Non-equilibrium systems and irreversible processes: adventures in applied topology

Non-Equilibrium Systems and Irreversible Processes Adventures in Applied Topology Vol. 1

Non-Equilibrium Thermodynamics from a Perspective of Continuous Topological Evolution.

R. M. Kiehn Emeritus Professor of Physics University of Houston

December 5, 2006

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Print on Demand books published by Lulu Enterprises, Inc. 3131 RDU Center, Suite 210, Morrisville, NC 27560 See http://www.lulu.com/kiehn c Copyright CSDC INC. 2002 - 2006 ° ISBN 978-1-84728-193-7 Second Edition

CONTENTS 0.1 0.2 0.3 0.4

I

Preface . . . . . . . Points of Departure . Results . . . . . . . . Monograph Site Map

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Thermodynamics from a topological perspective

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1 Non-Equilibrium Thermodynamics 21 1.1 From the Topological perspective of Continuous Topological Evolution 21 1.2 Applied topology versus applied geometry . . . . . . . . . . . . . . . 26 1.3 Topological Universality . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4 Fundamental Axioms and Notable Results . . . . . . . . . . . . . . . 28 1.4.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4.2 Notable Results . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.5 Topological properties vs. Geometrical properties . . . . . . . . . . . 32 1.6 Pfaff Topological Dimension . . . . . . . . . . . . . . . . . . . . . . . 34 1.6.1 The Pfaff Sequence . . . . . . . . . . . . . . . . . . . . . . . . 34 1.6.2 Eigendirection fields of exterior 2-forms: Vectors vs. Spinors . 37 1.7 Evolutionary Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.7.1 Deformation Invariants as Topological Properties . . . . . . . 38 1.7.2 Absolute Integral Invariants . . . . . . . . . . . . . . . . . . . 39 1.7.3 Relative Integral Invariants . . . . . . . . . . . . . . . . . . . 40 1.7.4 Holder Norms, Period Integrals and Topological Quantization 41 1.8 Unique Continuous Evolutionary Processes . . . . . . . . . . . . . . . 42 1.9 The Arrow of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2 Topological Thermodynamics 2.1 Continuous Topological Evolution and the First Law . 2.1.1 Cartan’s Magic Formula . . . . . . . . . . . . . 2.1.2 An electromagnetic example . . . . . . . . . . . 2.2 Thermodynamic Systems . . . . . . . . . . . . . . . . . 2.2.1 Applications of the Pfaff Topological Dimension 2.2.2 Physical Systems: Equilibrium, Isolated, Closed

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49 49 49 50 52 52 53

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CONTENTS

2.3

2.4 2.5 2.6

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2.8

2.9 2.10

II

2.2.3 Equilibrium versus Non-Equilibrium Systems . . . . . . . . . . 2.2.4 Change of Pfaff Topological Dimension . . . . . . . . . . . . . 2.2.5 Systems with Multiple Components . . . . . . . . . . . . . . . Thermodynamic Processes . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Continuous Processes . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Reversible and Irreversible Processes . . . . . . . . . . . . . . 2.3.3 Adiabatic Processes - Reversible and Irreversible . . . . . . . 2.3.4 Processes classified by connected topological constraints on the Work 1-form, W . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Planck’s Harmonic Oscillator and Type B processes - How does energy get quantized ? . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Locally Adiabatic Processes . . . . . . . . . . . . . . . . . . . 2.3.7 Reversible processes when the Pfaff topological dimension of Work is 2 or 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . A Physical System with Topological Torsion . . . . . . . . . . . . . . The Lie differential L(V ) and the Covariant differential ∇(V ) . . . . . Topological Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 The Cartan-Hilbert Action 1-form . . . . . . . . . . . . . . . . 2.6.2 Thermodynamics and Topological Fluctuations of Work . . . . 2.6.3 Thermodynamic Potentials as Bernoulli evolutionary invariants Entropy of Continuous Topological Evolution and Equilibrium Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Extensions of the Cartan-Hilbert Action 1-form . . . . . . . . An Irreversible Example: The Sliding Bowling Ball . . . . . . . . . . 2.8.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 The Observation . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3 The Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . Second order versus first order ODE’s . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Applications to Physical Systems

3 The Ubiquitous Topological van der Waals gas 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Extensive and Intensive variables . . . . . . . 3.1.2 Lagrangian-Hamiltonian features . . . . . . . 3.2 The Phase function for a van der Waals Gas . . . . . 3.3 The Jacobian Matrix of the Action 1-form . . . . . . 3.3.1 Collineations and Coordinate Diffeomorphisms 3.3.2 Correlations and the 1-form of Action . . . . . 3.3.3 The Thermodynamic Phase function . . . . .

54 56 57 58 58 62 64 66 67 69 70 72 76 78 78 81 86 88 89 94 94 94 95 97 99

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103 103 109 111 114 119 119 120 121

CONTENTS

3.3.4 The details of the Universal Characteristic Phase Function 3.4 The Reduced Phase function in 4D . . . . . . . . . . . . . . . . . 3.4.1 The Higgs potential . . . . . . . . . . . . . . . . . . . . . . 3.4.2 The Binodal line . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 The Spinodal Line . . . . . . . . . . . . . . . . . . . . . . 3.5 Oscillations, Spinors and the Hopf bifurcation . . . . . . . . . . . 3.5.1 The Hopf Map and Hopf vectors . . . . . . . . . . . . . . . 3.5.2 Minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Singularities as defects of Pfaff dimension 3 . . . . . . . . 3.5.4 The Adjoint Current and Topological Spin . . . . . . . . . 3.5.5 Non-Equilibrium Examples . . . . . . . . . . . . . . . . . .

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122 126 126 130 130 131 132 134 143 144 146

4 Electrodynamic, Hydrodynamic, and Mechanical Thermodynamic systems 151 4.1 Physical (Contact) Systems of Pfaff Topological Dimension 3 . . . . . 152 4.1.1 The Vector Processes . . . . . . . . . . . . . . . . . . . . . . . 152 4.1.2 The Spinor processes . . . . . . . . . . . . . . . . . . . . . . . 153 4.1.3 Irreversible Spinor processes . . . . . . . . . . . . . . . . . . . 155 4.2 Physical (Symplectic) Systems of Pfaff Topological Dimension 4 . . . 156 4.3 Electromagnetism as a topological theory . . . . . . . . . . . . . . . . 159 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.3.2 The classical Maxwell-Faraday and the Maxwell-Ampere equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.3.3 The Fundamental Exterior Differential Systems . . . . . . . . 161 4.3.4 The Maxwell Ampere system: J − dG = 0 . . . . . . . . . . . 164 4.3.5 The Maxwell-Faraday system: F − dA = 0 . . . . . . . . . . . 166 4.3.6 The Lorentz force . . . . . . . . . . . . . . . . . . . . . . . . 169 4.3.7 Non-Equilibrium Features of Electromagnetism . . . . . . . . 169 4.3.8 The Poincare Topological 4-forms . . . . . . . . . . . . . . . . 171 4.3.9 Topological Torsion and Spin quanta . . . . . . . . . . . . . . 172 4.4 Hydrodynamics as a topological theory . . . . . . . . . . . . . . . . . 174 4.4.1 Euler flows and Hamiltonian fluids . . . . . . . . . . . . . . . 174 4.4.2 The Navier-Stokes fluid . . . . . . . . . . . . . . . . . . . . . . 175 4.5 Mechanics as a topological theory . . . . . . . . . . . . . . . . . . . . 177 4.5.1 Cartan’s development of Hamiltonian Mechanics . . . . . . . . 177 4.5.2 A Generalized Hamiltonian Formalism . . . . . . . . . . . . . 180 4.5.3 A Generalized Lagrangian Formalism . . . . . . . . . . . . . . 181 4.5.4 Physical Systems on four dimensions again . . . . . . . . . . . 183 4.5.5 The Harmonic Oscillator (W is of Pfaff Topological Dimension 0)185 4.5.6 The Damped Harmonic Oscillator - a Sol Geometry . . . . . . 186 4.5.7 Rayleigh dissipation . . . . . . . . . . . . . . . . . . . . . . . 188 4.5.8 Wave propagation . . . . . . . . . . . . . . . . . . . . . . . . . 189

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CONTENTS

4.5.9 4.5.10 4.5.11 4.5.12

Helmholtz processes with variable connectivity The Master Equation . . . . . . . . . . . . . . The Isovector Class of Irreversible Processes . Mass in symplectic systems . . . . . . . . . .

