Geometrical Dynamics of Complex Systems
A Unified Modelling Approach to Physics, Control, Biomechanics, Neurodynamics and Psycho-Socio-Economical Dynamics
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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 K¿er Manifolds; 1.2.13 Conformal Killing¿Riemannian Geometry; ...
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 K¿er 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.
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