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| Preface | |
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| Acknowledgments | |
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| Glossary of Frequently Used Symbols | |
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| Modern Geometrical Machinery | |
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| Introduction | |
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| Smooth Manifolds | |
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| Intuition Behind a Smooth Manifold | |
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| Definition of a Smooth Manifold | |
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| Smooth Maps Between Manifolds | |
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| (Co)Tangent Bundles of a Smooth Manifold | |
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| Tensor Fields and Bundles of a Smooth Manifold | |
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| Lie Derivative on a Smooth Manifold | |
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| Lie Groups and Associated Lie Algebras | |
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| Lie Symmetries and Prolongations on Manifolds | |
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| Riemannian Manifolds | |
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| Finsler Manifolds | |
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| Symplectic Manifolds | |
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| Complex and K�hler Manifolds | |
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| Conformal Killing-Riemannian Geometry | |
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| Fibre Bundles | |
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| Intuition Behind a Fibre Bundle | |
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| Definition of a Fibre Bundle | |
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| Vector and Affine Bundles | |
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| Principal Bundles | |
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| Multivector-Fields and Tangent-Valued Forms | |
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| Jet Spaces | |
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| Intuition Behind a Jet Space | |
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| Definition of a 1-Jet Space | |
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| Connections as Jet Fields | |
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| Definition of a 2-Jet Space | |
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| Higher-Order Jet Spaces | |
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| Jets in Mechanics | |
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| Jets and Action Principle | |
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| Path Integrals: Extending Smooth Geometrical Machinery | |
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| Intuition Behind a Path Integral | |
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| Path Integral History | |
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| Standard Path-Integral Quantization | |
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| Sum over Geometries and Topologies | |
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| TQFT and Stringy Path Integrals | |
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| Dynamics of Complex Systems | |
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| Mechanical Systems | |
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| Autonomous Lagrangian/Hamiltonian Mechanics | |
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| Non-Autonomous Lagrangian/Hamiltonian Mechanics | |
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| Semi-Riemannian Geometrical Dynamics | |
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| Relativistic and Multi-Time Rheonomic Dynamics | |
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| Geometrical Quantization | |
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| Physical Field Systems | |
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| n-Categorical Framework | |
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| Lagrangian Field Theory on Fibre Bundles | |
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| Finsler-Lagrangian Field Theory | |
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| Hamiltonian Field Systems: Path-Integral Quantization | |
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| Gauge Fields on Principal Connections | |
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| Modern Geometrodynamics | |
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| Topological Phase Transitions and Hamiltonian Chaos | |
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| Topological String Theory | |
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| Turbulence and Chaos Field Theory | |
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| Nonlinear Control Systems | |
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| The Basis of Modern Geometrical Control | |
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| Geometrical Control of Mechanical Systems | |
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| Hamiltonian Optimal Control and Maximum Principle | |
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| Path-Integral Optimal Control of Stochastic Systems | |
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| Life: Complex Dynamics of Gene Regulatory Networks | |
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| Human-Like Biomechanics | |
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| Lie Groups and Symmetries in Biomechanics | |
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| Muscle-Driven Hamiltonian Biomechanics | |
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| Biomechanical Functors | |
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| Biomechanical Topology | |
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| Neurodynamics | |
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| Microscopic Neurodynamics and Quantum Brain | |
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| Macroscopic Neurodynamics | |
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| Oscillatory Phase Neurodynamics | |
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| Neural Path-Integral Model for the Cerebellum | |
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| Intelligent Robot Control | |
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| Brain-Like Control Functor in Biomechanics | |
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| Concurrent and Weak Functorial Machines | |
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| Brain-Mind Functorial Machines | |
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| Psycho-Socio-Economic Dynamics | |
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| Force-Field Psychodynamics | |
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| Geometrical Dynamics of Human Crowd | |
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| Dynamical Games on Lie Groups | |
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| Nonlinear Dynamics of Option Pricing | |
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| Command/Control in Human-Robot Interactions | |
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| Nonlinear Dynamics of Complex Nets | |
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| Complex Adaptive Systems: Common Characteristics | |
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| FAM Functors and Real-Life Games | |
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| Riemann-Finsler Approach to Information Geometry | |
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| Appendix: Tensors and Functors | |
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| Elements of Classical Tensor Analysis | |
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| Transformation of Coordinates and Elementary Tensors | |
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| Euclidean Tensors | |
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| Tensor Derivatives on Riemannian Manifolds | |
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| Tensor Mechanics in Brief | |
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| The Covariant Force Law in Robotics and Biomechanics | |
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| Categories and Functors | |
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| Maps | |
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| Categories | |
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| Functors | |
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| Natural Transformations | |
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| Limits and Colimits | |
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| The Adjunction | |
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| n-Categories | |
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| Abelian Functorial Algebra | |
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| References | |
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| Index | |