Geometrical Dynamics of Complex Systems: A Unified Modelling Approach to Physics, Control, Biomechanics, Neurodynamics and Psycho-Socio-Economical Dynamics ... and Intelligent Systems Engineering)

842 Pages·2006·7.12 MB·English

by  Vladimir G. Ivancevic

Geometrical Dynamics of Complex Systems
International Series on
MICROPROCESSOR-BASED AND
INTELLIGENTSYSTEMS ENGINEERING
VOLUME 31
Editor
Jacobs
Editorial Advisory Board
Professor C. S. Chen, University of Akron, Ohio, U.S.A.
Professor T. Fokuda, Nagoya University, Japan
Professor F. Harashima, University of Tokyo, Tokyo, Japan
Professor G. Schmidt, Technical University of Munich, Germany
Professor N. K. Sinha, McMaster University, Hamilton, Ontario, Canada
Professor D. Tabak, George Mason University, Fairfax, Virginia, U.S.A.
Professor K. Valavanis, University of Southern Louisiana, Lafayette, U.S.A.
Geometrical Dynamics of
Complex Systems
A Unified Modelling Approach to Physics,
Control, Biomechanics, Neurodynamics and
Psycho-Socio-Economical Dynamics
editedby
VLADIMIR G. IVANCEVIC
Defence Science and Technology Organisation, Adelaide, SA, Australia
and
TIJANA T. IVANCEVIC
The University of Adelaide, SA, Australia
AC.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 1-4020-4544-1 (HB)
ISBN-13 978-1-4020-4544-8 (HB)
ISBN-10 1-4020-4545-X (e-book)
ISBN-13 978-1-4020-4545-5 (e-book)
Published by Springer,
P.O. Box 17, 3300 AADordrecht, The Netherlands.
www.springer.com
Printed on acid-free paper
All Rights Reserved
© 2006 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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or otherwise, without written permission from the Publisher, with the exception
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and executed on a computer system, for exclusive use by the purchaser of the work.
Printed in the Netherlands.
Dedicated to Nitya, Atma and Kali
Contents
Preface...................................................... XI
Acknowledgments........................................... XVII
Glossaryof Frequently Used Symbols........................ XIX
1 Modern Geometrical Machinery ........................... 1
1.1 Introduction ............................................ 1
1.2 Smooth Manifolds ....................................... 9
1.2.1 Intuition Behind a Smooth Manifold................. 9
1.2.2 Definition of a Smooth Manifold .................... 11
1.2.3 Smooth Maps Between Manifolds.................... 12
1.2.4 (Co)Tangent Bundles of a Smooth Manifold .......... 15
1.2.5 Tensor Fields and Bundles of a Smooth Manifold...... 18
1.2.6 Lie Derivative on a Smooth Manifold ................ 43
1.2.7 Lie Groups and Associated Lie Algebras.............. 51
1.2.8 Lie Symmetries and Prolongations on Manifolds....... 59
1.2.9 Riemannian Manifolds ............................. 78
1.2.10 Finsler Manifolds.................................. 97
1.2.11 Symplectic Manifolds ..............................103
1.2.12 Complex and Ka¨hler Manifolds .....................108
1.2.13 Conformal Killing–Riemannian Geometry ............116
1.3 Fibre Bundles...........................................119
1.3.1 Intuition Behind a Fibre Bundle.....................119
1.3.2 Definition of a Fibre Bundle ........................120
1.3.3 Vector and Affine Bundles..........................124
1.3.4 Principal Bundles .................................134
1.3.5 Multivector–Fields and Tangent–Valued Forms........136
1.4 Jet Spaces..............................................143
1.4.1 Intuition Behind a Jet Space........................144
1.4.2 Definition of a 1–Jet Space .........................147
1.4.3 Connections as Jet Fields...........................151
1.4.4 Definition of a 2–Jet Space .........................161
1.4.5 Higher–Order Jet Spaces ...........................164
1.4.6 Jets in Mechanics .................................166
1.4.7 Jets and Action Principle...........................171
1.5 Path Integrals: Extending Smooth Geometrical Machinery ....176
VIII Contents
1.5.1 Intuition Behind a Path Integral ....................177
1.5.2 Path Integral History ..............................189
1.5.3 Standard Path–Integral Quantization ................197
1.5.4 Sum over Geometries and Topologies ................204
1.5.5 TQFT and Stringy Path Integrals ...................216
2 Dynamics of Complex Systems ............................231
2.1 Mechanical Systems .....................................231
2.1.1 Autonomous Lagrangian/Hamiltonian Mechanics......231
2.1.2 Non–Autonomous Lagrangian/Hamiltonian Mechanics .268
2.1.3 Semi–Riemannian Geometrical Dynamics.............300
2.1.4 Relativistic and Multi–Time Rheonomic Dynamics ....306
2.1.5 Geometrical Quantization ..........................314
2.2 Physical Field Systems ...................................322
2.2.1 n−Categorical Framework..........................322
2.2.2 Lagrangian Field Theory on Fibre Bundles ...........324
2.2.3 Finsler–Lagrangian Field Theory ....................337
2.2.4 Hamiltonian Field Systems: Path–Integral Quantization 338
2.2.5 Gauge Fields on Principal Connections...............355
2.2.6 Modern Geometrodynamics.........................368
2.2.7 Topological Phase Transitions and Hamiltonian Chaos .414
