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E-grāmata: Geometric Methods in Modern Biomechanics

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  • Formāts: 451 pages
  • Izdošanas datums: 01-Jun-2013
  • Izdevniecība: Nova Science Publishers Inc
  • ISBN-13: 9781619423046
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Geometric Methods in Modern BiomechanicsPalielināt
  • Formāts: 451 pages
  • Izdošanas datums: 01-Jun-2013
  • Izdevniecība: Nova Science Publishers Inc
  • ISBN-13: 9781619423046
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This book gives a comprehensive introduction into geometrical methodsof modern biomechanics. This book is designed for a rigorous, one-semester course at the graduate level. The intended audience includes mechanical, control and biomedical engineers (with stronger mathematical background), mathematicians, physicists, computer scientists and mathematical biologists, as well as all researchers and technical professionals interested in modelling and control of biomechanical systems and humanoid robots.
Preface xiii
1 Introduction
1(8)
1.1 Different Approaches to Biomechanics
2(1)
1.2 Biomechanical Configuration and Phase-Space Manifolds
3(1)
1.3 Hamiltonian Musculoskeletal Dynamics
4(1)
1.4 Muscular Excitation and Contraction Dynamics
5(1)
1.5 Neuro-Muscular Hamiltonian Control
6(3)
2 Geometrical Background: Manifolds and Tensors
9(46)
2.1 Sets, Maps and Diagrams
9(6)
2.1.1 Sets
9(1)
2.1.2 Maps
10(1)
2.1.3 Commutative Diagrams
11(1)
2.1.4 Notes from General Topology
12(3)
2.2 Manifolds
15(9)
2.2.1 Definition of a Manifold
17(1)
2.2.2 Formal Definition of a Smooth Manifold
18(1)
2.2.3 Smooth Maps between Smooth Manifolds
19(1)
2.2.4 Tangent Bundle and Lagrangian Dynamics
20(3)
2.2.5 Cotangent Bundle and Hamiltonian Dynamics
23(1)
2.3 Exterior Geometrical Machinery
24(4)
2.3.1 From Green's to Stokes' Theorem
24(1)
2.3.2 Exterior Derivative
25(2)
2.3.3 Exterior Forms
27(1)
2.3.4 Stokes Theorem
28(1)
2.4 Tensor Fields on a Smooth Manifold
28(8)
2.4.1 The Pull-Back and Push-Forward
29(1)
2.4.2 Dynamical Evolution and Flow
30(1)
2.4.3 Vector-Fields and Their Flows
31(5)
2.5 Lie Derivative on a Smooth Manifold
36(7)
2.5.1 Lie Derivative Operating on Functions
37(1)
2.5.2 Lie Bracket
38(3)
2.5.3 Derivative of the Evolution Operator
41(1)
2.5.4 Lie Derivative of Differential Foms
41(1)
2.5.5 Lie Derivative of Various Tensor Fields
42(1)
2.6 de Rham-Hodge Theory Basics
43(12)
2.6.1 Exact and Closed Forms and Chains
44(1)
2.6.2 de Rham Duality of Forms and Chains
45(1)
2.6.3 de Rham Cochain and Chain Complex
45(1)
2.6.4 de Rham Cohomology vs. Chain Homology
46(2)
2.6.5 Hodge Star Operator
48(1)
2.6.6 Hodge Inner Product
48(1)
2.6.7 Hodge Codifferential Operator
49(1)
2.6.8 Hodge Laplacian Operator
50(1)
2.6.9 Hodge Adjoints and Self-Adjoints
51(1)
