Continuous symmetries, Lie algebras, differential equations, and computer algebra

473 Pages·2007·1.37 MB·English

by  Steeb Willi-hans

CONTINUOUS SYMMETRIES,
LIE ALGEBRAS,
DIFFERENTIAL EQUATIONS
AND COMPUTER ALGEBRA
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CONTINUOUS SYMMETRIES,
LIE ALGEBRAS,
DIFFERENTIAL EQUATIONS
AND COMPUTER ALGEBRA
2nd Edition
Willi-Hans Steeb
University of Iohannesburg, South Africa
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World Scientific
- - - -
NEW JERSEY LONDON SINGAPORE BElJlNG SHANGHAI HONG KONG TAIPEI CHENNAI
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
CONTINUOUS SYMMETRIES, LIE ALGEBRAS, DIFFERENTIAL EQUATIONS
AND COMPUTER ALGEBRA
Second Edition
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN-13 978-981-270-809-0
ISBN-10 981-270-809-X
Printed in Singapore.
LaiFun - Continuous Symmetries.pmd 1 6/20/2007, 10:49 AM
Preface
Thepurposeofthisbookistoprovideacomprehensiveintroductiontothe
application of continuous symmetries and their Lie algebras to ordinary
and partial di(cid:11)erential equations. The study of symmetries of di(cid:11)erential
equations provides important information about the behaviour of di(cid:11)eren-
tial equations. The symmetries can be used to (cid:12)nd exact solutions. They
can be applied to verify and develop numerical schemes. One can also ob-
tain conservation laws of a given di(cid:11)erential equation with the help of the
continuoussymmetries. Gaugetheoryisalsobasedonthecontinuoussym-
metries of certain relativistic (cid:12)eld equations.
Apartfromthestandardtechniquesinthestudyofcontinuoussymmetries,
thebookincludes: thePainlev(cid:19)etestandsymmetries,invertiblepointtrans-
formationandsymmetries,Liealgebravalueddi(cid:11)erentialforms,gaugethe-
ory,Yang-Millstheoryandchaos,self-dualYang-Millsequationandsoliton
equations, Ba(cid:127)cklund transformation, Lax representation, Bose operators
and symmetries, Hirota technique, Sato’s theory, discrete systems and in-
variants and string theory.
Each chapter includes computer algebra applications. Examples are the
(cid:12)nding of the determining equation for the Lie symmetries, (cid:12)nding the
curvature for a given metric tensor (cid:12)eld and calculating the Killing vector
(cid:12)elds for a metric tensor (cid:12)eld. Each chapter also includes exercises.
The book is suitable for use by students and research workerswhose main
interestliesin(cid:12)ndingsolutionsofdi(cid:11)erentialequations. Itthereforecaters
for readersprimarily interested in applied mathematics and physics rather
thanpuremathematics. Thebookprovidesanapplicationfocusedtextthat
is self-contained. A large number of worked examples have been included
in the text to help the readers working independently of a teacher. The
advance of algebraic computation has made it possible to write programs
v
vi Preface
for the tedious calculations in this research (cid:12)eld. Thus the last chapter
gives a survey on computer algebra packages. Each chapter also includes
useful SymbolicC++ programs.
End of proofs are indicated by . End of examples are indicated by .
(cid:127) |
I wish to express my gratitude to Yorick Hardy for discussion on this re-
search (cid:12)eld and the support for SymbolicC++.
Any useful suggestions and comments are welcome.
ThebookcoversthecourseonLiegroupsandLiealgebrasprovidedbythe
InternationalSchool for Scienti(cid:12)c Computing. If you are interest in such a
course please contact the author.
The header (cid:12)les for SymbolicC++ and example programs can be down-
loaded from the home page of the author:
http://issc.uj.ac.za
Email addresses of the author:
[email protected]
[email protected]
[email protected]
Contents
1 Introduction 1
2 Groups 7
2.1 De(cid:12)nitions and Examples . . . . . . . . . . . . . . . . . . . 7
2.2 Computer Algebra Applications . . . . . . . . . . . . . . . . 17
2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Lie Groups 21
3.1 De(cid:12)nitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Concepts for Lie Groups . . . . . . . . . . . . . . . . . . . . 24
3.3 Haar Measure . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Computer Algebra Applications . . . . . . . . . . . . . . . . 32
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Lie Transformation Groups 35
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.1 A(cid:14)ne Groups . . . . . . . . . . . . . . . . . . . . . . 45
4.2.2 Group Actions and Group Representations . . . . . 48
4.3 Computer Algebra Applications . . . . . . . . . . . . . . . . 53
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 In(cid:12)nitesimal Transformations 57
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 In(cid:12)nitesimal Generators . . . . . . . . . . . . . . . . . . . . 61
5.3 Multi-Parameter Lie Groups of Transformations . . . . . . 66
5.4 Computer Algebra Applications . . . . . . . . . . . . . . . . 70
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
vii
viii Contents
6 Lie Algebras 75
6.1 De(cid:12)nition and Examples . . . . . . . . . . . . . . . . . . . . 75
6.2 Lie Algebras and Bose Operators . . . . . . . . . . . . . . . 78
6.3 Lie Algebras and Fermi Operators . . . . . . . . . . . . . . 78
6.4 Lie Algebras and Lie Groups . . . . . . . . . . . . . . . . . 79
6.5 Classical Lie Algebras . . . . . . . . . . . . . . . . . . . . . 84
6.6 Important Concepts . . . . . . . . . . . . . . . . . . . . . . 88
6.7 Adjoint Representation. . . . . . . . . . . . . . . . . . . . . 93
6.8 Casimir Operators . . . . . . . . . . . . . . . . . . . . . . . 96
6.9 Cartan-Weyl Basis . . . . . . . . . . . . . . . . . . . . . . . 97
6.10 Computer Algebra Applications . . . . . . . . . . . . . . . . 99
6.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7 Introductory Examples 103
7.1 One-Dimensional Linear Wave Equation . . . . . . . . . . . 103
7.2 One-Dimensional Di(cid:11)usion Equation . . . . . . . . . . . . . 107
7.3 Stationary Flow. . . . . . . . . . . . . . . . . . . . . . . . . 108
7.4 Gauge Transformation . . . . . . . . . . . . . . . . . . . . . 111
7.5 Computer Algebra Applications . . . . . . . . . . . . . . . . 118
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8 Di(cid:11)erential Forms and Tensor Fields 121
8.1 Vector Fields and Tangent Bundles . . . . . . . . . . . . . . 121
8.2 Di(cid:11)erential Forms and Cotangent Bundles . . . . . . . . . . 123
8.3 Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . 127
8.4 Pull-Back Maps . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.5 Interior Product or Contraction . . . . . . . . . . . . . . . . 130
8.6 Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . 131
8.7 Hodge Star Operator . . . . . . . . . . . . . . . . . . . . . . 134
8.8 Pseudospherical Surfaces . . . . . . . . . . . . . . . . . . . . 138
8.9 Computer Algebra Applications . . . . . . . . . . . . . . . . 139
8.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
9 Lie Derivative and Invariance 145
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2 De(cid:12)nitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.3 Invariance and Conformal Invariance . . . . . . . . . . . . . 149
9.4 Computer Algebra Applications . . . . . . . . . . . . . . . . 159
9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160