Mathematical mechanics : from particle to muscle

This unprecedented book offers all the details of the mathematical mechanics underlying modern modeling of skeletal muscle contraction. The aim is to provide an integrated vision of mathematics, physics, chemistry and biology for this one understanding. The method is to take advantage of latest mathematical technologies - Eilenberg-Mac Lane category theory, Robinson infinitesimal calculus and Kolmogorov probability theory - to explicate Particle Mechanics, The Theory of Substances (categorical thermodynamics), and computer simulation using a diagram-based parallel programming language (stochastic timing machinery). Proofs rely almost entirely on algebraic calculations without set theory. Metaphors and analogies, and distinctions between representational pictures, mental model drawings, and mathematical diagrams are offered.
AP level high school calculus students high school science teachers, undergraduates and graduate college students, and researchers in mathematics, physics, chemistry, and biology may use this integrated publication to broaden their perspective on science, and to experience the precision that mathematical mechanics brings to understanding the molecular mechanism vital for nearly all animal behavior.

Acknowledgments	p. xv
Introduction	p. 1
1 Introduction	p. 3
1.1 Why Would I Have Valued This Book in High School?	p. 4
1.2 Who Else Would Value This Book?	p. 5
1.3 Physics & Biology	p. 6
1.4 Motivation	p. 7
1.5 The Principle of Least Thought	p. 10
1.6 Measurement	p. 11
1.7 Conceptual Blending	p. 11
1.8 Mental Model of Muscle Contraction	p. 13
1.9 Organization	p. 15
1.10 What is Missing?	p. 18
1.11 What is Original?	p. 19
Mathematics	p. 21
2 Ground & Foundation of Mathematics	p. 23
2.1 Introduction	p. 23
2.2 Ground: Discourse & Surface	p. 26
2.2.1 Symbol & Expression	p. 27
2.2.2 Substitution & Rearrangement	p. 28
2.2.3 Diagrams Rule by Diagram Rules	p. 30
2.2.4 Dot & Arrow	p. 30
2.3 Foundation: Category & Functor	p. 36
2.3.1 Category	p. 38
2.3.2 Functor	p. 40
2.3.3 Isomorphism	p. 40
2.4 Examples of Categories & Functors	p. 41
2.4.1 Finite Set	p. 41
2.4.2 Set	p. 43
2.4.3 Exponentiation of Sets	p. 50
2.4.4 Pointed Set	p. 51
2.4.5 Directed Graph	p. 53
2.4.6 Dynamic System	p. 54
2.4.7 Initialized Dynamic System	p. 56
2.4.8 Magma	p. 59
2.4.9 Semigroup	p. 60
2.4.10 Monoid	p. 61
2.4.11 Group	p. 63
2.4.12 Commutative Group	p. 63
2.4.13 Ring	p. 64
2.4.14 Field	p. 65
2.4.15 Vector Space over a Field	p. 66
2.4.16 Ordered Field	p. 67
2.4.17 Topology	p. 68
2.5 Constructions	p. 69
2.5.1 Magma Constructed from a Set	p. 70
2.5.2 Category Constructed from a Directed Graph	p. 71
2.5.3 Category Constructed from a Topological Space	p. 74
3 Calculus as an Algebra of Infinitesimals	p. 75
3.1 Real & Hyperreal	p. 76
3.2 Variable	p. 79
3.2.1 Computer Program Variable	p. 79
3.2.2 Mathematical Variable	p. 79
3.2.3 Physical Variable	p. 80
3.3 Right, Left & Two-Sided Limit	p. 82
3.4 Continuity	p. 83
3.5 Differentiable, Derivative & Differential	p. 83
3.5.1 Partial Derivative	p. 86
3.6 Curve Sketching Reminder	p. 88
3.7 Integrability	p. 89
3.8 Algebraic Rules for Calculus	p. 92
3.8.1 Fundamental Rule	p. 92
3.8.2 Constant Rule	p. 92
3.8.3 Addition Rule	p. 92
3.8.4 Product Rule	p. 92
3.8.5 Scalar Product Rule	p. 93
3.8.6 Chain Rule	p. 93
3.8.7 Exponential Rule	p. 94
3.8.8 Change-of-Variable Rule	p. 94
3.8.9 Increment Rule	p. 94
3.8.10 Quotient Rule	p. 94
3.8.11 Intermediate Value Rule	p. 94
3.8.12 Mean Value Rule	p. 95
3.8.13 Monotoniaty Rule	p. 95
3.8.14 Inversion Rule	p. 95
3.8.15 Cyclic Rule	p. 97
3.8.16 Homogeneity Rule	p. 99
3.9 Three Gaussian Integrals	p. 99
3.10 Three Differential Equations	p. 101
3.11 Legendre Transform	p. 103
3.12 Lagrange Multiplier	p. 106
4 Algebra of Vectors	p. 111
4.1 Introduction	p. 111
4.2 When is an Array a Matrix?	p. 112
4.3 List Algebra	p. 113
4.3.1 Abstract Row List	p. 114
4.3.2 Set of Row Lists	p. 114
4.3.3 Inclusion of Row Lists	p. 115
4.3.4 Projection of Row Lists	p. 115
4.3.5 Row List Algebra	p. 115
4.3.6 Monoid Constructed from a Set	p. 117
4.3.7 Column List Algebra & NaturalTransformation	p. 119
4.3.8 Lists of Lists	p. 122
4.4 Table Algebra	p. 124
4.4.1 The Empty and Unit, Tables	p. 124
4.4.2 The Set of All Tables	p. 124
