Classical Mathematical Physics

www.pdfgrip.com


Springer Science+Business Media, LLC

www.pdfgrip.com


Waher Thirring

Classical Mathematical
Physics
Dynamical Systems
and. Field Theories
Third Edition

Translated by Evans M. Harrell II
With 146 Illustrations

,

Springer
www.pdfgrip.com


Walter Thirring
Institute for Theoretical Physics
University ofVienna

Vienna, A-I 090
Austria

Library ofCongress Cataloging-in-Publication Data
Thirring, Walter E., 1927Classical mathematical physics: dynamical systems and field
theories/Walter Thirring. - 3rd ed.
p.
cm.
Rev. ed. of: A course in mathematical physics 1 and 2. 2nd ed.
1992
Inc1udes bibliographical references and index.
1. Mathematical physics. 2. Dynamics.
3. Field Theory (Physics).
1. Thirring, Walter E., 1927- . Lehrbuch der mathematischen Physik.
English. II. Title.
QC20.T4513 1997
530.1 '5-----iic20
96-32751

ISBN 978-0-387-40615-2
ISBN 978-1-4612-0681-1 (eBook)
DOI 10.1007/978-1-4612-0681-1
Printed on acid-free paper.
First softcover printing, 2003.
© 1997, 1992 Springer Science+Business Media New York
Originally published by Springer-V erlag New York, Inc. in 1997
AlI rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in
connection with reviews or scholarly analysis. Use in connection with any form ofinformation storage
and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now

known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not they are
subject to proprietary rights.

Volume 1 (now Part 1) 1992, 1978 by Springer-VerlaglWien.
Volume 2 (now Part II) 1986, 1979 by Springer-VerlaglWien.
987654321

SPIN 10947258

www.springer-ny.com

www.pdfgrip.com


Preface to the Third Edition

This edition combines the earlier two volumes on Classical Dynamical Systems
and on Classical Field Theory, thus including in a single volume the material for
a two-semester course on classical physics. .
In preparing this new edition, I have once again benefited from valuable suggestions and corrections made by M. Breitenecker.
Vienna, Austria, February 1997

www.pdfgrip.com

Walter Thirring


Preface to the Second Edition:

Classical Dynamical Systems

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 one-semester
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 all

www.pdfgrip.com


viii


Preface to the Second Edition: Classical Dynamical Systems

M. Breitenecker. Once again the quality of the production has benefited from
drawings by R. Bertlmann and J. Ecker and the outstanding word processing 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, Austria, July 1988

Walter Thirring

www.pdfgrip.com


Preface to the Second Edition:
Classical Field Theory

In the past decade, the language and methods of modem 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-Jensen
for redoing some of the figures on the computer.
Vienna, Austria, December 1985

www.pdfgrip.com

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 that 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 student majoring in physics or mathematics.

www.pdfgrip.com


xii

Preface to the First Edition

Some terminology of the relevant mathematical background is collected in the
Glossary near the beginning of the book. 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 near 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.
Vienna, Austria, February 1977


www.pdfgrip.com

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.
Atlanta, Georgia, USA
Vienna, Austria

Evans M. Harrell II
Walter Thirring

www.pdfgrip.com


Contents

Preface to the Third Edition

v


Preface to the Second Edition: Classical Dynamical Systems

vii

Preface to the Second Edition: Classical Field Theory

ix

Preface to the First Edition

xi

Note About the Translation

xiii

Glossary

xix

Symbols Defined in the Text

xxv

Part I Classical Dynamical Systems
1 Introduction
1.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . ..
1.2 The Mathematical Language . . . . . . . . . . . . . . . . . ..
1.3 The Physical Interpretation. . . . . . . . . . . . . . . . . . ..
2 Analysis on Manifolds

2.1 Manifolds . . . . . . . . . .
2.2 Tangent Spaces. . . . . . . . . . . . . . . . .

www.pdfgrip.com

3
3
6
7

11
11

23


xvi

Contents
2.3 Flows . . . .
2.4 Tensors....
2.5 Differentiation
2.6 Integrals . . . .

36
45
65
77

3 Hamiltonian Systems

3.1 Canonical Transformations .
3.2 Hamilton's Equations
3.3 Constants of Motion . . . .
3.4 The Limit t ---+ ±oo . . . .
3.5 Perturbation Theory: Preliminaries .
3.6 Perturbation Theory: The Iteration.

89
89
96
105
122
145
157

4 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 . . . . . . . .