2.6.10 Hodge Decomposition Theorem
52(3)
3 Lie Groups and Their Mechanical Applications
55(36)
3.1 Definition of a Group
57(2)
3.2 Definition of a Lie Group
59(1)
3.3 Lie Algebra
60(1)
3.4 One-Parameter Subgroup
61(1)
3.5 Exponential Map
62(1)
3.6 Adjoint Representation
63(1)
3.7 Actions of Lie Groups on Smooth Manifolds
64(1)
3.8 Basic Tables of Lie Groups and Their Lie Algebras
65(2)
3.9 Representations of Lie Groups
67(1)
3.10 Root Systems and Dynkin Diagrams
68(3)
3.10.1 Definitions
68(1)
3.10.2 Classification
68(1)
3.10.3 Dynkin Diagrams
69(2)
3.10.4 Irreducible Root Systems
71(1)
3.11 Simple and Semisimple Lie Groups and Algebras
71(2)
3.12 Mechanical Examples of Lie Groups
73(18)
3.12.1 Galilei Group
73(1)
3.12.2 General Linear Group
74(1)
3.12.3 Rotational Lie Groups in Human/Humanoid Biomechanics
75(4)
3.12.4 Euclidean Groups of Rigid Body Motion
79(4)
3.12.5 Basic Mechanical Examples
83(2)
3.12.6 Newton-Euler SE(3)-Dynamics
85(3)
3.12.7 Symplectic Group in Hamiltonian Mechanics
88(3)
4 Basics of Nonlinear Dynamics and Chaos Theory
91(32)
4.1 Poincare's Qualitative Dynamics, Topology and Chaos
92(7)
4.2 Smale's Horseshoe: Chaos of Stretching and Folding
99(7)
4.3 Lorenz' Weather Prediction and Chaos
106(2)
4.4 Feigenbaum's Constant and Universality
108(1)
4.5 May's Population Modelling and Chaos
109(3)
4.6 Henon's 2D Map and Its Strange Attractor
112(4)
4.6.1 Other Famous 2D Chaotic Maps
114(2)
4.7 Nonlinear Neurodynamics
116(7)
4.7.1 Integrate-and-Fire Neuron
116(1)
4.7.2 Integrate-and-Fire Neuron with Adaptation
116(1)
4.7.3 Integrate-and-Fire-or-Burst Neuron
117(1)
4.7.4 Complex-Valued Resonate-and-Fire Neuron
117(1)
4.7.5 Quadratic Integrate-and-Fire Neuron
117(1)
4.7.6 Hindmarsh-Rose Neuron
118(1)
4.7.7 Morris-Lecar Neuron
118(1)
4.7.8 Wilson-Cowan Neuronal Population Model
118(2)
4.7.9 Classical Neural Models
120(3)
5 Basics of Riemannian and Symplectic Geometry
123(22)
5.1 Riemannian Manifolds
123(12)
5.1.1 Riemannian Metric on M
124(4)
5.1.2 Geodesies on M
128(1)
5.1.3 Riemannian Curvature on M
129(2)
5.1.4 Riemann and Ricci Curvatures on a Smooth Manifold
131(4)
5.2 Group Structure of the Biomechanical Manifold
135(3)
5.2.1 Purely Rotational Biomechanical Manifold
135(1)
5.2.2 Reduction of the Rotational Biomechanical Manifold
136(1)
5.2.3 The Complete Biomechanical Manifold
137(1)
5.2.4 Realistic Human Spine Manifold
137(1)
5.3 Symplectic Manifolds
138(7)
5.3.1 Symplectic Algebra
138(1)
5.3.2 Symplectic Geometry
139(1)
5.3.3 Momentum Map and Symplectic Reduction
140(2)
5.3.4 Poisson Manifolds
142(3)
6 Basics of Lagrangian and Hamiltonian Mechanics
145(20)
6.1 Basics of Lagrangian Dynamics
145(3)
6.1.1 Basics of Poincare Dynamics
146(1)
6.1.2 The Lagrangian-Poincare Equations
147(1)
6.2 Basics of Hamiltonian Mechanics
148(8)
6.2.1 1DOF Hamiltonian Dynamics
149(7)
6.3 Library of Basic Hamiltonian Systems
156(9)