4.4.3 Juxtaposition of Tables is a Table	p. 125
4.4.4 Outer Product of Two Lists is a Table	p. 126
4.5 Vector Algebra	p. 127
4.5.1 Category of Vector Spaces & Vector Operators	p. 128
4.5.2 Vector Space Isomorphism	p. 129
4.5.3 Inner Product	p. 133
4.5.4 Vector Operator Algebra	p. 134
4.5.5 Dual Vector Space	p. 135
4.5.6 Double Dual Vector Space	p. 137
4.5.7 The Unique Extension of a Vector Operator	p. 137
4.5.8 The Vector Space of Matrices	p. 139
4.5.9 The Matrix of a Vector Operator	p. 139
4.5.10 Operator Composition & Matrix Multiplication	p. 140
4.5.11 More on Vector Operators	p. 141
Particle Mechanics	p. 145
5 Particle Universe	p. 147
5.1 Conservation of Energy & Newton's Second Law	p. 149
5.2 Lagrange's Equations & Newton's Second Law	p. 150
5.3 The Invariance of Lagrange's Equations	p. 152
5.4 Hamilton's Principle	p. 155
5.5 Hamilton's Equations	p. 160
5.6 A Theorem of George Stokes	p. 162
5.7 A Theorem on a Series of Impulsive Forces	p. 163
5.8 Langevin's Trick	p. 164
5.9 An Argument due to Albert Einstein	p. 165
5.10 An Argument clue to Paul Langevin	p. 167
Timing Machinery	p. 173
6 Introduction to Timing Machinery	p. 175
6.1 Blending Time & State Machine	p. 177
6.2 The Basic Oscillator	p. 178
6.3 Timing Machine Variable	p. 179
6.4 The Robust Low-Pass Filter	p. 180
6.5 Frequency Multiplier & Differential Equation	p. 180
6.6 Probabilistic Timing Machine	p. 181
6.7 Chemical Reaction System Simulation	p. 182
6.8 Computer Simulation	p. 183
7 Stochastic Timing Machinery	p. 187
7.1 Introduction	p. 187
7.1.1 Syntax for Drawing Models	p. 189
7.1.2 Semantics for Interpreting Models	p. 190
7.2 Examples	p. 192
7.2.1 The Frequency Doubler of Brian Stromquist	p. 192
7.3 Zero-Order Chemical Reaction	p. 193
7.3.1 Newton's Second Law	p. 194
7.3.2 Gillespie Exact Stochastic Simulation	p. 195
7.3.3 Brownian Particle in a Force Field	p. 196
Theory of Substances	p. 203
8 Algebraic Thermodynamics	p. 205
8.1 Introduction	p. 205
8.2 Chemical Element, Compound & Mixture	p. 207
8.3 Universe	p. 209
8.4 Reservoir & Capacity	p. 224
8.5 Equilibrium & Equipotentiality	p. 225
8.6 Entropy & Energy	p. 229
8.7 Fundamental Equation	p. 234
8.8 Conduction & Resistance	p. 238
9 Clausius, Gibbs & Duhem	p. 241
9.1 Clausius Inequality	p. 241
9.2 Gibbs-Duhem Equation	p. 244
10 Experiments & Measurements	p. 247
10.1 Experiments	p. 247
10.1.1 Boyle, Charles & Gay-Lussac Experiment	p. 247
10.1.2 Rutherford-Joule Friction Experiment	p. 251
10.1.3 Joule-Thomson Free Expansion of an Ideal Gas	p. 252
10.1.4 Iron-Lead Experiment	p. 254
10.1.5 Isothermal Expansion of an Ideal Gas	p. 258
10.1.6 Reaction at Constant Temperature & Volume	p. 260
10.1.7 Reaction at Constant Pressure& Temperature	p. 261
10.1.8 Théophile de Donder & Chemical Affinity	p. 265
10.1.9 Gibbs Free Energy	p. 268
10.2 Measurements	p. 271
10.2.1 Balance Measurements	p. 273
11 Chemical Reaction	p. 275
11.1 Chemical Reaction Extent, Completion & Realization	p. 279
11.2 Chemical Equilibrium	p. 281
11.3 Chemical Formations & Transformations	p. 285
11.4 Monoidal Category & Monoidal Functor	p. 286
11.5 Hess' Monoidal Functor	p. 289
Muscle Contraction Research	p. 291
12 Muscle Contraction	p. 293
12.1 Muscle Contraction: Chronology	p. 293
12.1.1 19th Century	p. 293
12.1.2 1930-1939	p. 293
12.1.3 1940-1949	p. 294
12.1.4 1950-1959	p. 296
12.1.5 1960-1969	p. 299
12.1.6 1970-1979	p. 301
12.1.7 1980-1989	p. 304
12.1.8 1990-1999	p. 305
12.1.9 2000-2010	p. 311
12.2 Conclusion	p. 325
Appendices	p. 327
Appendix A Exponential & Logarithm Functions	p. 329
Appendix B Recursive Definition of Stochastic Timing Machinery	p. 331
B.1 Ordinary Differential Equation: Initial Value Problem	p. 331
B.2 Stochastic Differential Equation: A Langevin Equation without Inertia	p. 332
B.3 Gillespie Exact Stochastic Simulation: Chemical Master Equation	p. 333
B.4 Stochastic Timing Machine: Abstract Theory	p. 334
Appendix C MATLAB Code	p. 335
C.1 Stochastic Timing Machine Interpreter	p. 335
C.2 MATLAB for Stochastic Timing Machinery Simulations	p. 338
C.3 Brownian Particle in Force Field	p. 339
C.4 Figures. Simulating Brownian Particle in Force Field	p. 344
Appendix D Fundamental Theorem of Elastic Bodies	p. 347
Bibliography	p. 353
Index	p. 363

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