Applied differential geometry. A modern introduction PDF
Preview Applied differential geometry. A modern introduction
APPLIED DIFFERENTIAL GEOMETRY A Modern Introduction TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk APPLIED DIFFERENTIAL GEOMETRY A Modern Introduction Vladimir G Ivancevic Defence Science and Technology Organisation, Australia Tijana T Ivancevic The University of Adelaide, Australia World Scientific N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I (cid:13)(cid:10) 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. APPLIED DIFFERENTIAL GEOMETRY A Modern Introduction 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-13 978-981-270-614-0 ISBN-10 981-270-614-3 Printed in Singapore. Rhaimie - AppliedDifferential.pmd 1 6/8/2007, 2:41 PM April19,2007 16:57 WSPC/BookTrimSizefor9inx6in ApplDifGeom Dedicated to: Nitya, Atma and Kali v April19,2007 16:57 WSPC/BookTrimSizefor9inx6in ApplDifGeom TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk April19,2007 16:57 WSPC/BookTrimSizefor9inx6in ApplDifGeom Preface Applied Differential Geometry: A Modern Introduction is a graduate–level monographictextbook. Itisdesignedasacomprehensiveintroductioninto methods and techniques of modern differential geometry with its various physical and non–physical applications. In some sense, it is a continuation of our previous book, Natural Biodynamics (World Scientific, 2006), which containsallthenecessarybackgroundforcomprehensivereadingofthecur- rent book. While the previous book was focused on biodynamic applica- tions,thecoreapplicationsofthenewbookareintherealmofmoderntheo- reticalphysics,mainlyfollowingitscentralline: Einstein–Feynman–Witten. Other applications include (among others): control theory, robotics, neu- rodynamics, psychodynamics and socio–economical dynamics. The book has six chapters. Each chapter contains both ‘pure mathe- matics’ and related ‘applications’ labelled by the word ‘Application’. Thefirstchapterprovidesasoft(‘plain–English’)introductionintoman- ifoldsandrelatedgeometricalstructures,foralltheinterestedreaderswith- out the necessary background. As a ‘snap–shot’ illustration, at the end of the first chapter, a paradigm of generic differential–geometric modelling is given, which is supposed to fit all above–mentioned applications. Thesecondchaptergivestechnicalpreliminariesfordevelopmentofthe modernapplieddifferentialgeometry. Thesepreliminariesinclude: (i)clas- sical geometrical objects – tensors, (ii) both classical and modern physical objects – actions, and modern geometrical objects – functors. Thethirdchapterdevelopsmodernmanifoldgeometry,togetherwithits main physical and non–physical applications. This chapter is a neccessary background for comprehensive reading of the remaining chapters. Thefourthchapterdevelopsmodernbundle geometry, togetherwithits main physical and non–physical applications. vii April19,2007 16:57 WSPC/BookTrimSizefor9inx6in ApplDifGeom viii Applied Differential Geometry: A Modern Introduction The fifth chapter develops modern jet bundle geometry, together with its main applications in non–autonomous mechanics and field physics. All material in this chapter is based on the previous chapter. ThesixthchapterdevelopsmoderngeometricalmachineryofFeynman’s path integrals, together with their various physical and non–physical appli- cations. For most of this chapter, only the third chapter is a neccessary background, assuming a basic understanding of quantum mechanics (as provided in the above–mentioned World Scientific book, Natural Biody- namics). The book contains both an extensive Index (which allows easy connec- tions between related topics) and a number of cited references related to modern applied differential geometry. Our approach to dynamics of complex systems is somewhat similar to the approach to mathematical physics used at the beginning of the 20th Century by the two leading mathematicians: David Hilbert and John von Neumann–theapproachofcombiningmathematicalrigorwithconceptual clarity, or geometrical intuition that underpins the rigor. The intended audience includes (but is not restricted to) theoreti- cal and mathematical physicists; applied and pure mathematicians; con- trol, robotics and mechatronics engineers; computer and neural scientists; mathematically strong chemists, biologists, psychologists, sociologists and economists – both in academia and industry. Compared to all differential–geometric books published so far, Applied Differential Geometry: A Modern Introduction has much wider variety of both physical and non–physical applications. After comprehensive read- ing of this book, a reader should be able to both read and write journal papersinsuchdiversefieldsassuperstring&topologicalquantumfieldthe- ory,nonlineardynamics&control,robotics,biomechanics,neurodynamics, psychodynamics and socio–economical dynamics. V. Ivancevic Defence Science & Technology Organisation, Australia e-mail: [email protected] T. Ivancevic School of Mathematics, The University of Adelaide e-mail: [email protected] Adelaide May, 2006 April19,2007 16:57 WSPC/BookTrimSizefor9inx6in ApplDifGeom ix Acknowledgments The authors wish to thank Land Operations Division, Defence Science & Technology Organisation, Australia, for the support in developing the HumanBiodynamicsEngine (HBE)andallHBE–relatedtextinthismono- graph. Finally, we express our gratitude to the World Scientific Publishing Company, and especially to Ms. Zhang Ji and Mr. Rhaimie Wahap.
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