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189 190 191 193

5 The Thermodynamics of Dynamical Systems of Non-Linear ODE’s195 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.2 The Dynamical System as an Exterior Differential System . . . . . . 196 5.2.1 Frenet-Serret theory . . . . . . . . . . . . . . . . . . . . . . . 198 5.2.2 Implicit theory . . . . . . . . . . . . . . . . . . . . . . . . . . 199 5.2.3 The dynamical system defines an adiabatic process . . . . . . 200 5.2.4 Conservative distributions . . . . . . . . . . . . . . . . . . . . 200 5.2.5 Characteristic Polynomial . . . . . . . . . . . . . . . . . . . . 201 5.3 The van der Waals Gas as a Dynamical System . . . . . . . . . . . . 202 5.4 The Similarity and Brand Scalar Invariants for an arbitrary 3D vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 5.4.1 Vector Fields in 3D . . . . . . . . . . . . . . . . . . . . . . . . 209 5.4.2 Chirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.4.3 The Gauss map normalization and the Shape matrix for a projection 3D to 2D . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.5 Universal Cayley-Hamilton polynomials and 3D Thermodynamics . . 212 5.5.1 Lessons from the van der Waals gas . . . . . . . . . . . . . . . 213 5.5.2 Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 5.5.3 Bifurcations and Thermodynamics . . . . . . . . . . . . . . . 220 5.5.4 Antisymmetry in the Jacobian matrix, vorticity, and torsion . 226 5.6 Examples of the thermodynamics of Dynamical Systems . . . . . . . 229 5.7 2D-Limit Cycles and 3D-Limit Surfaces . . . . . . . . . . . . . . . . . 243 5.7.1 Closed 2D Limit Curves . . . . . . . . . . . . . . . . . . . . . 243 5.7.2 Closed 3D Limit Bounding Surfaces . . . . . . . . . . . . . . . 246 5.7.3 The Saddle Node Hopf Solution . . . . . . . . . . . . . . . . . 250 5.7.4 The Hysteretic Hopf Solution . . . . . . . . . . . . . . . . . . 252 5.7.5 Anisotropic Modifications of the SNH flow . . . . . . . . . . . 253 5.7.6 Phase Transitions in Dynamical Systems . . . . . . . . . . . . 254 5.7.7 Phase transitions and topological evolution . . . . . . . . . . . 257 5.8 The Falaco Soliton - A Topological String in a Swimming Pool . . . 257 5.8.1 The Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . 257 5.8.2 A Visual Topologically Coherent Defect in a Fluid . . . . . . . 258 5.8.3 Falaco Surface dimples are of zero mean curvature . . . . . . . 260 5.8.4 Falaco Surfaces are related to Harmonic vector fields . . . . . 263 5.8.5 Spinors and zero mean curvature surfaces . . . . . . . . . . . . 265 5.8.6 Topological Universality independent from scales . . . . . . . 265 5.8.7 The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 266

CONTENTS

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5.9 Bifurcation Processes and the Production of Topological Defects . . . 5.9.1 Lessons from the bifurcation to Hopf Solitons . . . . . . . . . 5.9.2 The bifurcation to a Falaco Soliton . . . . . . . . . . . . . . . 5.9.3 Falaco Solitons as Landau Ginsburg structures in micro, macroscopic and cosmological systems . . . . . . . . . . . . . . . . 5.9.4 Wheeler Wormholes and Falaco Strings between Branes . . . . 5.9.5 A Cosmological Conjecture . . . . . . . . . . . . . . . . . . . 5.10 Details of the Langford tertiary Hopf Bifurcations . . . . . . . . . . . 5.11 Emergence of Contact structures from Symplectic domains . . . . . . 5.11.1 Effect of Constraints on the Reduction of the Pfaff Topological Dimension from 4 to 3 . . . . . . . . . . . . . . . . . . . . . . 5.11.2 Emergence of Lagrangian submanifolds from Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III

Topological Aspects of Exterior Differential Forms

267 267 272 275 277 279 282 293 293 296 297

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6 Cartan’s Topological Structure 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 A Point Set Topology Example . . . . . . . . . . . . . . . . . 6.1.2 Algebraic and Differential Closure . . . . . . . . . . . . . . . . 6.1.3 The Exterior Product and Set Intersection . . . . . . . . . . . 6.1.4 The Exterior Differential and Limit Points . . . . . . . . . . . 6.2 The Cartan "Point Set" Topology . . . . . . . . . . . . . . . . . . . . 6.3 Topological Torsion, Connected vs. Non-Connected Cartan topologies 6.4 Applications of Cartan’s Topological Structure . . . . . . . . . . . . . 6.4.1 Continuous Processes . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Uniform Continuity, Frozen - in Fields, the Master Equation .

301 301 303 306 308 309 311 315 317 317 319

7 Continuous Topological Evolution 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . 7.2 Continuity . . . . . . . . . . . . . . . . . . . . . . 7.3 The evolutionary process . . . . . . . . . . . . . . 7.3.1 Cartan’s Magic formula . . . . . . . . . . 7.3.2 The Lie differential and C2 continuity 7.3.3 C1 Continuity . . . . . . . . . . . . . . . . 7.4 Topological Evolution . . . . . . . . . . . . . . . . 7.4.1 Evolutionary Invariants . . . . . . . . . . . 7.4.2 Deformation Invariants . . . . . . . . . . . 7.5 Simple Systems . . . . . . . . . . . . . . . . . . . 7.5.1 The Action 1-form and its Pfaff Sequence .

323 323 325 327 327 328 329 330 330 331 333 333

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CONTENTS

7.5.2 The Action 1-form and Topological Fluctuations . 7.6 Continuous Processes . . . . . . . . . . . . . . . . . . . . 7.6.1 Uniform and Non-Uniform Continuity . . . . . . . 7.6.2 Continuous Hydrodynamic Processes . . . . . . . 7.6.3 DeRham categories of Continuous Vector Fields . 7.6.4 The Hamiltonian Extremal Sub-Category . . . . . 7.6.5 The Bernoulli-Euler Sub-Category . . . . . . . . . 7.6.6 The Stokes Sub-Category . . . . . . . . . . . . . 7.7 Global Conservation Laws . . . . . . . . . . . . . . . . . 7.7.1 First Variation . . . . . . . . . . . . . . . . . . . 7.7.2 Second Variation . . . . . . . . . . . . . . . . . . 7.7.3 Continuity and the Integers . . . . . . . . . . . . 7.8 Pfaff’s Problem, Characteristics, and the Torsion Current 7.8.1 The Euler index . . . . . . . . . . . . . . . . . . . 7.8.2 Evolution of Topological Torsion . . . . . . . . . 7.8.3 Thermodynamic processes . . . . . . . . . . . . . 7.8.4 The Kinematic Topological Base . . . . . . . . . . 7.8.5 Comments 2006. . . . . . . . . . . . . . . . . . .

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335 336 336 337 338 341 342 342 343 343 345 346 346 348 350 351 352 352

8 Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals 353 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 8.1.1 Historical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 8.1.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 8.1.3 Exterior differential p-forms are not p-tensors. . . . . . . . . 359 8.1.4 Exterior differential p-forms are pullback Scalars or pullback Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 8.1.5 Closed differential forms . . . . . . . . . . . . . . . . . . . . . 363 8.1.6 Pullbacks, Immersions and Submersions . . . . . . . . . . . . 364 8.2 Homogeneous Functions . . . . . . . . . . . . . . . . . . . . . . . . . 368 8.3 Homogeneous differential forms . . . . . . . . . . . . . . . . . . . . . 370 8.4 Some applications of closed but not exact p-forms . . . . . . . . . . . 372 8.4.1 Closed but not exact 1-forms . . . . . . . . . . . . . . . . . . 372 8.4.2 Closed 1-forms in 3D and 4D . . . . . . . . . . . . . . . . . . 375 8.4.3 Closed 2-forms in 3D and 4D . . . . . . . . . . . . . . . . . . 376 8.4.4 Closed 3-forms in 4D . . . . . . . . . . . . . . . . . . . . . . . 376 8.4.5 Direction Fields with Zero Divergence . . . . . . . . . . . . . . 378 8.5 The Gauss Integrals (2-forms) . . . . . . . . . . . . . . . . . . . . . . 379 8.5.1 Example 1. The Gauss Link Integral . . . . . . . . . . . . . . 382 8.5.2 Example 2: Flat tangential developables . . . . . . . . . . . . 383 8.5.3 Example 3. Scrolls . . . . . . . . . . . . . . . . . . . . . . . . 384 8.6 Braids, Spin and Torsion-Helicity (3-forms) . . . . . . . . . . . . . . . 385

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8.6.1 Chaos and the Unknot . . . . . . . . . . . . . . . . . . 8.6.2 The Torsion 3-form and the Braid integral . . . . . . . 8.6.3 Braids . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Electrodynamic Applications . . . . . . . . . . . . . . . . . . . 8.7.1 Phase-Polarization and Orientability in 4D . . . . . . . 8.7.2 Topological Torsion and the Polarization quantum - a 3-form . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.3 The Flux or Circulation Integral 1-form . . . . . . . . . 8.7.4 The Charge Integral 2-form . . . . . . . . . . . . . . . 8.7.5 Hedgehog fields, Rotating plasmas, Accretion discs . . 9 Topology and the Cartan Calculus 9.1 Why differential forms? . . . . . . . . . . . . . . . . . . . 9.1.1 Pair and Impair exterior differential forms . . . . 9.1.2 Functional Substitution and the pullback . . . . . 9.1.3 Summary: Pair forms and Impair forms . . . . . . 9.2 Why Topology? . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Geometry and Physics . . . . . . . . . . . . . . . 9.2.2 Topological Physics . . . . . . . . . . . . . . . . . 9.3 Cartan’s Exterior Calculus . . . . . . . . . . . . . . . . . 9.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 9.3.2 The exterior product and the exterior differential 9.3.3 The exterior algebra . . . . . . . . . . . . . . . . 9.3.4 The Exterior Differential . . . . . . . . . . . . . 9.3.5 The Interior Product . . . . . . . . . . . . . . . . 9.3.6 The Lie Differential . . . . . . . . . . . . . . . . . 9.3.7 The Pullback with examples . . . . . . . . . . . 9.3.8 Some Topological Features . . . . . . . . . . . . . 9.4 Closure and Continuity . . . . . . . . . . . . . . . . . . . 9.4.1 Closure . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Continuity . . . . . . . . . . . . . . . . . . . . . . 9.5 Point Set Topology . . . . . . . . . . . . . . . . . . . . . 9.5.1 Closed and Open Sets . . . . . . . . . . . . . . . 9.5.2 Limit Points . . . . . . . . . . . . . . . . . . . . . 9.5.3 Closure . . . . . . . . . . . . . . . . . . . . . . . 9.5.4 Continuity . . . . . . . . . . . . . . . . . . . . . . 9.5.5 Interior . . . . . . . . . . . . . . . . . . . . . . . 9.5.6 Exterior . . . . . . . . . . . . . . . . . . . . . . . 9.5.7 The Boundary . . . . . . . . . . . . . . . . . . . . 9.6 Mappings, Basis Frames, and Connections . . . . . . . . 9.6.1 From the point of view of topology . . . . . . . . 9.6.2 The Jacobian matrix as a Basis Frame . . . . . .

