Geometrical Dynamics of Complex Systems

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Unified to complex systems of various natures

Design of hyper-complex robots of the future

Predictive power of geometry and topology in real life

Path integrals in biophysics, psychophysics, sociophysics and econophysics


From the reviews:

"As it is mentioned in the preface this is 'a graduate-level monographical textbook. It represents a comprehensive introduction into rigorous geometrical dynamics of complex systems of various natures'. ... This book has to be recommended for graduates in applied mathematics who are interested in basics of modern mathematical methods mostly based on geometry." (Iskander A. Taimanov, Zentralblatt MATH, Vol. 1092 (18), 2006)


Modern Geometrical Machinery; 1 .1 Introduction; 1 .2 Smooth Manifolds; 1.2.1 Intuition Behind a Smooth Manifold; 1.2.2 Definition of a Smooth Manifold;
1.2.3 Smooth Maps Between Manifolds; 1.2.4 (Co)Tangent Bundles of a Smooth Manifold; 1.2.5 Tensor Fields and Bundles of a Smooth Manifold; 1.2.6 Lie Derivative on a Smooth Manifold; 1.2.7 Lie Groups and Associated Lie Algebras; 1.2.8 Lie Symmetries and Prolongations on Manifolds ;1.2.9 Riemannian Manifolds; 1.2.10 Finsler Manifolds; 1.2.11 Symplectic Manifolds; 1.2.12 Complex and Kähler Manifolds; 1.2.13 Conformal KillingRiemannian Geometry; 1.3 Fibre Bundles; 1.3.1 Intuition Behind a Fibre Bundle; 1.3.2 Definition of a Fibre Bundle ;1.3.3 Vector and Affine Bundles; 1.3.4 Principal Bundles; 1.3.5 MultivectorFields and TangentValued Forms; 1.4 Jet Spaces; 1.4.1 Intuition Behind a Jet Space ; 1.4.2 Definition of a 1Jet Space; 1.4.3 Connections as Jet Fields; 1.4.4 Definition of a 2Jet Space; 1.4.5 HigherOrder Jet Spaces; 1.4.6 Jets in Mechanics;1.4.7 Jets and Action Principle; 1.5 Path Integrals: Extending Smooth Geometrical Machinery; 1.5.1 Intuition Behind a Path Integral; 1.5.2 Path Integral History; 1.5.3 Standard PathIntegral Quantization; 1.5.4 Sum over Geometries/Topologies; 1.5.5 TQFT and Stringy Path Integrals; 2 Dynamics of HighDimensional Nonlinear Systems; 2.1 Mechanical Systems; 2.1.1 Autonomous Lagrangian/Hamiltonian Mechanics; 2.1.2 NonAutonomous Lagrangian/Hamiltonian Mechanics; 2.1.3 SemiRiemannian Geometrical Dynamics; 2.1.4 Relativistic and MultiTime Rheonomic Dynamics; 2.1.5 Geometrical Quantization; 2.2 Physical Field Systems; 2.2.1 nCategorical Framework; 2.2.2 Lagrangian Field Theory on Fibre Bundles; 2.2.3 FinslerLagrangian Field Theory; 2.2.4 Hamiltonian Field Systems: PathIntegral Quantization; 2.2.5 Gauge Fields on Principal Connections; 2.2.6 Modern Geometrodynamics; 2.2.7 Topological Phase Transitions and Hamiltonian Chaos; 2.2.8 Topological Superstring Theory; 2.2.9 Turbulence and Chaos Field Theory; 2.3 Nonlinear Control Systems; 2.3.1 The Basis of Modern Geometrical Control;2.3.2 Geometrical Control of Mechanical Systems ;2.3.3 Hamiltonian Optimal Control and Maximum Principle; 2.3.4 PathIntegral Optimal Control of Stochastic Systems; 2.4 HumanLike Biomechanics; 2.4.1 Lie Groups and Symmetries in Biomechanics; 2.4.2 MuscleDriven Hamiltonian Biomechanics; 2.4.3 Biomechanical Functors; 2.4.4 Biomechanical Topology; 2.5 Neurodynamics; 2.5.1 Microscopic Neurodynamics and Quantum Brain; 2.5.2 Macroscopic Neurodynamics; 2.5.3 Oscillatory Phase Neurodynamics ;2.5.4 Neural PathIntegral Model for the Cerebellum;
2.5.5 Intelligent Robot Control; 2.5.6 BrainLike Control Functor in Biomechanics; 2.5.7 Concurrent and Weak Functorial Machines; 2.5.8 BrainMind Functorial Machines; 26 PsychoSocioEconomic Dynamics; 2.6.1 ForceField Psychodynamics; 2.6.2 Geometrical Dynamics of Human Crowd; 2.6.3 Dynamical Games on Lie Groups ; 2.6.4 Nonlinear Dynamics of Option Pricing; 2.6.5 Command/Control in HumanRobot Interactions; 2.6.6 Nonlinear Dynamics of Complex Nets; 2.6.7 Complex Adaptive Systems: Common Characteristics; 2.6.8 FAM Functors and RealLife Games; 2.6.9 RiemannFinsler Approach to Information Geometry; 3 Appendix: Tensors and Functors; 3.1 Elements of Classical Tensor Analysis; 3.1.1 Transformation of Coordinates and Elementary Tensors; 3.1.2 Euclidean Tensors; 3. 1 .3 Tensor Derivatives on Riemannian Manifolds; 3.1.4 Tensor Mechanics in Brief ; 3.1.5 The Covariant Force Law in Robotics and Biomechanics; 3.2 Categories and Functors; 3.2.1 Maps; 3.2.2 Categories; 3.2.3 Functors; 3.2.4 Natural Transformations; 3.2.5 Limits and Colimits; 3.2.6 The Adjunction; 3.2.7 riCategories; 3.2.8 Abelian Functorial Algebra; References; Index.

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Geometrical Dynamics of Complex Systems
A Unified Modelling Approach to Physics, Control, Biomechanics, Neurodynamics and Psycho-Socio-Economical Dynamics
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Springer Netherlands
Anzahl Seiten
H235mm x B155mm x T45mm
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