Geometrical Dynamics of Complex Systems: A Unified Modelling Approach to Physics, Control, Biomechanics, Neurodynamics and Psycho-Socio-Economical DynPaperback - Softcover Reprint of the Original 1st 2006, 23 August 2016
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 Killing-Riemannian 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
Multivector-Fields and Tangent-Valued Forms; 1.4 Jet Spaces; 1.4.1
Intuition Behind a Jet Space; 1.4.2 Definition of a 1-Jet Space; 1.4.3
Connections as Jet Fields; 1.4.4 Definition of a 2-Jet Space; 1.4.5
Higher-Order 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 Path-Integral Quantization; 1.5.4 Sum over
Geometries/Topologies; 1.5.5 TQFT and Stringy Path Integrals; 2 Dynamics
of High-Dimensional Nonlinear Systems; 2.1 Mechanical Systems; 2.1.1
Autonomous Lagrangian/Hamiltonian Mechanics; 2.1.2 Non-Autonomous
Lagrangian/Hamiltonian Mechanics; 2.1.3 Semi-Riemannian Geometrical
Dynamics; 2.1.4 Relativistic and Multi-Time Rheonomic Dynamics; 2.1.5
Geometrical Quantization; 2.2 Physical Field Systems; 2.2.1
n-Categorical Framework; 2.2.2 Lagrangian Field Theory on Fibre Bundles;
2.2.3 Finsler-Lagrangian Field Theory; 2.2.4 Hamiltonian Field Systems:
Path-Integral 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 Path-Integral Optimal Control of Stochastic Systems;
2.4 Human-Like Biomechanics; 2.4.1 Lie Groups and Symmetries in
Biomechanics; 2.4.2 Muscle-Driven 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
Path-Integral Model for the Cerebellum;
2.5.5 Intelligent Robot Control; 2.5.6 Brain-Like Control Functor in
Biomechanics; 2.5.7 Concurrent and Weak Functorial Machines; 2.5.8
Brain-Mind Functorial Machines; 26 Psycho-Socio-Economic Dynamics; 2.6.1
Force-Field 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 Human-Robot Interactions; 2.6.6
Nonlinear Dynamics of Complex Nets; 2.6.7 Complex Adaptive Systems:
Common Characteristics; 2.6.8 FAM Functors and Real-Life Games; 2.6.9
Riemann-Finsler 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 ri-Categories; 3.2.8 Abelian
Functorial Algebra; References; Index.