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9.6.3 9.6.4 9.6.5 9.6.6

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9.8

9.9

9.10

The Right and Left Cartan Connections . . . . . . . . . . . . 449 The vector of Cartan Torsion 2-forms . . . . . . . . . . . . . . 450 The vector of Affine Torsion 2-forms, |Ga i (Not Cartan Torsion)452 Vectors of 3-form Currents, Topological Spin, and Topological Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 9.6.7 Matrices of Cartan Curvature 2-forms [Φ] . . . . . . . . . . . 454 9.6.8 Matrices of Bianchi 3-forms . . . . . . . . . . . . . . . . . . . 455 9.6.9 Decomposition of the Cartan Connection . . . . . . . . . . . . 455 The Strong Principle of Equivalence . . . . . . . . . . . . . . . . . . 455 9.7.1 Examples using Maple . . . . . . . . . . . . . . . . . . . . . . 456 9.7.2 The Particle (Contravariant) Affine Connection . . . . . . . . 456 9.7.3 The Wave (Covariant) Affine Connection . . . . . . . . . . . . 458 9.7.4 The Schwarzschild Metric embedded in a Basis Frame . . . . . 459 9.7.5 The Basis Frame and Physical Vacuums . . . . . . . . . . . . 462 Partitioned internal structure of Cartan Basis Frames . . . . . . . . . 463 9.8.1 The Structural equations . . . . . . . . . . . . . . . . . . . . . 465 9.8.2 Exterior Algebraic and Interior Differential Curvatures . . . . 467 9.8.3 Non-Integrable Frames . . . . . . . . . . . . . . . . . . . . . . 468 Distributions and the Adjoint Field . . . . . . . . . . . . . . . . . . . 473 9.9.1 The implicit algebraic Frame field . . . . . . . . . . . . . . . . 474 9.9.2 The parametric differential Frame field . . . . . . . . . . . . . 475 9.9.3 Projective Frames . . . . . . . . . . . . . . . . . . . . . . . . . 477 Intersection, Envelopes and Topological Torsion . . . . . . . . . . . . 478 9.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 9.10.2 A Family of Curves in the Plane ((2+1)-space) . . . . . . . . . 479 9.10.3 Singular and stationary points . . . . . . . . . . . . . . . . . . 480 9.10.4 Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 9.10.5 A Family of Surfaces in (3+1)-space . . . . . . . . . . . . . . . 483 9.10.6 The edge of regression . . . . . . . . . . . . . . . . . . . . . . 484 9.10.7 Examples of Envelopes of families of surfaces . . . . . . . . . . 484 9.10.8 The Jacobian cubic characteristic polynomial . . . . . . . . . 486 9.10.9 The General Theory . . . . . . . . . . . . . . . . . . . . . . . 492

10 References and Addenda 10.1 Acknowledgments and Index 10.2 Symbols . . . . . . . . . . . 10.3 About the Cover Picture . . 10.4 About the Author . . . . . . 10.5 Other Volumes in the Series

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Preface

0.1

11

Preface

Felix Klein, in describing how real research takes place, said "You often hear from non-mathematicians, especially from philosophers, that mathematics consists of drawing conclusions from clearly stated premises....... The investigator himself, however, in mathematics, as in every other science, does not work in this rigorous deductive fashion. On the contrary, he makes essential use of his phantasy and proceeds inductively, aided by heuristic expedients" — From his Erlangen program: Elementary Mathematics from an Advanced Standpoint : Arithmetic, Algebra, Analysis, Dover, NY, 1962. Remark 1 Update Notes as of November 15, 2006: More than a year after the first draft of this monograph, an extraordinary breakthrough in perspective was achieved when it was realized (in the first part of 2003, and then developed more thoroughly in the period 2005 - 2006) that thermodynamic fluctuations and irreversible processes in non-equilibrium systems must be due to macroscopic isotropic Spinor direction fields (not vector direction fields). These Spinor direction fields can always be associated with the eigendirections of antisymmetric matrices (see Chapters 2.3 and 4.1-4.2). Without Spinors there would be no irreversible turbulence. Classic analysis has focused attention on symmetric matrices of stress and strain, for which Spinors, as complex isotropic vectors (of zero quadratic form, or length), do not exist. In the future, this recognition that topological fluctuations are to be associated with (macroscopic) Spinor direction fields will be studied in more detail [277]. Also it should be noted that not until early 2006 was it appreciated fully that the projective geometry concepts of correlations are related to the Jacobian matrix of the coefficients, Ak , of that 1-form of Action (per unit source) that encodes a thermodynamic system. On the other hand, projective concepts of collineations are related to the Jacobian matrix of the coefficients of a direction field, V k , that encodes a thermodynamic process, and an (N-1)-form current. The purpose of this monograph is to present features of non-equilibrium and irreversible physical systems that can be understood in terms of applied topology. The monograph is not intended to be a text book in Topology nor a textbook in Tensor Analysis and/or Differential Geometry, although the methods developed in these disciplines will be employed in that which follows. The objective is to display and apply techniques of continuous topological evolution in order to gain a better understanding of non-equilibrium physical systems and irreversible processes. If you are not sure of what continuous topological evolution entails, you can pick up the details and a more formal description in Chapter 7. However, it is not necessary to jump to Chapter 7, for many of the concepts are explained by application and example in the earlier chapters. Classic equilibrium thermodynamics utilizes statistical methods, strongly

12

CONTENTS

influenced by properties of deterministic geometric evolution. These classic methods have limited (though useful) success when applied to non-equilibrium systems and irreversible processes. In this monograph it is demonstrated that fundamental thermodynamic principles can be extended to describe non-equilibrium systems and irreversible processes, when such concepts are described in terms of topological, not geometrical, evolution. Evolutionary processes of the type that can be described by C2 differentiable maps from initial to final state will be at the foundations of the methods to be developed. When the inverse processes do not exist, or are not continuous, such irreversible processes permit description of topological change, but they do not admit representations in terms of linear groups of motions. A projection from a space of (N+M)-dimensions to a space of M-dimensions is an example of a non-invertible, but continuous map. These processes of continuous topological change are not diffeomorphisms. Diffeomorphisms, which form a differentiable subset of homeomorphisms, do not describe topological evolution. A basic axiom is that thermodynamic irreversibility requires topological change. The historical use of a geometric diffeomorphic approach (tensor analysis), with emphasis on uniqueness, symmetries and conservation laws, to solve problems in physics has heretofore constrained, if not eliminated, the stated objective of understanding non-equilibrium systems and irreversible processes. However, geometric methods, borrowing the words of Eugene Wigner, have been "unreasonably effective" in understanding physical phenomena - at least for phenomena that can be approximated by isolated-equilibrium systems and statistical averages. The geometric methods developed historically (and based upon geometry) are time reversal invariant. However, thermodynamic irreversible continuous processes require that the topology of the initial state and the topology of the final state are not the same. Paraphrasing Eddington: Remark 2 analysis.

Aging and the arrow of time have slipped through the net of geometric

Most of the references to my earlier publications have been compiled for convenience in Volume 7 "Selected Publications", which is available in paper back form, or in PDF file download format. See http://www.lulu.com/kiehn. 0.2

Points of Departure

Herein it is demonstrated that these concepts of thermodynamic irreversibility can be captured in terms of continuous topological evolution. Remark 3 Geometric based, diffeomorphic, processes (describing invariant topology and invariant geometry) and group based symmetry methods (with their inherent inverse operations leading to unique invertible solutions) are not explicitly useful in describing the non-equilibrium systems and irreversible processes of interest herein.

Points of Departure

13

As the development progresses, it may come as a surprise to many readers to find that the theoretical basis of thermodynamics and electromagnetism indicate that these disciplines are topological, not geometrical, physical theories. In this monograph, a perspective of topological evolution and change is subsumed from the outset. Topological properties and features of physical systems and processes are emphasized, and their evolutionary change becomes the point of departure from classical physical theories. Physical properties of size and shape (though useful and interesting, indeed) are intentionally suppressed, in favor of topologically coherent deformable features. Such topologically coherent, but deformable structures, appear to self organize themselves during thermodynamically irreversible processes of topological change. It was recognized by Tisza (1961) [259] that metrical based properties can not be used to distinguish between the two classes of intensive and extensive thermodynamic variables. Thermodynamic features appear immediately in terms of the topological properties of isolated-equilibrium, closed, and open physical systems. Caratheodory pointed out that a thermodynamic physical system in isolatedequilibrium admitted description in terms of a Pfaffian form constructed from at most two independent functions (but with arguments over perhaps N geometric variables and parameters). Such Pfaff systems are said to be of (Pfaff) topological dimension 2∗ , and are uniquely integrable in the sense of Frobenius. Such uniquely integrable systems consist of a single topologically connected and topologically coherent component. In other words they are systems of a single phase. Once the integrating factor for an isolated system is specified, the Pfaff topological dimension is reduced from 2 to 1, which defines the state of equilibrium. It is remarkable that the Cartan topology constructed from an integrable Pfaffian 1-form is a connected (but not necessarily simply connected) isolated topology. On the other hand, irreducible, non-equilibrium thermodynamic systems are of Pfaff topological dimension 3 or more. The Frobenius theorem of unique integrability fails. Even more remarkably, the Cartan topology for such systems, of Pfaff topological dimension greater than 2, is a disconnected topology and may have many components (mixed phases). Another way of describing such a topologically disconnected system is that if solutions exist, there may be more than one solution (non-uniqueness) at any geometric point, leading to the notion of envelopes, Huygen wavelets, and edges of regression representing stability limits and the possibility of thermodynamic phase change. Pfaff topological dimension 3 (or more) systems are non-equilibrium systems of multiple topological components. Pfaff dimension 3 systems can be chaotic, but the chaotic processes can be reversible in a thermodynamic sense. However, Pfaff dimension 3 (in general, 2n+1) systems, always admit a unique extremal vector direction field which can be interpreted as long-lived kinematic evolution - neglecting topological fluctuations. Such extremal fields do not exist in ∗

1.6.