169
169
173
182
190
204

5 Relativistic Motion

5.1 The Hamiltonian Formulation of the Electrodynamic Equations
of Motion . . . . .
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

213

6 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.

265
265
267
269
271
277

213
219
226
232

237
242
248
257

Part II Classical Field Theory
7 Introduction to Classical Field Theory
7.1 Physical Aspects of Field Dynamics
7.2 The Mathematical Formalism . . .
7.3 Maxwell's and Einstein's Equations

www.pdfgrip.com

285
285
294
312


Contents

8 The Electromagnetic Field of a
Known Charge Distribution
8.1
8.2
8.3
8.4

The Stationary-Action Principle and
Conservation Theorems . .

The General Solution . . .
The Field of a Point Charge
Radiative Reaction . . . . .

xvii

329
329
340
352
370

9 The Field in the Presence of Conductors
9.1 The Superconductor . . . . . . . . .
9.2 The Half-Space, the Wave-Guide, and
the Resonant Cavity . . .
9.3 Diffraction at a Wedge ..
9.4 Diffraction at a Cylinder .

383

10 Gravitation
10.1 Covariant Differentiation and
the Curvature of Space . . . .
10.2 Gauge Theories and Gravitation
10.3 Maximally Symmetric Spaces .
10.4 Spaces with Maximally Symmetric Submanifolds
10.5 The Life and Death of Stars .
10.6 The Existence of Singularities


433

Bibliography

529

Index

539

www.pdfgrip.com

383
393
405
417

433
452
468
481
499
512


Glossary

Logical Symbols
V


3

.lI

3!

a=}b

iff

for every
there exist(s)
there does not exist
there exists a unique
if a then b
if and only if

Sets
aeA
a¢A

AUB
AnB

CA

A\B
All.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 ¢ AD
{a : a e A, 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,
beB

www.pdfgrip.com


xx

Glossary

Important Families of Sets
open sets
closed sets
measurable
sets

Borelmeasurable
sets
null sets, or
sets of
measure
zero

contains 0 and the universal set and some other specified sets,
such that the open sets are closed under union and finite
intersection
the complements of open sets
contains 0 and some other specified sets, and closed under
complementation and countable intersection
the smallest family of measurable sets that contains the open
sets
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 nonintersecting family of
subsets. a "" b means that a and b are in the same subset. An equiValence relation
has the following properties: (i) a "" a for all a; (ii) a '" b ~ b '" a; (iii) a '" b,
b""c ~a ""c.

Numbers
N
Z
IR
1R+ (IR-)

C

sup
inf
I
(a, b)
[a, b]
(a, b] and
[a, b)

JRn

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
~

x .: . x

~

This is a vector space with the scalar product

Ntimes


(YI,""

YN

I XI,···, XN)

= Li=1 YiXi

Maps (= Mappings, Functions)
f:A-+B
f(A)
I-I (b)

I-I

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 : I(a) = b}
inverse mapping to I. Warning: (1) it is not necessarily a
function, and (2) distinguish from 1/1 when B = JR.

www.pdfgrip.com


Glossary

I-I(B)

I is injective


(one-to-one)
I is surjective
(onto)
I is bijective

xxi

inverse image of B : UbeB I-I(b)
al =F a2 ~ I(al) =F l(a2)
I(A)

=B

I is injective and surjective. Only in this case is I-I a true

function
the function defined from Al x A2 to BI X B2 , so that
(alo a2) -+ (/1(al), h(a2»
II composed with h: if II : A -+ B and h : B -+ C, then
h 0 II : A -+ C so that a -+ h(/l(a»
identity map, when A = B; i.e., a -+ a. Warning: do not
1
confuse with a -+ 1 when A = B = R
I
restricted
to a subset U C A
Ilu
evaluation of the map I at the point a; i.e., I(a)
Iia

I is continuous the inverse image of any open set is open
I is measurable the inverse image of any measurable set is measurable
support of I: the smallest closed set on whose complement
suppI

1=0

cr

the set of r times continuously differentiable functions
the set of C' functions of compact (see below) support
characteristic function of A : XA(a) = 1 ...