7 Geometrical Dynamics and Control...
165(26)
7.1 Introduction
165(1)
7.2 Configuration Manifold and the Covariant Force Law
166(4)
7.3 Lagrangian vs. Hamiltonian Approach to HumanoidRobotics
170(6)
7.4 Generalization to Human Biodynamics
176(5)
7.4.1 Realistic Configuration Manifold of Human Motion
176(1)
7.4.2 Brief on Time-Dependent Biomechanics
177(4)
7.5 Hierarchical Control of Humanoid Robots
181(5)
7.5.1 Spinal Control Level
181(1)
7.5.2 Cerebellum-Like Velocity and Jerk Control
181(2)
7.5.3 Cortical-Like Fuzzy-Topological Control
183(3)
7.6 Simulation Examples
186(5)
8 Medical Application 1: Prediction of Injuries
191(14)
8.1 Unified Mechanics of All Neuro-Musculo-Skeletal Injuries
191(5)
8.2 Analytical Mechanics of Traumatic Brain Injury (TBI)
196(9)
8.2.1 The SE(3)---jolt: The Cause of TBI
196(1)
8.2.2 SE(3)---group of Brain's Micro-motions within the CSF
197(1)
8.2.3 Brain's Natural SE(3)---Dynamics
197(3)
8.2.4 Brain's Traumatic Dynamics Caused by SE(3)---jolt
200(1)
8.2.5 Brain's Dislocations and Disclinations Caused by SE(3)---jolt
201(4)
9 Medical Application 2: Electrical Muscular Stimulation
205(20)
9.1 Introduction to EMS
205(2)
9.2 Neuro-Muscular EMS-Physiology
207(3)
9.2.1 Hodgkin-Huxley Models
207(2)
9.2.2 Bioelectrical Diffusion Cascade
209(1)
9.3 Quantum EMS-Physics
210(8)
9.3.1 Electrical Stimulation Fields: EMSfields
212(4)
9.3.2 Stimulated Muscular Contraction Paths: EMSpaths
216(2)
9.4 Quantum EMS---Control
218(2)
9.5 EMS-Treatment for the Back-Pain
220(4)
9.5.1 Common Back-Pain Conditions
221(3)
9.5.2 EMS-Treatment for the Back-Pain with Sciatica
224(1)
9.6 EMS Summary
224(1)
10 Control Strategies in Human Operator Modeling
225(38)
10.1 Introduction to Human Control
225(3)
10.2 Nonlinear Control Modeling of the Human Operator
228(9)
10.2.1 Graphical Techniques for Nonlinear Systems
228(2)
10.2.2 Feedback Linearization
230(4)
10.2.3 Controllability
234(3)
10.3 Fuzzy-Logic Control Modeling of the Human Operator
237(14)
10.3.1 Fuzzy Inference Engine
239(3)
10.3.2 Fuzzy Decision Making
242(1)
10.3.3 Fuzzy Logic Control
243(7)
10.3.4 Neuro-Fuzzy Hybrid Systems
250(1)
10.4 Adaptive Control Modeling of the Human Operator
251(8)
10.4.1 Lie-Adaptive Operator Control
252(1)
10.4.2 Neuro-Fuzzy-Fractal Operator Control
253(6)
10.5 Sports Application: Fuzzy-Control Based Tennis Simulator
259(3)
10.6 Conclusion on Human Control
262(1)
11 Cerebellar Controller for Human Biomechanics
263(10)
11.1 Introduction
263(2)
11.2 Sub-Cerebellar Biodynamics and Its Spinal Reflex Control
265(2)
11.2.1 Local Muscle-Joint Mechanics
265(2)
11.2.2 Hamiltonian Biodynamics and Its Reflex Servo-Control
267(1)
11.3 Cerebellum: The Adaptive Path-Integral Comparator
267(6)
11.3.1 Cerebellum as a Neural Controller
267(2)
11.3.2 Hamiltonian Action and Neural Path Integral
269(1)
11.3.3 Entropy and Motor Control
270(3)
12 Biomechanical Bundles and Jets
273(12)
12.1 Bundle, Forms and Vector-Fields