A simple method to determine the Pfaff topological dimension of any 1-form is given in Section

14

CONTENTS

domains that are of Pfaff topological dimension 4 (or 2n+2). Such four-dimensional (2n+2) topological spaces are the domain of thermodynamic irreversible processes. Self organized topologically coherent structures are the domains of Pfaff topological dimension 2n+1. The topological perspective of thermodynamics used in this monograph is based upon Cartan’s theory of exterior differential forms, which can be utilized to describe continuous topological evolution. A fundamental example of continuous topological evolution is described by the evolutionary change of Pfaff topological dimension. The topological perspective is founded on the idea that thermodynamic physical systems can be encoded in terms of a 1-form of covariant Action Potentials, Ak (x, y, z, t), on a four-dimensional abstract variety of ordered independent variables, {x, y, z, t}. The variety supports a volume element Ω4 = dxˆdyˆdzˆdt. It is also assumed that thermodynamic processes can be encoded, to within a factor, ρ(x, y, z, t), in terms of contravariant vector direction fields, V4 (x, y, z, t). In printings of this monograph starting in 2005, it was appreciated that direction fields representing thermodynamic processes should include classical Spinors, as well as Vectors. Spinors, which behave as vectors with respect to affine translations, do not behave as vectors with respect to rotations (see Chapters 1 and 4). Variational principles are not used to define "equations" of motion. Instead, continuous topological evolution of the thermodynamic system and its system of differential forms is encoded in terms of Cartan’s magic formula (see p. 122 in [148]), L(ρV4 ) A = i(ρV4 )dA + d(i(ρV4 )A).

(1)

The motivation for this departure from classical theories is that the Lie differential, when applied to a exterior differential 1-form of Action (per unit source), A = Ak dxk , is equivalent abstractly to the first law of thermodynamics. Hence, the first law of thermodynamics is a topological, not a geometrical idea. Remarkably, physical systems and processes can be put into equivalence classes defined by the concept of Pfaff topological dimension. These concepts will be presented in detail in the chapters that follow. Discontinuous processes and statistical methods are, more or less, ignored. However, it is important to remember (and for some - a surprising fact) that continuous evolution in a topological sense can cause discrete changes in the topological properties of a given system. Indeed, an important topological property is the number of disconnected parts, which in this treatment of thermodynamics will be related to the mole number, n. 0.3

Results

The original motivation for this monograph was based upon the goal of developing analytical methods which can decide if a given physical system was an equilibrium system or a non-equilibrium system. If a specific analytic process was applied to the

Results

15

physical system the methods should be able to decide if that process was thermodynamically reversible or irreversible. It is remarkable that by using a topological perspective and the axioms for continuous processes, given in detail below, these goals have been achieved without the use of probability or statistical methods, and without the use of metric constraints and linear connections. The topological method, constructed on a Cartan system of exterior differential forms which are inherently antisymmetric, emphasizes the antisymmetric properties of a physical system, where the more geometric and statistical methods, based upon quadratic metric forms and symmetric averages, tend to obscure the antisymmetry properties. Perhaps one of the most significant properties of antisymmetric matrices (associated with exterior differential 2-forms) is the fact that their eigendirection fields (spinors and vectors) are related to real eigenvalues which are zero, or to pure imaginary numbers. The eigenvectors which are associated with the complex eigenvalues have complex components whose squares add up to zero. That is, these complex eigenvectors are what have been called null isotropic vectors, and are the generators of Spinors. Remark 4 The Cartan method of exterior differential forms incorporates Spinors in a natural way, without allusions to microphysics or relativity theory. Spinors play a role in classical physics and dominate the theory of minimal surfaces. It is further remarkable that the Jacobian matrix of the coefficients of the 1form of Action (per unit source) - for those non-equilibrium turbulent physical systems of Pfaff topological dimension 4 - leads to a universal thermodynamic phase function represented by a polynomial equation of 4th degree. The universality is related to the singularity theory of non-degenerate systems which are equivalent under (small) deformations. The Phase function is constructed in terms of the symmetric similarity invariants of the Jacobian matrix of the component functions that encode the 1form of Action (per unit source), A. The resultant Phase function brings attention to thermodynamic phases that have equivalent (symmetry) structures other than those depending upon size and shape. In general, the exterior differential form method focuses attention on thermodynamic phases that have equivalent deformable topological structures (equivalent Pfaff topological dimension), and which are the result of continuous topological evolution. This resultant universal fourth order Phase function result matches the concepts of Landau Ψ4 mean field theory and phase transitions on one hand, and on the other hand makes contact with the non-equilibrium expansion of the universe described by "inflation" and dark matter and dark energy concepts due to a "Higgs" quartic potential below the critical point of a deformable van der Waals gas. The concepts of surface tension (or string theory) can be related to the mean curvature (induced by the molar density) of the universal phase surface, the concepts of temperature and entropy are related to the quadratic or Gauss curvature (induced by the molar density), while the concepts of pressure (of either sign) and interactions are

16

CONTENTS

related to the cubic curvatures (induced by the molar density). The theory as presented herein is far from being complete, yet the methods offer a new perspective for analyzing thermodynamic problems. Moreover, the techniques appear to solve the problem of making a marriage between mechanical dynamics and thermodynamics; the methods can be quite useful in the design of new applications previous excluded by assumptions of equilibrium and uniqueness. The historical limitations of geometric (metric-size-and-shape) and topological (deformation) invariance usually imposed upon theoretical descriptions of nature (especially in relativity theories) are abandoned herein in favor of studying those properties that are homeomorphic invariants in odd topological dimensions, and yet permit description of topological, as well as geometric, change relative to continuous transformations in even topological dimensions. The methods which are presented herein are based upon Cartan’s calculus of exterior differential forms [73], [40]. Exterior differential forms are objects, which, in contrast to tensors, are well behaved with respect to differentiable (continuous) mappings that do not have an inverse (and therefore do not preserve topological properties), and are also well behaved with respect to diffeomorphisms, which are differentiable invertible continuous mappings (and which preserve topological properties). Evolutionary processes will be defined in terms of the action of the Lie differential with respect to vector direction fields acting on differential forms [148]. The Lie differential acting on differential forms is not confined by the diffeomorphic constraints of tensor analysis, and can treat problems of topological change. The method goes beyond the more standard "extremal" techniques based upon the calculus of variations. In most of that which follows, the functions used to define the physical systems will be assumed to be C2 differentiable. The functions that describe processes most often will be assumed to be C2 differentiable as well, but certain C1 processes (inducing tangential discontinuities and wakes) and C0 processes (inducing shocks and first order phase transitions) are of physical interest. A fundamental result can be expressed by the statement: Remark 5 Topological change is a necessary condition for a continuous thermodynamic process to be irreversible. Irreversible processes, related to the arrow of time and the biological aging process, require topological evolution and topological change. Current physical theories that describe evolutionary processes (for example, Hamiltonian or Unitary dynamics) usually are formulated in terms of homeomorphisms that emphasize geometrical properties, but do not permit topological change. Hence all such homeomorphic continuous processes are thermodynamically reversible. 0.4

Monograph Site Map

The Monograph consists of three parts. Part I deals with "Thermodynamics from a Topological Perspective". A rather long (and sometimes repetitive) introductory

Monograph Site Map

17

Chapter 1 describing how a "Topological Perspective" can be useful to the applied sciences. Read the chapter rapidly, and several times. Terms that may be new to some readers, and a few equations, are introduced without apology or tutorial description. Chapter 9 acts as a terse appendix which covers most of the topological features utilized, as well as the basic features and notation of Cartan’s theory of exterior differential forms. A number of textbooks are available for those who want more detail [73], [12], [140], [7]. Chapter 2 of Part I goes directly to the heart of the theory of Topological Thermodynamics, and uses the concept of Pfaff topological dimension to distinguish between equilibrium and non-equilibrium systems. In addition, the concept of Frobenius integrability is used to distinguish between reversible and irreversible processes. Certain topological features of non-equilibrium thermodynamics lead to subtle differences relative to the concepts presented in the study of equilibrium thermodynamics. For example, from the topological point of view of non-equilibrium thermodynamics, internal energy, as well as work and heat, are properties of both the physical system and its evolutionary dynamics. A non-equilibrium adiabatic process is not necessarily an isentropic process, but describes a path that resides on a surface of constant internal energy. A key topological property is the concept of the Pfaff topological dimension. In Part I, the fundamental antisymmetrical topological features of a thermodynamic system are encoded by a 1-form of Action (per unit source), A, and other exterior differential forms obtained by studying the dynamics of continuous topological evolution. The theory of continuous topological evolution is encoded in terms of the Lie differential, LV , acting on differential forms, with respect to a process direction field, V . Part II considers "Applications to Physical Systems". In Part II, the topological properties of exterior differential forms will be refined by recognizing £ ¤ k that the Jacobian matrix, [J] = ∂A/∂x , of the coefficients of the 1-form of Action (per unit source), A, lead to correlation mappings and other features of a projective geometry. the Jacobian matrix, [J] , can have symmetric as well as antisymmetric features that will refine the topology of completely antisymmetric differential form methods used in Part 1. The rank of the Jacobian matrix defines a "Projective topological" dimension, which is to be compared to the "Pfaff topological" dimension generated by the antisymmetric components of the Jacobian matrix† . The 4th degree Cayley-Hamilton polynomial of [J], when expressed in terms of the matrix similarity invariants, generates a universal thermodynamic phase function, Θ, with envelopes that have correspondence with the Higgs function, the Landau-Ginsburg theory of order parameters, and the equation of state of van der Waals gas. The idea of universality is due to the fact that the features of these structures have properties of deformation invariance; that is, they are topological properties. Chapter 3 describes the thermodynamic phase features of the classic and ex†

The interplay between between the Pfaff topological dimension and the Projective topological dimension is a theory that is yet to be thoroughly understood, and exploited.