2.2.8 Topological String Theory ..........................422
2.2.9 Turbulence and Chaos Field Theory .................470
2.3 Nonlinear Control Systems ...............................488
2.3.1 The Basis of Modern Geometrical Control ............488
2.3.2 Geometrical Control of Mechanical Systems...........502
2.3.3 Hamiltonian Optimal Control and Maximum Principle .514
2.3.4 Path–Integral Optimal Control of Stochastic Systems ..518
2.3.5 Life: Complex Dynamics of Gene Regulatory Networks.523
2.4 Human–Like Biomechanics ...............................531
2.4.1 Lie Groups and Symmetries in Biomechanics..........532
2.4.2 Muscle–Driven Hamiltonian Biomechanics ............545
2.4.3 Biomechanical Functors ............................550
2.4.4 Biomechanical Topology............................564
2.5 Neurodynamics .........................................587
2.5.1 Microscopic Neurodynamics and Quantum Brain ......587
2.5.2 Macroscopic Neurodynamics ........................600
2.5.3 Oscillatory Phase Neurodynamics ...................618
2.5.4 Neural Path–Integral Model for the Cerebellum .......623
2.5.5 Intelligent Robot Control...........................629
2.5.6 Brain–Like Control Functor in Biomechanics..........631
2.5.7 Concurrent and Weak Functorial Machines ...........643
2.5.8 Brain–Mind Functorial Machines ....................655
2.6 Psycho–Socio–Economic Dynamics.........................662
2.6.1 Force–Field Psychodynamics........................662
Contents IX
2.6.2 Geometrical Dynamics of Human Crowd .............676
2.6.3 Dynamical Games on Lie Groups....................680
2.6.4 Nonlinear Dynamics of Option Pricing ...............687
2.6.5 Command/Control in Human–Robot Interactions .....697
2.6.6 Nonlinear Dynamics of Complex Nets................700
2.6.7 Complex Adaptive Systems: Common Characteristics ..702
2.6.8 FAM Functors and Real–Life Games.................705
2.6.9 Riemann–Finsler Approach to Information Geometry ..712
3 Appendix: Tensors and Functors...........................723
3.1 Elements of Classical Tensor Analysis ......................723
3.1.1 Transformation of Coordinates and Elementary Tensors 723
3.1.2 Euclidean Tensors .................................729
3.1.3 Tensor Derivatives on Riemannian Manifolds..........730
3.1.4 Tensor Mechanics in Brief ..........................735
3.1.5 The Covariant Force Law in Robotics and Biomechanics744
3.2 Categories and Functors..................................746
3.2.1 Maps ............................................747
3.2.2 Categories........................................758
3.2.3 Functors .........................................761
3.2.4 Natural Transformations ...........................763
3.2.5 Limits and Colimits ...............................766
3.2.6 The Adjunction ...................................767
3.2.7 n−Categories .....................................768
3.2.8 Abelian Functorial Algebra .........................775
References.....................................................777
Index..........................................................813
Preface
Geometrical Dynamics of Complex Systems is a graduate–level monographic
textbook.Itrepresentsacomprehensiveintroductionintorigorousgeometrical
dynamicsofcomplexsystemsofvariousnatures.By‘complexsystems’,inthis
book are meant high–dimensional nonlinear systems, which can be (but not
necessarily are) adaptive. This monograph proposes a unified geometrical ap-
proachtodynamicsofcomplexsystemsofvariouskinds:engineering,physical,
biophysical,psychophysical,sociophysical,econophysical,etc.Astheir names
suggest, all these multi–input multi–output (MIMO) systems have something
incommon:theunderlyingphysics.However,insteadofdealingwiththepop-
ular ‘soft complexity philosophy’,1 we rather propose a rigorous geometrical
and topological approach. We believe that our rigorous approach has much
greater predictive power than the soft one. We argue that science and tech-
nology is all about prediction and control. Observation, understanding and
explanationareimportantineducationatundergraduatelevel,butafterthat
it should be all prediction and control. The main objective of this book is to
show that high–dimensional nonlinear systems and processes of ‘real life’ can
be modelled and analyzed using rigorous mathematics, which enables their
complete predictability and controllability, as if they were linear systems.
It is well–known that linear systems, which are completely predictable
and controllable by definition – live only in Euclidean spaces (of various di-
mensions). They are as simple as possible, mathematically elegant and fully
elaboratedfromeitherscientificorengineeringside.However,innature,noth-
ing is linear. In reality, everything has a certain degree of nonlinearity, which
means:unpredictability,withsubsequentuncontrollability.So,oursimpleand
elegant linear systems, that cover almost all of our university textbooks in
1 Itiswell–knownthatthe‘softcomplexityphilosophy’,whichhasbeenproclaimed
and developed in the famous Santa Fe Institute, actually advocates ‘simplicity’
bymeansof‘reduction’underthename‘complexity’.Assuch,itisverydifferent
fromthe‘generalsystemtheory’whichstatesthatacomplexsystemismorethan
a sum of its components.