Co
XA

Topological Concepts

topology
compact set
connected set
discrete topology
trivial topology
simply connected set
(open) neighborhood of

aeA

(open) neighborhood of
BCA

p is a point of

B

accumulation
(= cluster point) of B

B is dense in A

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 that are
both open and closed
the topology for which every set is an open set
the topology for which the only open sets are 0 and

C0

a set in which every closed path (loop) can be
continuously deformed to a point
any open subset of A containing a. Usually denoted
byUorV

any open subset of A containing B
for any neighborhood U containing p, un B \ {p} =F 13
closure of B: the smallest closed set containing B

B=A


www.pdfgrip.com


xxii

Glossary

B is nowhere dense in
A
metric (distance function) for A

separable space
homeomorphism
product topology on
Al x A2

A\

iJ is dense in A

a map d : A x A ~ lR such that d(a, a) = 0;
d(a, b) = d(b, a) > 0 for b =j:. a; and d(a, c) :::::
d(a, b) + d(b, 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) < 1/} are open
a space with a countable dense subset
a continuous bijection with a continuous inverse
the family of open sets of the form UI x U2, where UI
is open in A I and U2 is open in A 2 , and unions of
such sets


Mathematical Conventions

Ii

af/aqi
dq(t)/dt
determinant of the matrix Mij

TrM
8~,8ij

Li Mii
1 if i = j, otherwise 0
the totally antisymmetric tensor of degree m, with values
transposed matrix: (Mb)ij
Mji
Hermitian conjugate matrix: (M*)ij
(Mji)*

q(t)
det IMijl

Bi) •... ,im

M

b

M*


v· W, (v I w),
or (v· w)
v x wor
[YAW]

VI

V x f
V ·f
IIvll (in three
dimensions,
Ivl)
ds

dS
dmq
.1

II

L
dQ
Matn{lR.)
O(x)

=

±l


=

scalar (inner, dot) product
cross product
gradient of I
curl off
divergence of f
length of the vector v:

IIvll

= (Li=1 vl)I/2 = d(O,

v)

differential line element
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, LikXk stands for Lk LikXk.

www.pdfgrip.com



Glossary

xxiii

Groups

GL n
On
SOn
En
Sn
Un

group of n X n matrices with nonzero determinant
group of n x n matrices M with M Mt = 1 (unit matrix)
subgroup of 0" with determinant 1
Euclidean group
group of permutations of n elements
group of complex n x n matrices M with M M* = 1 (unit
matrix)
group of symplectic n x n matrices

Physical Symbols
mi
Xi

t=

XO je


S

qi
Pi
ei
IC

e
Ii

= hj2rr

Fap

gap
E
B

»

mass of the ith particle
Cartesian coordinates of the i th particle
time
proper time
generalized coordinates
generalized momenta
charge of the ith particle
gravitational constant
speed of light

Planck's constant divided by 2rr
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

www.pdfgrip.com


Symbols Defined in the Text

Df
(V,cI»
Tn
sn
aM
8c(q)

Tq(M)
Tq(f)
T(M)

n

T(f)
Tc}(M)
cI>*
Lx


ai

cI>Xt
rXt
W
L
H
T;(M)
e*I
df
T;s(M)

derivative of f : IR n ~ IRm
chart
n-dimensional torus
n-dimensional sphere
boundary of M
mapping of the tangent space into IRm
tangent space at the point q
derivative of f at the point q
tangent bundle
projection onto a basis
derivative of f : MJ ~ M2
set of vector fields
induced mapping on T/
Lie derivative
natural basis on the tangent space
flow
automorphism of a flow

action
Lagrangian
Hamiltonian
cotangent space
dual basis
differential of a function
space of tensors

www.pdfgrip.com

(2.1.1)
(2.1.3)
(2.1.7; 2)
(2.1.7; 2)
(2.1.20)
(2.2.1)
(2.2.4)
(2.2.7)
(2.2.12)
(2.2.15)
(2.2.17)
(2.2.19)
(2.2.21)
(2.2.25; 1), (2.5.7)
(2.2.26)
(2.3.7)
(2.3.8)
(2.3.16)
(2.3.17)
(2.3.26)

(2.4.1)
(2.4.2; 1)
(2.4.3; 1)
(2.4.4)


xxvi

Symbols Defined in the Text

®
/\

ix
*
T;(M)
g

T/(M)
Ep(M)
7r

x

T*(
<1>*
d
[]
S,w

Q

XH
b
{}

Me

1£

(I, rp)
Q±

S
du
L
K
T/a{3

y
F
A

A

ro

eili2 ...ip

Ep(M)

d
WIN

E~(U)
(e i (x) I ek(x»)

iv

*I)
f).