18

CONTENTS

tended van der Waals gas. Remarkably, thermodynamic systems of Pfaff topological dimension 4 lead to a universal Phase function as a polynomial equation of 4th order, from which the "Higgs" potential and the realization of the Spinodal and Binodal lines appear in a natural universal manner. Chapter 4 describes applications to nonequilibrium systems and irreversible processes in electrodynamics, hydrodynamics, and mechanics; applications to dynamical systems are found in Chapter 5. The idea of a dynamical system having component condensations as topological defect structures is given particular attention. The Falaco Solitons are introduced as experimental examples of topological coherent defect structures that any one can create in a swimming pool. More detail about Falaco Solitons is to be found in [274]. Part III considers "Topological Aspects of Exterior Differential Forms". The details of Cartan’s Topological Structure are presented in Chapter 6 in a more or less self contained manner. That is, you can read Chapter 6 without reference to the rest of the text. The key feature is that the Cartan Topological structure of isolated or equilibrium thermodynamic systems is a connected topology, but nonequilibrium systems are represented by a disconnected topology. Similarly, Chapter 7 contains a more formal and detailed presentation of the general theory of Continuous Topological Evolution in terms of Cartan’s theory of exterior differential systems. Chapter 8 develops the topic of closed and homogeneous p-forms, which form the basis of topological "quantum" numbers. Chapter 9 contains a bit of philosophy, a series of terse examples demonstrating the machinery of the Cartan methods and point set topology, and includes a development of Cartan’s structural equations in terms of the connections associated with a Physical Vacuum. Such configurations are in essence "voids", but with internal topological and geometric structure. A subsection is devoted to the general theory of envelopes, which are important to the study of non-equilibrium systems, in terms of exterior differential forms.

Part I Thermodynamics from a topological perspective

19

Chapter 1 NON-EQUILIBRIUM THERMODYNAMICS 1.1

From the Topological perspective of Continuous Topological Evolution

Much of modern physical theory is based upon geometry, especially Riemannian metric geometry, which dominates the theory of Relativity and Gravity. Remark 6 "However, the Riemannian metric does have one property which does not seem quite appropriate to physical space-time, and that is the perfect symmetry between opposite directions for any coordinate interval. Perhaps the most characteristic property of the physical world is the unidirection of time-like intervals..." Randers (as quoted in [5], p. 41). It would seem that some other basic mathematical tool that goes beyond tensor analysis, diffeomorphic connections, and geometry of scales is required to understand thermodynamic irreversibility. In this volume, the new paradigm is based upon continuous topological evolution. According to many authors [259] [26] [270], the connection between deterministic predictive mechanics and thermodynamics remains an open problem. The topological relationships and constraints that make up the laws of equilibrium thermodynamics heretofore have resisted analysis in terms of the geometrical and deterministic methods of classical mechanics. A fundamental issue is that the concept of intensive and extensive variables in thermodynamics is not compatible with Riemannian geometries built on quadratic forms [238]. Although non-Riemannian Finsler spaces (projectivized spaces that support torsion and distinguish between functions that are homogeneous of degree 1 and degree 0) appear to be the natural domain for equilibrium thermodynamics, little has been done in this area [52] [5]. Statistical methods appear to lead to reasonable values for equilibrium properties [183] of physical systems, but neither the statistical or deterministic prediction methods of mechanics say anything about the details of the processes - especially of the irreversible processes that can take place. Other authors have emphasized the topological foundations of thermodynamics [130], and from the time of Caratheodory have noted the connection to Pfaff systems [110]. However, these authors did not have access to, or did not utilize, the Cartan topology and deRham cohomology. A remark by Tisza, greatly stimulated the early developments of the theory presented in this monograph:

22

Non-Equilibrium Thermodynamics

Remark 7 "..the main content of thermostatic phase theory is to derive the topological properties of the sets of singular points in Gibbs phase space." (see p.195 [259].) It has been demonstrated [192] that for continuous but non-homeomorphic maps (C1 maps without continuous inverse) it is impossible to predict the functional form of either covariant or contravariant vector fields. That is, the functional form of the field on the final state is not well defined in terms of the functional form of the field on the initial state, if the map from initial to final state is continuous but not homeomorphic. On the other hand it can be shown that exterior differential forms, with coefficients composed of (antisymmetric) covariant tensor fields or contravariant tensor densities, are deterministic in a retrodictive sense, even though the continuous maps from initial to final state are not reversible. That is, the functional form of the components of differential forms defined on the final state are well defined on the initial state even if the map from the initial state to the final state is C1 smooth, but not a homeomorphism. With respect to continuous topological evolution there exists a natural, logical, arrow of time, which is not observable with respect to diffeomorphic geometric evolution. Therefore, to understand irreversible phenomena, a retrodictive point of view seems to be of some value and it is this non-statistical retrodictive point of view constructed on exterior differential systems that is the point of departure in this monograph. The methods will be restricted at first to those evolutionary processes which are C2 continuous. It is appreciated that this restriction does not cover all physical situations, where in my opinion, "true" discontinuities, not just mathematical artifacts, are possible. The continuous evolutionary processes to be considered will permit topology to change in a continuous but irreversible manner (example: the pasting together of two blobs, or the collapse of a hole). Discontinuous processes are at first excluded. This first introductory chapter will introduce and describe many of the ideas, and the general philosophy, associated with a topological view of thermodynamics. The reader is to be alerted to the fact that details, derivations and examples are to be found in the subsequent chapters. As implied in the preface, a major objective of this monograph is to establish a topological, non-statistical, link between thermodynamics and mechanical, electrical, or hydrodynamic physical systems. A particular goal is to develop a method of describing the differences, and how and when such differences occur, between: • Equilibrium and non-equilibrium physical systems. • Reversible and irreversible evolutionary processes acting on such systems. Warning! The topology of interest is that generated by Cartan’s Topological structure, which is defined in terms of his theory of exterior differential forms (see Chapter 6). The topology is NOT a metric topology, NOT a Hausdorf topology, and

From the Topological perspective of Continuous Topological Evolution

23

even does NOT satisfy the separation axioms required to be a T1 topology∗ . Yet all of the pertinent topological ideas, including the non-intuitive ones, are easy to grasp from the simple example of the T 4 point set topology. Perhaps the most important topological property is that of Pfaff topological dimension. A simple method for computing the Pfaff topological dimension of any exterior differential 1-form, say A, is given in section 1.6. In effect, the Pfaff topological dimension determines the irreducible minimum number of functions required to express the topological features of a region without topological defects. Of utmost importance, for physical systems of topological dimension greater than 2, the Cartan topology associated with a 1-form is not a connected topology. Such systems can have several connected components. As will be discussed below, when the topological dimension is greater than 2, the physical system is a nonequilibrium physical system. The importance of this result resides with the topological theorem that mappings from a disconnected topology to a connected topology can be continuous, but continuous maps from a connected topology to a disconnected topology are impossible. In other words continuous maps can be used to describe the decay of a turbulent state, but not its creation [217]. However, the presentation herein is not meant to be a textbook on Cartan’s theory of exterior differential forms, nor a textbook on abstract topology. Instead an effort has been made to meld Cartan’s methods and topological ideas in a manner that would be useful to the applied researcher and engineer. At the time of writing, not too many physicists, and almost no engineers, are conversant in either the language of the exterior calculus or the language of topology. Of course, some familiarity with the fundamentals of each said discipline is required, and to that end several terse, to the point, presentations (with examples) are given in Chapter 9. Most of the useful topological ideas can be rapidly absorbed in terms of point set topology with its metric de-emphasis. In fact, one of the beauties of using the Cartan calculus is that it constructs a differential topology that is free of the metric and connection constraints of differential geometry. It is extraordinary that the Maxwell theory of Electromagnetism (when based on the fields, E and B, distinct from the fields, D and H) was one of the first physical theories to be recognized as being a topological theory [271], independent from a choice of metric, or connection. Geometric constraints (such as constitutive constraints between the two distinct sets of fields) merely refine the topological features of the fundamental theory. In this monograph, it will be demonstrated how thermodynamics may be considered fundamentally as a topological theory, also independent from metric and connection. Pick up a modern text in classical thermodynamics and note the appearance of the following words used in describing fundamental thermodynamic concepts: 1. Isolated 2. Closed ∗

For those not familiar with point set topology, chapter 5 in Schaum’s outline [141] can be useful.

