stationary-Action Principle and Conservation Theorems 46 2.2 The General Solution 57 2.3 The Field of a Point Charge 71 2.4 Radiative Reaction 89
The Field in the Presence of Conductors 102
3.1 The Superconductor 102 3.2 The Half-Space, the Wave-Guide, and the Resonant Cavity 113 3.3 Diffraction at a Wedge 125 3.4 Diffraction at a Cylinder 138
Gravitation
4.1 Covariant Differentiation and the Curvature of Space 4.2 Gauge Theories and Gravitation
155
155 174
v
VI
4.3 Maximally Symmetric Spaces 4.4 Spaces with Maximally Symmetric Submanifolds 4.5 The Life and Death of Stars 4.6 The Existence of Singularities
Bibliography
Index
Contents
191 204 223 236
253
259
Symbols Defined in the Text
ei1i2 , .. ip basis of the p-forms ( 1.2.3) Ep(M) linear space of the p-forms (1.2.5; 2)
d exterior differential (1.2.6) (1)[,'1 restriction of a form (1.2.7; 3)
E~(U) space of m-forms with compact support ( 1.2.9)
<ei(x)Id'(x» scalar product ( 1.2.14)
iv interior product (1.2.16)
* isomorphism between Ep and Em - p ( 1.2.17)
6 codifferential (1.2.19) .1 Laplace-Beltrami operator ( 1.2.20)
Lv Lie derivative ( 1.2.23)
w\ affine connection ( 1.2.25)
(Uik affine connection ( 1.2.25) 0(x) Heaviside step function ( 1.2.31) 6(x) Dirac delta function (1.2.31)
6 .. Dirac delta form ( 1.2.33)
Gx Green function ( 1.2.35)
E,B,F electric and magnetic fields (1.3. J) A vector potential (1.3. 7)
1\ gauge function (1.3.10; I)
J current (1.3.12)
Q total charge (1.3.18; 2) T'P energy-momentum tensor ( 1.3.20) p' total energy-momentum (1.3.21 )
.OJ' energy-momentum form of the field ( 1.3.22) z(s) world-line (1.3.25; 2) t' energy-momentum form of matter (1.3.25; 2) !f Lagrangian (2.1.1 )
W action (2.1.1)
vii
viii Symbols Defined in the Text
9' Poynting's vector (2.1.13) D±(N) domains of influence (2.1.15)
Di/ Green function (2.2.5) Drel(x) retarded Green function (2.2.7) G~el
x retarded Green function (form) (2.2.7) Frel retarded field strength (2.2.9) Fin incoming field strength (2.2.15) Foul outgoing field strength (2.2.15) Frad radiation field (2.2.21) D(x) D-function (2.2.22) (jE energy loss per period (2.4.4; 2)
j specified current (3.1.7) e(k) dielectric constant (3.1.19; 3) S superpotential (3.1.21; 1)
F(z) Fresnel's integral (3.3.10)
<I) scalar product (4.1.3) Sp Sections (4.1.9) D exterior covariant derivative (4.1.10) Dx covariant derivative (4.1.15) n curvature form (4.1.19) R curvature in space-time (4.1.20; 1)
f ijk Christoffel symbol (4.1.36) Rijkm Riemann-Christoffel tensor (4.1.44; 2) C jk Weyl forms (4.1.44; 3)
ro Schwarzschild radius (4.4.41) K curvature parameter (4.4.42) c rate of convergence (4.6.8) rex) future of x (4.6.18(a» r(x) past of x (4.6. I 8(a» C(x, S) set of causal curves (4.6.18(b» C1(x; S) set of differentiable causal curves (4.6. 18(b» d(.l.) length of .l. (4.6.l8(c»