Lv

tensor product
wedge (outer, exterior) product
interior product
*-mapping
tensor bundle
pseudo-Riemannian metric
set of tensor fields
set of p-forms
fiber product
transposed derivative
pull-back, or inverse image of the covariant
tensors
exterior derivative
Lie bracket
canonical forms
Liouville measure
Hamiltonian vector field

bijection associated with w
Poisson brackets
generalized configuration space
Hamiltonian on Me
action-angle variables
M011er transformations
scattering matrix
differential scattering cross-section
angular momentum
boost
Minkowski space metric
1/..; I - v2 / c2 (relativistic dilatation)
electromagnetic 2-form
I-form of the potential
Lorentz transformation
Schwarzschild radius
basis of the p-forms
linear space of the p-forms
exterior differential
restriction of a form
space of m-forms with compact support
scalar product
interior product
isomorphism between Ep and Em- p
codifferential
Laplace-Beltrami operator
Lie derivative

www.pdfgrip.com

(2.4.5)
(2.4.7)
(2.4.9), (2.4.16)
(2.4.18)
(2.4.25)
(2.4.27)
(2.4.28)
(2.4.28)
(2.4.34)
(2.4.34)
(2.4.41)
(2.5.1)
(2.5.9; 6)
(3.1.1)
(3.1.2; 3)
(3.1.9)
(3.1.9)
(3.1.11)
(3.2.12)
(3.2.12)
(3.3.14)
(3.4.4)
(3.4.9)
(3.4.15)
(4.1.3)
(4.1.9)
(5.1.2)
(5.1.4; 2)
(5.1.10; 1)
(5.1.10; 1)

(5.1.12)
(5.7.1)
(7.2.3)
(7.2.5; 2)
(7.2.6)
(7.2.7; 3)
(7.2.9)
(7.2.14)
(7.2.16)
(7.2.17)
(7.2.19)
(7.2.20)
(7.2.23)


Symbols Defined in the Text
wi
k
Wik

8(x)
8(x)
8-x
Gi

E,B,F
A
A
J

Q
Tap
pa
Ta
z(s)

ta

£
W

S
D±(N)
Di
Drel(x)
Gf.:1
x
F rel
Fin
Foul
Frad
D(x)
8E
j
e(k)

S
F(z)
(I)
Sp

D
Dx

n

R

r ijk
Rijkm
Cjk

K
c

affine connection
affine connection
Heaviside step function
Dirac delta function
Dirac delta form
Green function
electric and magnetic fields
vector potential
gauge function
current
total charge
energy-momentum tensor
total energy-momentum
energy-momentum form of the field
world-line
energy-momentum form of matter

Lagrangian
action
Poynting's vector
domains of influence
Green function
retarded Green function
retarded Green function (form)
retarded field strength
incoming field strength
outgoing field strength
radiation field
D-function
energy loss per period
specified current
dielectric constant
superpotential
Fresnel's integral
scalar product
Sections
exterior covariant derivative
covariant derivative
curvature form
curvature in space-time
Christoffel symbol
Riemann-Christoffel tensor
Weyl forms
curvature parameter
rate of convergence

www.pdfgrip.com

xxvii

(7.2.25)
(7.2.25)
(7.2.31)
(7.2.31)
(7.2.33)
(7.2.35)
(7.3.1)
(7.3.7)
(7.3.10; 1)
(7.3.12)
(7.3.18; 2)
(7.3.20)
(7.3.21)
(7.3.22)
(7.3.25; 2)
(7.3.25; 2)
(8.1.1)
(8.1.1)
(8.1.13)
(8.1.15)
(8.2.5)
(8.2.7)
(8.2.7)
(8.2.9)
(8.2.15)
(8.2.15)
(8.2.21)

(8.2.22)
(8.4.4; 2)
(9.1.7)
(9.1.19; 3)
(9.1.21; 1)
(9.3.10)
(10.1.3)
(10.1.9)
(10.1.10)
(10.1.15)
(10.1.19)
(10.1.20; 1)
(10.1.36)
(10.1.44; 2)
(10.1.44; 3)
(10.4.42)
(10.6.8)


XXVlll

J+(x)

r(x)
C(x, S)
C 1(x, S)

d(A)

Symbols Defined in the Text

future of x
past of x
set of causal curves
set of differentiable causal curves
length of A

www.pdfgrip.com

(1O.6.l8( a»
(lO.6.l8(a»
(10.6.18(b»
(10.6. 18(b»
(lO.6.18(c»


Part I

Classical Dynamical Systems

www.pdfgrip.com