24

Non-Equilibrium Thermodynamics

3. Open 4. Number of disconnected parts (moles) 5. Closed Cycles 6. Integrability 7. Extensive (homogeneous degree 1) variables 8. Intensive (homogeneous degree 0) variables Now go to Schaum’s Outline, "General Topology" [141], or some other textbook on elementary topology, and check the index for these terms. All of these terms have precise definitions in topology, without the imposition of geometric constraints of size, shape or scales. In short, it would appear that Thermodynamics has its foundations in topology, and should be treated as a topological theory from the outset. This is the topological perspective of thermodynamics adopted herein. In 1974 it was suggested that a certain extension to Hamilton’s principle [185], [189] could be made such that the evolutionary processes considered would describe dissipative mechanical systems. Cartan had proved that extremal vector fields, which satisfy the Cartan-Hamilton equation, i(V)dA = 0, are generators of Hamiltonian dynamical processes [43]. Rather than study such "extremal" vector fields, it was suggested to consider those processes that satisfy the extended equation, i(V)dA = ΓA + dθ. Throughout this current presentation, and in the older articles, it is subsumed that a physical system may be described adequately by a 1-form of Action (per unit source), A, and a physical process may be defined in terms of a dynamical system generated by a vector field, V. It was not appreciated in 1974 that the topological domain of the extremal conservative (Hamiltonian) systems was a contact manifold of odd topological dimension, while the topological domain of the suggested dissipative extension was a symplectic manifold of even topological dimension. (Extremal solutions do not exist on even-dimensional manifolds of maximal rank.) Currently, the concept of topological dimension seems to be intimately related to the differences between conservative and dissipative processes. In fact, as is developed in that which follows, irreversible turbulent thermodynamic processes are artifacts of Pfaff topological dimension 4 (or more). Irreversibility requires that the evolutionary topology of the initial state is not the same as the topology of the final state. It sometimes comes as a surprise to realize that such topological changes can occur continuously. These features of continuous topological evolution (and an explanation of the symbols) are presented in detail in Chapter 6 and Chapter 7, below. A symplectic manifold is defined by the non-zero domain of an exact 2-form, F = dA. The concept of Hamiltonian mechanics can be extended to symplectic (even dimensional) manifolds, where the Bernoulli-Hamiltonian constraint is of the form

From the Topological perspective of Continuous Topological Evolution

25

i(V)dA = −dB. The Bernoulli constraint also satisfies the more general Helmholtz constraint, d(i(V)dA) = 0. Such Helmholtz processes (although reversible) are of interest for they admit the evolution of topological defects of the Bohm-Aharanov type. These topological defects are related to the Work 1-form, W = i(V)A, which is produced by the process V acting on the physical system, A. These Work related defects are due to the Work 1-form, W , and are not due to the closed, but not exact, parts of the 1-form of Action (per unit source). Almost symplectic manifolds are defined by a 2-form G which is closed but not exact. These even-dimensional manifolds G can have compact defect domains without boundary. Such topological structures have been studied by Fomenko [28]. From electromagnetic theory, it becomes apparent that the non-exact 2-form (almost symplectic) G is to be associated with the defect structures called charge, and extensive thermodynamic variables, D and H, while the exact 2-form (symplectic) F is to be associated with intensive thermodynamic variables, E and B. The 1-form of Action (per unit source) (Lagrangian) point of view has its advantages, for the fundamental 2-form of a symplectic domain is deduced by construction, A ⇒ F = dA. The disadvantage is that almost all symplectic domains so constructed are not compact without boundary. This apparent flaw becomes an advantage when it is appreciated that such non-compact domains are precisely that which is needed to describe closed (but not isolated) or open thermodynamic systems. Most classical "laws of physics" are based upon the dogma that a useful physical theory of evolution must give a unique prediction starting from a given set of initial conditions. Combining this constraint with a mathematical description of physical systems in terms of geometrical tensor fields leads to evolutionary processes which preserve topological properties and are (therefore) "reversible". The ubiquitous assumption of uniqueness of predicted solutions, and/or homeomorphic evolution, are topological constraints on "classical mechanics" that eliminates any time asymmetry. The point of departure in this monograph assumes that reasonable physical laws must be capable of describing topological change, and when this feature is encoded in mathematical form, the laws of physics are no longer necessarily reversible. Hence, in this monograph, the Boltzmann paradox will be resolved in terms of a theory based upon Continuous Topological Evolution [Chapter 7]. It is presumed that the presence of a physical system establishes a Topological Structure [Chapter 6] on a base space of independent (but ordered) variables. When a specific evolutionary process is applied to this physical system, the topology becomes refined. It will become evident that physical systems require two topological structures, one based upon an exact 2-form, F = dA, and its associated symplectic manifold structure, and the other based upon a non-exact 2-form, G , which may or may not be closed, but when closed, dG = 0, leads to an almost symplectic structure with compact topological defects.

26

1.2

Non-Equilibrium Thermodynamics

Applied topology versus applied geometry

It was mentioned above that the presentation herein is not meant to be a textbook on Cartan’s theory of exterior differential forms, nor a textbook on abstract topology. Instead, this monograph is an attempt to use the simpler features of topology contained in the exterior calculus and apply the Cartan methods of continuous topological evolution to interesting non-equilibrium problems in the disciplines of mechanics, electrodynamics, or hydrodynamics, without the use of probability theory or statistics. Although the concept of equilibrium has had many useful scientific successes, the truth of the matter is that the observable parts of the real world are rarely in equilibrium. The historical theories and methods of describing evolution that have been developed so far give no details as how the change from non-equilibrium to equilibrium takes place. Even the much touted Quantum Mechanics fails to describe the details of the evolutionary decay of an excited state to the ground state. Paraphrasing Bohr, "a miracle takes place" is not a satisfying answer. The topological methods employed herein can be used to determine when a physical system is in an equilibrium or non-equilibrium state. The topological methods employed herein can be used to distinguish thermodynamically irreversible from reversible evolutionary processes. The topological methods employed herein can be used to describe the irreversible dissipative decay processes from open systems into excited stationary states far from equilibrium, and the further decay from excited states into equilibrium ground states. Since before the beginning of the 20th century, advances in physical theories have been predicated upon a geometrical approach. (It should be mentioned that another interesting point of view about thermodynamics based upon algebra was presented by Zeleznik [297].) Several attempts to better understand thermodynamics in terms of geometrical ideas have been attempted [23], [291], but without notable success. It was pointed out by Tisza [259] that metrical based properties can not be used to distinguish between the two classes of intensive and extensive thermodynamic variables, and the hint was offered that perhaps topological methods, rather than geometrical methods, might prove to be suitable. In fact, as mentioned above, Tisza states the "main content of thermostatic phase theory is to derive the topological properties of the sets of singular points in Gibbs space". These ideas will be examined in detail and extended in sections 2.10 and Chapter 3. Finsler spaces have been examined by Anotelli and Ingevarden [5]. At the Aspen conference in 1977, I suggested that the methods based upon the first fundamental form (metric) should be replaced by methods based on the second fundamental form (the shape matrix). In Chapter 3, these ideas will be exploited in describing how in four-dimensional space, the van der Waals gas is a universal topological artifact. The same ideas apply to the symplectic dual vector fields that encode dynamical systems. In 4D, the features of the van der Waals gas are universal. Much of the motivation for development of a topological view of thermodynamics was based upon the concept of topological defects being related to domains

Topological Universality

27

or points where topological change took place. The idea that a phase transition was a realization of topological evolution and change was very influential in the struggle over the years to develop a dynamics of such a thermodynamics process. The work of van der Kulk and Schouten inspired the use of the concept of Pfaff topological dimension, and its change, as being one of the key tools for use in topological thermodynamics. The later work Martinet and Zhitomirski [147] [299] has yet to be fully exploited. The concept that smoothness could influence thermodynamic evolution has only recently been appreciated. How C1 smoothness interplays with the Nash and Gromov axioms has yet to be exploited, but it was a surprise to find out that C1 translational sequences of transitive processes in 3D could be reversible, where C2 intransitive processes of rotation and expansion could be thermodynamically irreversible (see Chapter 4). Herein, the emphasis is on topological properties and features of physical systems, and moreover, how the topology of a physical system can change in a continuous evolutionary manner. What is meant by this statement is that the topology of the initial state need not be the same as the topology of the final state. Such topological change can take place either continuously (pasting) or discontinuously (cutting). The result to be demonstrated is that topological change is a necessary artifact of continuous thermodynamic irreversibility. 1.3

Topological Universality

It is a remarkable fact that the physical theories of Thermodynamics, Electrodynamics and Hydrodynamics all have similar topological foundations. These similarity features become evident, and useful, when the different disciplines are expressed in the universal language of Cartan’s theory of exterior differential forms. 1. Each discipline utilizes the concept that a physical system can be encoded in terms of an exterior differential 1-form of Action† (per unit source), A. 2. Each discipline utilizes the concept that a process, or current, acting on the physical system, can be encoded to within a factor, ρ, by a contravariant direction field, V . 3. Each discipline has a dynamics that can be expressed in terms of continuous topological evolution based upon the Lie differential with respect to V . Warning: this topological dynamics is not always fully equivalent to that dynamics generated by the covariant differential of tensor analysis. The geometric dynamics of tensor analysis is a subset of the topological dynamics. †

From here on, in this volume, the words (per unit source) will be assumed to be adjoined to the phrase, 1-form of Action 1-form (per unit source), and will not be written explicitly, unless otherwise noted.

28

Non-Equilibrium Thermodynamics

The arguments of the functions that define the physical system, the process, and the induced additional 1-forms, in this section are limited (with some exceptions) to an ordered variety of n = 4 independent base variables, abstractly specified as {x, y, z, t}, and their differentials, {dx, dy, dz, dt}. It is presumed that other varieties of base variables {ξ 1 , ξ 2 , ξ 3 , ξ 4 } can be represented in terms of diffeomorphic maps from {ξ 1 , ξ 2 , ξ 3 , ξ 4 } to {x, y, z, t}. To a physicist, the base variables play the role of admissible coordinates if they are diffeomorphically related. However no specific geometric metric or connection is (necessarily) imposed on these varieties of pregeometric dimension n = 4 base variables. The main thrust of this monograph is to study useful applications of Cartan’s theory of exterior differential systems to problems of non-equilibrium thermodynamics systems. Some knowledge of Cartan’s’ topological structure, and the topological properties of the continuum are required. This material has been developed in Chapter 6 and Chapter 7.

