Contents -- Preface -- Glossary of Frequently Used Symbols -- 1. Introduction -- 1.1 The Problem of Natural Biodynamics -- 1.2 A Brief History of Biodynamics -- 1.3 Mechanical Basis of Biodynamics -- 1.3.1 Natural Galilei Group -- 1.3.2 Newtonian Equations of Motion -- 1.3.3 Calculus of Variations -- 1.3.4 Lagrangian Equations of Motion -- 1.3.5 Hamiltonian Equations of Motion -- 1.3.6 Lagrangian Flows on Biodynamic Manifolds -- 1.4 Conservative versus Dissipative Hamiltonian Dynamics -- 1.4.1 Dissipative Systems -- 1.4.2 Thermodynamic Equilibrium -- 1.4.3 Nonlinearity -- 1.4.4 The Second Law of Thermodynamics -- 1.4.5 Geometry of Phase Space -- 1.5 Neural Basis of Biodynamics -- 2. Natural Language of Biodynamics -- 2.1 Categorical Metalanguage -- 2.1.1 Preliminaries from Calculus, Algebra and Topology -- 2.1.1.1 Notes From Calculus -- 2.1.1.2 Notes from Set Theory -- 2.1.1.3 Notes from General Topology -- 2.1.1.4 Commutative Diagrams -- 2.1.1.5 Groups and Related Algebraic Structures -- 2.1.2 Categories -- 2.1.3 Functors -- 2.1.4 Natural Transformations -- 2.1.5 Limits and Colimits -- 2.1.6 The Adjunction -- 2.1.7 n-Categories -- 2.1.7.1 Generalization to 'Big' n-Categories -- 2.1.7.2 Topological Structure of n-Categories -- 2.1.8 Algebra in Abelian Categories -- 2.1.9 Fundamental Biodynamic Adjunction -- 2.2 The Basics of Dynamics -- 2.2.1 Ordinary Differential Equations -- 2.2.2 Linear Autonomous Dynamics -- 2.2.2.1 The Flow of a Linear ODE -- 2.2.2.2 Canonical Linear Flows in R2 -- 2.2.2.3 Topological Equivalence -- 2.3 Chaos and Synergetics in Biodynamics -- 2.3.1 Prototype of Chaotic and Synergetic Systems -- 2.3.2 Chaotic Systems and Biomorphs -- 2.3.2.1 Simulation Examples: Chaotic Systems -- 2.3.2.2 Simulation Examples: Biomorphic Systems -- 2.3.3 Controlling Chaos within the Chaos Theory -- 2.3.3.1 Exploiting Critical Sensitivity.

2.3.3.2 Lyapunov exponents and KY-dimension -- 2.3.3.3 Kolmogorov-Sinai entropy -- 2.3.3.4 Chaos Control by Ott, Grebogi and Yorke -- 2.3.3.5 Floquet Stability Analysis and OGY Control -- 2.3.3.6 Jerk Functions of Simple Chaotic Flows -- 2.3.4 The Basic Hamiltonian Model of Biodynamics -- 2.3.5 The Basics of Haken's Synergetics -- 2.3.5.1 Phase Transitions -- 2.3.5.2 Mezoscopic Derivation of Order Parameters -- 2.3.6 Macro-Synergetic Control of Biodynamics -- 3. Natural Geometry of Biodynamics -- 3.1 Motivation for Geometry in Biodynamics -- 3.2 Biodynamic Manifold M -- 3.2.1 Definition of the Manifold M -- 3.2.2 Smooth Maps Between Manifolds -- 3.3 Biodynamic Bundles -- 3.3.1 The Tangent Bundle of the Manifold M -- 3.3.2 The Cotangent Bundle of the Manifold M -- 3.4 Sections of Biodynamic Bundles -- 3.4.1 Biodynamic Evolution and Flow -- 3.4.2 Vector-Fields and Their Flows -- 3.4.2.1 Vector-Fields on M -- 3.4.2.2 Integral Curves as Biodynamic Trajectories -- 3.4.2.3 Biodynamic Flows on M -- 3.4.2.4 Categories of ODES -- 3.4.3 Differential Forms on M -- 3.4.3.1 1-Forms on M -- 3.4.3.2 &Forms on M -- 3.4.3.3 Exterior Differential Systems -- 3.4.3.4 Exterior Derivative on M -- 3.4.3.5 De Rham Complex and Homotopy Operators -- 3.4.3.6 Stokes Theorem and De Rham Cohomology -- 3.4.3.7 Euler-Poincare Characteristics of M -- 3.4.3.8 Duality of Chains and Forms on M -- 3.4.3.9 Other Exterior Operators on M -- 3.4.4 Geometry of Nonlinear Dynamics -- 3.5 Lie Categories in Biodynamics -- 3.5.1 Lie Derivative in Biodynamics -- 3.5.1.1 Lie Derivative on Functions -- 3.5.1.2 Lie Derivative of Vector Fields -- 3.5.1.3 Derivative of the Evolution Operator -- 3.5.1.4 Lie Derivative of Differential Forms -- 3.5.1.5 Lie Derivative of Various Tensor Fields -- 3.5.1.6 Lie Algebras -- 3.5.2 Lie Groups in Biodynamics -- 3.5.2.1 Lie Groups and Their Lie Algebras.

