Classical Dynamical Systems

271 Pages·1978·5.881 MB·English

by  Dr. Walter Thirring (auth.)

Walter Thirring
A Course
in Mathematical Physics
1
Classical Dynamical Systems
Translated by Evans M. Harrell
lS]
Springer Science+Business Media, LLC
Dr. WalterThirring Dr. EvansHarrell
InstituteforTheereticalPhysics MassachusettsInstituteofTechnology
UniversityofVienna Cambridge,Massachusetts
Austria USA
LibraryofCongressCataloging inPublication Data
Thirring, WalterE 1927-
Acourse inmathematical physics.
TranslationofLehrbuchdermathematischen Physik.
Bibliography:p.
Includes index,
CONTENTS:[v.]1.Classical dynamical systems.
1.Mathematical physics. I. Title.
QC20.T4513 530.\'5 78-16172
With58Figures
Allrightsreserved.
Nopartofthisbookmaybetranslatedorreproduced
inanyformwithoutwrittenpermissionfromSpringer-Verlag.
Copyright© 1978by SpringerScience+BusinessMedia New York
Originallypublishedby Springer-VerlagNew YorkInc. in 1978.
Softcoverreprintofthehardcover 1stedition 1978
987654321
ISBN 978-3-662-38942-3 ISBN 978-3-662-39892-0(eBook)
DOI 10.1007/978-3-662-39892-0
Preface
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 intro
ductory 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 pre
dictions 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 rela
tivistic quantum theory, have not yet matured from the stage of rules for
calculations to mathematically well understood disciplines, and are there
fore 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 mathe
matics. Some terminology of the relevant mathematical background is
11l
IV Preface
collected in the glossary at the beginning. More specialized tools are intro
duced 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 treat
ment 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
Glossary vii
Symbols Defined in the Text
Xl
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 Integration 66
3 Hamiltonian Systems 76
3.1 Canonical Transformations 76
3.2 Hamilton's Equations 83
3.3 Constants of Motion 92
3.4 The Limit t -> I ± 00 108
3.5 Perturbation Theory: Preliminaries 124
3.6 Perturbation Theory: The Iteration 133
V
vi Contents
4 Nonrelativistic Motion 142
4.1 Free Particles 142
4.2 The Two-Body Problem 146
4.3 The Problem of Two Centers of Force 154
4.4 The Restricted Three-Body Problems 163
4.5 The N-body Problem 176
5 Relativistic Motion 185
5.1 The Hamiltonian Formulation of the Electrodynamic
Equation of Motion 185
5.2 The Constant Field 190
5.3 The Coulomb Field 197
5.4 The Betatron 203
5.5 The Traveling Plane Disturbance 208
5.6 Relativistic Motion in a Gravitational Field 214
5.7 Motion in the Schwarzschild Field 220
5.8 Motion in a Gravitational Plane Wave 229
6 The Structure of Space and Time 235
6.1 The Homogeneous Universe 235
6.2 The Isotropic Universe 237
6.3 Me according to Galileo 239
6.4 Me as Minkowski Space 241
6.5 M. as a Pseudo-Riemannian Space 248
References 251
Index 255
Glossary
Logical Symbols
V for every
j there exist(s)
~ there does not exist
j! there exists a unique
a => b if a then b
iff if and only if
Sets
aEA a is an element of A
a¢A a is not an element of A
AuB union of A and B
AnB intersection A and B
CA complement of A (In a larger set B: {a: a E B, a ¢ A})
A\B {a:aEA,a¢B}
A~B symmetric difference of A and B: (A\B) u (B\A)
o
empty set
C0 universal set
AxB 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
vii
Vlll Glossary
closed sets the complements of open sets
measurable 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 prop
erties: i) a - a for all a. ii) a - b => b - a. iii) a - b, b - c => a-c.
Numbers
N natural numbers
71. integers
IR real numbers
IR+(IR-) positive (negative) numbers
IC complex numbers
sup supremum, or lowest upper bound
inf infimum, or greatest lower bound
I any open interval
(a, b) the open interval from a to b
[a,b] the closed interval from a to b
(a,b] and [a, b) half-open intervals from a to b
IR" IRx···xlR
'-.,-'
!Ii times
This is a vector space with the scalar product
(YI'···' YNlxl'···' xN) = Li= I YiXi
Maps ( = Mappings, Functions)
f:A -> B for every a E A an element f(a) E B is specified
f(A) image of A, i.e., if f:A -> B, {f(a) E B:a E A}
rl(b) inverse image of b, i.e. {a E A: f(a) = b}
r l inverse mapping to f. Warning: 1) it is not necessarily a
function 2) distinguish from I/f when B = IR
rl(B) inverse image of B: UbEB f-I(b)
f is injective (one-to-one) al # a2 => f(al) # f(a2)
f is surjective (onto) f(A) = B
f is bijective f is injective and surjective. Only in this case is f -I a
true function
the function defined from Al x A2 to BI X B2, so that
(aI' a2) -> (f1(al),fia2»
fl composed withf2: iffl:A -> B andf2:B -> C, then
f2 0 fl:A -> C so that a -> f2(fI(a»
Glossary ix
1 identity map, when A = B; i.e., a -+ a. Warning: do
not confuse with a -+ 1 when A = B = R
Ilu I restricted to a subset V c A
Iia evaluation of the map I at the point a; i.e.,f(a).
I is continuous the inverse image of any open set is open
I is measurable the inverse image of any measurable set is measurable
suppi support of f: the smallest closed set on whose
complement I = 0
C' the set of r times continuously differentiable functions
C:; the set of C functions of compact (see below) support
XA characteristic function of
A:xia) = I. ..
Topological Concepts
Topology any family of open sets, as defined above
compact set a set for which any covering with open sets has
a finite subcovering
connected set a set for which there are no proper subsets which
are both open and closed
discrete topology the topology for which every set is an open set
trivial topology the topology for which the only open sets are
0andC0
simply connected set a set in which every path can be continuously
deformed to a point
(open) neighborhood of a E A any open subset of A containing a. Usually
denoted by V or V.
(open) neighborhood of B c A any open subset of A containing B
p is a point of accumulation for any neighborhood V containing p,
(= cluster point) V n B of. {p} or 0
B closure of B: the smallest closed set containing B
B is dense in A B=A
B is nowhere dense in A A\B is dense in A
metric (distance function) for A a map d: A x A -+ IR such that dCa, a) = 0;
dCa, b) = deb, a) > 0 for b of. a; and
dCa, c) S 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) < '1} are open.
separable space a space with a countable dense subset
homeomorphism a continuous bijection with a continuous inverse
product topology on AI x A2 the family of open sets of the form V I X V 2,
where VI is open in AI and V2 is open in A2,
and unions of such sets
Mathematical Conventions
f,i
q(t)