1.4 1.4.1

Fundamental Axioms and Notable Results Axioms

The theory of non-equilibrium thermodynamics from the perspective of continuous topological evolution, as utilized in this monograph, is based on four axioms.

Axiom 1. Thermodynamic physical systems can be encoded in terms of a 1-form of covariant Action‡ Potentials, Ak (x, y, z, t...), on a ≥ fourdimensional abstract variety of ordered independent variables, {x, y, z, t...}. The variety supports a differential volume element Ω4 = dxˆdyˆdzˆdt... Axiom 2. Thermodynamic processes are assumed to be encoded, to within a factor, ρ(x, y, z, t...), in terms of contravariant vector and/or complex isotropic Spinor direction fields, V4 (x, y, z, t...). Axiom 3. Continuous topological evolution of the thermodynamic system can be encoded in terms of Cartan’s magic formula (see p. 122 in [148]). The Lie differential, when applied to an exterior differential 1form of Action, A = Ak dxk , is equivalent abstractly to the first law of thermodynamics. ‡

It should be realized that the 1-form of Action has physical units of angular momentum per unit source.

Fundamental Axioms and Notable Results

Cartan’s Magic Formula L(ρV4 ) A First Law Inexact Heat 1-form Q Inexact Work 1-form W Internal Energy U

29

= : = = =

i(ρV4 )dA + d(i(ρV4 )A), W + dU = Q, W + dU = L(ρV4 ) A, i(ρV4 )dA, i(ρV4 )A.

(1.1) (1.2) (1.3) (1.4) (1.5)

Axiom 4. Equivalence classes of systems and continuous processes can be defined in terms of the Pfaff topological dimension of the 1-forms of Action, A, Work, W , and Heat, Q. In effect, Cartan’s methods can be used to formulate precise mathematical definitions for many thermodynamic concepts in terms of topological properties without the use of statistics or geometric constraints such as metric or connections. Moreover, the method applies to non-equilibrium thermodynamical systems and irreversible processes, again without the use of statistics or metric constraints. The fundamental tool is that of continuous topological evolution, which is distinct from the usual perspective of continuous geometric evolution. In order to make the equations more suggestive to the reader, the symbolism for the variety of independent variables has been chosen to be of the format {x, y, z, t}, but be aware that no geometric constraints of metric or connection are imposed upon this variety. For instance, it is NOT assumed that the base variety is Euclidean. It is emphatically stated that geometric notions of scale and metric are to be avoided in favor of topological properties, some of which are invariants of continuous topological evolution, and some of which are not. Those classical thermodynamic features which are diffeomorphic invariants (useful to many equilibrium applications) are ignored, while topological features which are invariants of continuous transformations (and therefore useful to non-equilibrium applications) are not. Topological evolution is understood to occur when topological features (not geometrical features of size and shape) change. The motivation for this perspective was based upon the goal of developing analytical methods which could decide if a given physical system was an equilibrium system or a non-equilibrium system, and, also, if a specific analytic process was applied to the physical system, was that process reversible or irreversible. 1.4.2 Notable Results Remarkably, utilization of these (topological) axioms leads to notable results that are not obtained by geometric methods: 1. Thermodynamics is a topological theory.

30

Non-Equilibrium Thermodynamics

2. Topological change is a necessary condition for thermodynamic irreversibility. 3. When the Pfaff topological dimension of the 1-form of Action that encodes a physical system is 2, or less, the system is topologically isolated. The topological structure on domains of topological dimension n ≤ 2 never admit a continuous process which is thermodynamically irreversible. Non-equilibrium systems are of Pfaff topological dimension > 2. 4. A 1-form of Action, A, with Pfaff topological dimension equal to 1, defines an equilibrium isolated system which has representation as a Lagrangian submanifold. 5. The topological structure of physical systems on domains (contact manifolds) of odd topological dimension n = 3, 5, 7.. > 2 are non-equilibrium systems. On such systems there exists (to within a factor) a unique continuous extremal process, VE , which may be chaotic, but nevertheless is thermodynamically reversible, and has a Hamiltonian generator. 6. The topological structure of physical systems on domains of even topological dimension n = 4, 6, 8... > 2 (symplectic manifolds) are non-equilibrium systems. Such systems support (to within a factor) a unique continuous process, VT , related to the concept of Topological Torsion. Continuous evolution in the direction of the topological torsion vector is thermodynamically irreversible. In this sense, thermodynamic irreversibility is an artifact of topological dimension n ≥ 4. 7. The change of the Pfaff topological dimension will produce topological defects and thermodynamic phase changes. 8. The assumption of uniqueness of evolutionary solutions (which implies a Pfaff Topological dimension equal to 2 or less), and homeomorphic evolution, are different, but ubiquitous, constraints imposed upon classical mechanics that eliminate any time asymmetry. 9. All Hamiltonian, Symplectic-Bernoulli and Helmholtz processes are thermodynamically reversible. In particular, the work 1-form, W , created by Hamiltonian processes is of Pfaff topological dimension 1 or less. 10. The functional forms of tensor fields (with arguments in terms of the base variables of the final state) are not uniquely predictable in a deterministic manner (in terms of the functional forms of tensor fields with arguments in terms of the base variables of the initial state), unless the map from initial to final state is a diffeomorphism (which preserves topology) [192]. On the other hand, the functional forms of those alternating tensor fields which are coefficients of exterior

Fundamental Axioms and Notable Results

31

differential forms, and with arguments in terms of the base variables of the initial state, are well defined in terms of the functional forms of tensor fields with arguments in terms of the base variables of the final state, even when the (C1) map from initial to final state describes topological evolution. In other words, retrodiction of differential forms is possible when topology changes, but prediction is impossible. Hence an Arrow of Time asymmetry is a logical result [221] when topological evolution is admitted, but does not appear if the evolution is restricted to be homeomorphic, and therefore topologically invariant. 11. The topological structure of domains of Pfaff dimension 2 or less creates a connected, but not necessarily simply connected topology. Evolutionary solution uniqueness is possible. 12. The topological structure of domains of Pfaff dimension 3 or more creates a disconnected topology of multiple components. If solutions to a particular evolutionary problem exist, then the solutions are not unique. Envelope solutions, such as Huygen wavelets and propagating tangential discontinuities (called signals) are classic examples of solution non-uniqueness. 13. Cartan’s Magic formula, in terms of the Lie differential acting on exterior differential 1-forms establishes the long sought for combination of dynamics and thermodynamics, enabling non-equilibrium systems and many irreversible processes to be computed in terms of continuous topological evolution, without resort to probability theory and statistics. 14. The Lie differential acting on differential forms is not necessarily the same as a linear affine covariant differential acting on differential forms. It is possible to demonstrate that if the process is locally adiabatic (no heat flow in the direction of the evolutionary process), then the Lie differential and the covariant differential can be made to coincide, as they both satisfy the Koszul axioms for an affine connection. This is a surprising result, for, when the argument is reversed, the theorem implies that the ubiquitous affine covariant differential of tensor analysis, acting on a 1-form of Action, can always be cast into a form representing an adiabatic process. However, such adiabatic processes need not be reversible. 15. If the evolutionary process described by the Lie differential, affine equivalent or not, leaves the 1-form of Action invariant, then the process is thermodynamically reversible. If the affine covariant differential of tensor analysis induces parallel transport (the covariant differential is zero), then the affine process is adiabatic and reversible. 16. The Lie differential can describe evolutionary processes which are not C2 differentiable, leading to a better understanding of wakes (tangential discontinuities)

32

Non-Equilibrium Thermodynamics

and shocks (longitudinal discontinuities). On odd-dimensional spaces, sequential C1 (translational) processes can be thermodynamically reversible, while intransitive C2 processes (rotation and expansion with a fixed point) can be thermodynamically irreversible. 17. On spaces of Pfaff topological dimension 4, the Cayley-Hamilton theorem produces a characteristic polynomial with similarity invariant coefficients which will generate the format of the Gibbs function for a (universal) van der Waals gas, with a well defined critical point and binodal and spinodal lines. The same technique can be applied to dynamical systems. The combined thermodynamic-topological perspective presented in this monograph uses the mathematical tools of exterior differential forms to describe the topological features of physical systems, and vector fields to describe the continuous evolutionary processes that may or may not change the topology of the physical system. Examples will demonstrate that topological change is a necessary condition for thermodynamic irreversibility. 1.5

Topological properties vs. Geometrical properties

The idea that the presence of a physical system establishes a topological structure on a base space of independent variables is different from, but similar to, the geometric perspective of general relativity, whereby the presence of a physical system is presumed to establish a metric on a base space of independent variables. The topological features of the physical system are presumed to be encoded in terms of exterior differential forms, which - unlike tensors - are functionally well behaved with respect to differentiable maps that are not invertible. Note that a given base may support many different topological structures; hence a given base may support many different physical systems. In particular, the topology associated with a 1-form of Action need not be the same as the topology associated with the 1-form of Heat, Q, or the 1-form of Work, W , even though the base variables are the same for each 1-form. The Pfaff topological dimension can be different for each of the 1-forms. For maps, between base sets, that are C1 differentiable§ , but are not invertible, it is impossible to predict uniquely the functional forms of covariant or contravariant vector fields, constructed over a final base set, in terms of functional forms given on an initial base set [192]. Point-wise (numeric) values of the tensor fields in certain cases may be predicted, but the functional forms describing neighborhoods are never predictable with respect to such non-invertible maps. Hence, classical theories based on tensor fields, which can describe geometrical evolution, will fail to describe topological evolution. It may be surprising to note that (with respect to noninvertible, non-homeomorphic, maps) it is possible to retrodict the functional forms of covariant vectors and contravariant vector densities on the initial base set in terms §

C1 implies connected, where C2 implies smooth.