3.5.2.2 Actions of Lie Groups on M -- 3.5.2.3 Basic Biodynamic Groups -- 3.5.2.4 Groups of Joint Rotations -- 3.5.2.5 Special Euclidean Groups of Joint Motions -- 3.5.3 Group Structure of the Biodynamic Manifold M -- 3.5.3.1 Purely Rotational Biodynamic Manifold -- 3.5.3.2 Reduction of the Rotational Manifold -- 3.5.3.3 The Complete Biodynamic Manifold -- 3.5.3.4 Realistic Human Spine Manifold -- 3.5.4 Lie Symmetries in Biodynamics -- 3.5.4.1 Lie Symmetry Groups -- 3.5.4.2 Prolongations -- 3.5.4.3 Special Biodynamic Equations -- 3.6 Riemannian Geometry in Biodynamics -- 3.6.1 Local Riemannian Geometry on M -- 3.6.1.1 Riemannian Metric on M -- 3.6.1.2 Geodesics on M -- 3.6.1.3 Riemannian Curvature on M -- 3.6.2 Global Riemannian Geometry on M -- 3.6.2.1 The Second Variation Formula -- 3.6.2.2 Gauss-Bonnet Formula -- 3.6.2.3 Ricci Flow on M -- 3.6.2.4 Structure Equations on M -- 3.6.2.5 Basics of Morse Theory -- 3.6.2.6 Basics of (Co)Bordism Theory -- 3.7 Symplectic Geometry in Biodynamics -- 3.7.1 Symplectic Algebra -- 3.7.2 Symplectic Geometry on M -- 3.8 Impulse Biodynamics and Synthetic Geometry -- 3.8.1 Delta Spikes -- 3.8.2 Kick Dynamics -- 3.8.2.1 Deterministic Delayed Kicks -- 3.8.2.2 Random Kicks and Langevin Equation -- 3.8.3 Distributions and Synthetic Differential Geometry -- 3.8.3.1 Distributions -- 3.8.3.2 Synthetic Calculus in Euclidean Spaces -- 3.8.3.3 Spheres and Balls as Distributions -- 3.8.3.4 Stokes Theorem for Unit Sphere -- 3.8.3.5 Time Derivatives of Expanding Spheres -- 3.8.3.6 The Wave Equation -- 3.9 A Quick Look at Modern Geometrodynamics -- 3.9.1 Einstein Equations -- 3.9.2 n-Categories in Physics -- 3.9.3 Quantum Geometry Framework -- 3.10 3D Modelling and Animation in Biodynamics -- 3.10.1 Essentials of Human Animation -- 3.10.1.1 Motion CaptureBased Human Animation -- 3.10.1.2 Virtual Muscular Dynamics in 3D-Graphics.