274(3)
12.2 p-Forms and Vector-Fields on Bundles
277(1)
12.3 Jet Manifolds
278(7)
12.3.1 First-Order Jet Manifolds
279(2)
12.3.2 Second-Order Jet Manifolds
281(1)
12.3.3 Bundle Connections
282(3)
13 Time-Dependent Lagrangian Biomechanics
285(8)
13.1 Exterior Lagrangian Biomechanics
285(3)
13.2 Jet Dynamics and Quadratic Equations
288(1)
13.3 Local Muscle-Joint Mechanics
289(4)
14 Time-Dependent Hamiltonian Biomechanics
293(24)
14.1 Symplectic Dynamics
293(4)
14.1.1 Jacobi Structure
293(1)
14.1.2 Contact Forms
294(1)
14.1.3 The Poisson Structure
294(2)
14.1.4 Symplectic Structure
296(1)
14.2 Polysymplectic Dynamics
297(8)
14.2.1 Lagrangian Dynamics
297(1)
14.2.2 The Legendre Morphisms
298(1)
14.2.3 Polysymplectic Structure
299(1)
14.2.4 Hamiltonian Connections
300(1)
14.2.5 Hamiltonian Forms
300(2)
14.2.6 Hamiltonian and Lagrangian Formalisms
302(1)
14.2.7 Vertical Extension of Polysymplectic Dynamics
303(2)
14.3 Time-Dependent Biomechanics
305(12)
14.3.1 n = 1 Reduction of Polysymplectic Dynamics
305(2)
14.3.2 Canonical Poisson Structure
307(2)
14.3.3 Presymplectic and Contact Structures
309(2)
14.3.4 Canonical Morphisms
311(3)
14.3.5 Vertical Extension of Hamiltonian Dynamics
314(3)
15 Hamiltonian vs. Lagrangian Time-Dependent Biomechanics
317(12)
15.1 The Poisson Structure
317(1)
15.2 Spray-Like Evolution Equations
318(4)
15.3 Hamiltonian vs Lagrangian Formalisms
322(1)
15.3.1 Examples
322(1)
15.4 Quadratic Lagrangians and Hamiltonians
323(2)
15.5 Unified Lagrangian + Hamiltonian Biomechanics
325(4)
16 Ricci Flow and Nonlinear Reaction-Diffusion Systems
329(26)
16.1 Introduction
329(3)
16.2 Bio-reaction-diffusion Systems
332(12)
16.2.1 1-component Systems
334(4)
16.2.2 2-Component Systems
338(4)
16.2.3 3-Component and Multi-component Systems
342(2)
16.3 Dissipative Evolution under the Ricci Flow
344(11)
16.3.1 Geometrization Conjecture
344(2)
16.3.2 Reaction-Diffusion-Type Evolution of Curvatures and Volumes
346(4)
16.3.3 Dissipative Solitons and Ricci Breathers
350(2)
16.3.4 Smoothing/Averaging Heat Equation and Ricci Entropy
352(3)
17 Brain Biomechanics in the Life Space Foam
355(28)
17.1 Introduction
355(2)
17.2 Classical versus Quantum Probability
357(6)
17.2.1 Classical Probability and Stochastic Dynamics
357(5)
17.2.2 Quantum Probability Concept
362(1)
17.3 Human Psychodynamics in the Life Space Foam
363(8)
17.3.1 Noisy Decision Making in the LSF
368(2)
17.3.2 The Evolution of Consciousness in the LSF
370(1)
17.4 Geometric Chaos and Topological Phase Transitions
371(5)
17.5 Joint Action of Several Agents
376(2)
17.6 Discussion
378(5)
18 Appendix
383(38)
18.1 Muscular System
383(10)
18.1.1 Muscular Histology
383(2)
18.1.2 Classical Theories of Muscular Contraction
385(6)
18.1.3 The Equivalent Muscular Actuator
391(1)
18.1.4 Biochemistry of Muscular Contraction
391(2)
18.2 Houk's Autogenetic Motor Servo
393(28)
Index 421
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