3.10.2 Curves and Surfaces in Geometric Modelling -- 3.10.2.1 Power Basis Form of a Curve -- 3.10.2.2 Bezier Curves -- 3.10.2.3 Rational Bezier Curves -- 3.10.3 B-Spline Basis Functions -- 3.10.3.1 DeBoor-Cox Recursive Definition -- 3.10.3.2 Derivatives of B-Spline Basis Functions -- 3.10.4 B-Spline Curves and Surfaces in Geometric Modelling -- 3.10.4.1 Definition of B-Spline Curves -- 3.10.4.2 Properties of B-Spline Curves -- 3.10.4.3 Derivatives of a B-Spline Curve -- 3.10.4.4 Definition of B-Spline Surfaces -- 3.10.4.5 Properties of B-Spline Surfaces -- 3.10.4.6 Derivatives of a B-Spline Surface -- 3.10.5 NURBS Curves and Surfaces -- 3.10.5.1 Definition of NURBS Curves -- 3.10.5.2 Properties of NURBS Curves -- 3.10.5.3 Definition of NURBS Surfaces -- 3.10.5.4 Properties of NURBS Surfaces -- 3.11 Kinematics of Biomechanical Chains -- 3.11.1 3D Transformation Matrix -- 3.11.2 A Multilink Kinematic Chain -- 3.11.3 CNS Representation of the Body Posture -- 3.11.4 Transformation Matrix Used in Computer Graphics -- 4. Natural Mechanics of Biodynamics -- 4.1 Lagrangian Formalism in Biodynamics -- 4.2 Hamiltonian Formalism in Biodynamics -- 4.2.1 Nonlinear Dynamics in Hamiltonian Form -- 4.2.1.1 Real 1-DOF Hamiltonian Dynamics -- 4.2.1.2 Complex One-DOF Hamiltonian Dynamics -- 4.2.1.3 Library of Basic Hamiltonian Systems -- 4.2.1.4 n-DOF Hamiltonian Dynamics -- 4.2.2 Hamiltonian Geometry in Biodynamics -- 4.2.3 Hamilton-Poisson Geometry in Biodynamics -- 4.2.3.1 Hamilton-Poisson Biodynamic Systems -- 4.2.4 Completely Integrable Hamiltonian Systems -- 4.2.4.1 Liouville Theorem -- 4.2.4.2 Action-Angle Variables -- 4.2.5 Ergodicity -- 4.2.5.1 Ergodicity in Hamiltonian Systems -- 4.2.5.2 Dynamical Systems and Hyperbolicity -- 4.2.5.3 Ergodic Theory and Nontrivial Recurrence -- 4.2.5.4 Nonuniformly Hyperbolic Trajectories.

4.2.5.5 Systems with Nonzero Lyapunov Exponents -- 4.3 Quantum Formalism in Nano-Biodynamics -- 4.3.1 Quantum Mechanics in Biological Matter -- 4.3.2 Dirac's Canonical Quantization -- 4.3.2.1 Quantum States and Operators -- 4.3.2.2 Quantum Pictures -- 4.3.2.3 Spectrum of a Quantum Operator -- 4.3.2.4 General Representation Model -- 4.3.2.5 Direct Product Space -- 4.3.2.6 State-Space for n Quantum Particles -- 4.3.2.7 Quantum Measurement and Penrose Paradox -- 4.3.3 The Problem of Quantum Measurement and Entropy -- 4.3.3.1 The Classical Apparatus -- 4.3.3.2 Quantum Object -- 4.3.3.3 Adiabatic Measurement Lagrangians -- 4.3.3.4 The Stern-Gerlach Experiment -- 4.3.3.5 Work and Heat -- 4.3.3.6 Statistical Thermodynamics -- 4.3.3.7 Friction in a Macroscopic Apparatus -- 4.3.3.8 Low Velocity Projective Measurements -- 4.3.3.9 Information and Entropy -- 4.3.4 Von Neumann's Density Matrix Quantization -- 4.3.4.1 Dissipative Quantum Formalism -- 4.3.5 Geometric Quantization -- 4.3.5.1 Motivation -- 4.3.5.2 Geometric Prequantization -- 4.4 Variational Formalism in Biodynamics -- 4.4.1 Biodynamic Action Functional -- 4.4.2 Lagrangian Action -- 4.4.3 Hamiltonian Action -- 4.4.4 Noether Theorem -- 4.4.5 Hamiltonian-Action Formulation of Biodynamics -- 4.4.6 Feynman Quantum Action -- 4.5 Nonholonomic Biodynamics -- 4.5.1 Lagrangian Approach -- 4.5.2 Hamiltonian Approach -- 4.5.3 Biodynamic Example: Bicycle Dynamics -- 4.6 Stochastic Formalism in Biodynamics -- 4.6.1 Markov Stochastic Processes -- 4.6.2 Statistical Mechanics of Oscillator Chains -- 4.7 Muscular Excitation-Contraction Dynamics -- 4.7.1 Human Musculo-Skeletal System -- 4.7.1.1 Human Skeleton -- 4.7.1.2 Human Joints -- 4.7.1.3 Human Muscular System -- 4.7.1.4 Human Energy Flow -- 4.7.1.5 Equivalent Muscular Actuator -- 4.7.2 Molecular Muscular Dynamics -- 4.7.3 Mezoscopic Muscular Dynamics.

4.7.3.1 Myocybernetics.