Topological properties vs. Geometrical properties

33

of the given functional forms on the final base set. For differentiable evolutionary processes that are diffeomorphisms, topology does not change and both prediction and retrodiction of tensor fields is possible. For differentiable evolutionary processes which are not homeomorphisms, topology changes, and deterministic prediction fails, but deterministic retrodiction remains possible. Hence the feature of topological evolution imposes a sense of asymmetry with respect to an evolutionary parameter the arrow of time is an artifact of topological change. Although C1 non-invertible maps are not homeomorphisms, and therefore the topology of the initial state and the topology of the final state are not the same, such maps can be continuous. Continuous topological evolution is not an oxymoron, for topological continuity is defined such that the limit points of every subset in the domain (relative to the topology on the initial state) permute into the closure of the subsets in the range (relative to the topology on the final state). The initial and final state topologies need not be the same! A physical property which is independent of continuous deformation, and is independent from geometric concepts of size and shape, is a primitive example of a topological property. However, not all topological properties fit this useful, but imprecise, description. As examples, note that the number of holes in a rubber sheet is a topological property, and is independent of the continuous deformation of the rubber sheet into different sizes and shapes. The Planck black body radiation distribution of a hot body is a topological property, for the distribution of radiation frequencies (in first approximation) depends only on temperature, but not on the size and shape of the heated sample. Deformation invariants often can be encoded in terms of multi-dimensional integrals. As the elements of the integrand and the integration chain evolve, the value of the integral may be an evolutionary invariant, even though the domain and integration chain are deformed by the evolutionary process. Of special interest are those integral deformation invariants where the integration chain is a closed cycle. Such objects lead to topological "quantum-like" concepts, for the values of the integrals of closed, but not exact, exterior differential forms over different cycles have (by deRham’s theorems) rational ratios. (In this volume, quantum effects are discussed only in terms of their topological origin and deRham period integrals.) If the evolutionary process causes the topological quantum number to change, then the process describes a topological quantum transition. Surprisingly, processes of topological evolution can change topology in a continuous manner. A soap film connected to a double loop of wire will form the non-orientable surface of a Moebius band. Deformation of the wire into a single loop will cause the soap film to form a disk surface which is orientable. The topological property of orientability has changed continuously in terms of the process described. More precisely, a topological property is defined as an invariant of a homeomorphism. A homeomorphism is a map from initial to final state, which is continuous and has a continuous inverse. If the homeomorphism is C1 differentiable both ways, then the map is called a diffeomorphism. Diffeomorphisms are the trans-

34

Non-Equilibrium Thermodynamics

formations used to define tensors and most geometric properties. Invariance with respect to diffeomorphisms is a constraint employed in many physical theories which are based upon tensor calculus and the calculus of variations. Recall Klein’s concept of a (Euclidean) geometric property as being defined in terms of the invariants of rotations and translations (which are diffeomorphisms). Yet diffeomorphisms are specialized homeomorphisms which preserve topology. It follows that tensor analysis, so useful in studying geometric concepts, cannot be used effectively to describe topological change, and therefore tensor analysis is inadequate to describe irreversible evolution, where topological change is a necessary condition. However, continuous C1 processes need not be homeomorphisms, and therefore can be used to describe topological change. Exterior differential forms are mathematical objects that are well behaved in a retrodictive sense with respect to functional substitution of C1 continuous, but not invertible, maps; tensor fields are not. It follows that Cartan’s exterior differential forms become the mathematical objects of choice for describing continuous topological evolution, and therefore Cartan’s mathematics is the mathematics of choice for a theory of Irreversible Thermodynamic processes. A key topological property is that of dimension. However, the concept of topological dimension is somewhat different from the concept of geometrical dimension. For purposes of the theory developed herein, the topological structure imposed upon a base variety of m independent variables can be used to determine the "Pfaff topological dimension", n, which is to be distinguished from the "geometric dimension" of the base variety, n ≤ m. The primary feature of a topological structure is that it can be used to determine when an evolutionary process involving topological change (such as the change in topological dimension) is continuous. Topological change can occur both continuously and discontinuously. However, in this monograph, the focus is on continuous topological evolution. Herein it will be demonstrated that thermodynamic "relaxation" from some initial configuration to a state of "equilibrium" can be described by a sequence of continuous processes that cause the topological dimension to change from some initial value n to a final value n ≤ 2. 1.6 1.6.1

Pfaff Topological Dimension The Pfaff Sequence

Perhaps one of the most important topological tools to be used within the theory of continuous topological evolution is the concept of Pfaff topological dimension. The maximum Pfaff dimension is equal to the number of independent variables in the base variety, which in this monograph has been limited (in most cases) to n = 4. For a given 1-form of Action, A = Ak (x, y, z, t)dxk , defined on the base variety of {x, y, z, t}, it is possible to ask what is the irreducible minimum number of independent functions, θ(x, y, z, t), required to describe the topological features (on a contractible domain) that can be generated by the specified 1-form, A. This irreducible number of functions is defined herein as the "Pfaff topological dimension" of the 1-form, A. For

Pfaff Topological Dimension

35

example, if A = Ak dxk ⇒ dθ(x, y, z, t)irreducible , such that Ak = ∂θ(x, y, z, t)/∂xk ,

(1.6) (1.7)

then only one function, θ(x, y, z, t), is required to describe the Action, not four. In this example the irreducible Pfaff topological dimension of the 1-form, A, is 1, although the geometric dimension of base variety is 4. In a sense, the Pfaff topological dimension defines the existence of a domain of "topological" base variables (topological coordinates) as submersions from the original base variety (geometric coordinates) to the irreducible base variety (topological coordinates). Differential forms constructed on the irreducible topological base variety of functions, are functionally well defined on the original geometric base variety by means of functional substitution, in terms of the submersive mapping. In addition, it is possible to use topological coordinates to define singular domains, where singular p-form components can have a zero exterior differential¶ , if the singularity is excised (like a hole removed from a piece of paper). These singular defect regions define topological obstructions; their representation in terms of homogeneous closed p-forms will be detailed below. Physically such p-forms can be related to "fixed" points of rotation or expansion, and tangential discontinuities such as found in hydrodynamic wakes, and propagating electromagnetic signals. Integrals of such singular p-forms around cycles lead to topological quantization. The values of such integrals are related to the integers. Relative to the Cartan topology [11], the "Pfaff topological dimension" can be generated by each of the Pfaffian forms associated with each discipline. The irreducible Pfaff topological dimension for any given 1-form A is readily computed by constructing the Pfaff sequence of forms, Pfaff sequence :{A, dA, AˆdA, dAˆdA}.

(1.8)

The Pfaff topological dimension is equal to the number of non-zero terms in the Pfaff sequence. For example, if the Pfaff sequence for a given 1-form A is {A, dA, 0, 0} in a region U ⊂ {x, y, z, t}, then the Pfaff topological dimension of A is 2 in the region, U. The 1-form A, in the region U, then admits description in terms of only 2, but not less than 2, independent variables, say {u1 , u2 }. For a differentiable map ϕ from {x, y, z, t} ⇒ {u1 , u2 }, the exterior differential 1-form defined on the target variety U of 2 pre-geometry dimensions as A(u1 , u2 ) = A1 (u1 , u2 )du1 + A2 (u1 , u2 )du2 ,

(1.9)

has a functionally well defined pre-image A(x, y, z, t) on the base variety {x, y, z, t} of 4 pre-geometric dimensions. This functionally well defined pre-image is obtained ¶

The p-form is said to be closed.

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Non-Equilibrium Thermodynamics

by functional substitution of u1 , u2 , du1 , du2 in terms of {x, y, z, t} as defined by the mapping ϕ. The process of functional substitution is called the pullback, A(x, y, z, t) = Ak (x)dxk ⇐ ϕ∗ (A(u1 , u2 )) ⇐ ϕ∗ (Aσ duσ ).

(1.10)

It may be true that the functional form of A yields a Pfaff topological dimension equal to 2 globally over the domain {x, y, z, t}, except for sub regions where the Pfaff dimension of A is 3 or 4. These sub regions represent topological defects in the almost global domain of Pfaff dimension 2. Conversely, the Pfaff dimension of A could be 4 globally over the domain, except for sub regions where the Pfaff dimension of A is 3, or less. These sub regions represent topological defects in the almost global domain of Pfaff dimension 4. Applications of both viewpoints will be described below. The important concept of Pfaff topological dimension also can be used to define equivalence classes of physical systems and processes. The concept defined herein as the "Pfaff topological dimension" was developed more than 110 years ago (see page 290 of Forsyth [77] ), and has been called the "class" of a differential 1-form in the mathematical literature. The term "Pfaff topological dimension" (instead of class) was introduced by me in order to emphasize the topological foundations of the concept. More mathematical developments can be found in Van der Kulk [238]. The method and its properties have been little utilized in the applied world of physics and engineering. Of key importance is the fact that the non-zero existence of the 3-form AˆdA, or, Topological Torsion = AˆF,

(1.11)

implies that the Pfaff topological dimension of the region is 3 or more, and the nonzero existence of the 4-form of Topological Parity, dAˆdA = F ˆF implies that the Pfaff topological dimension of the region is 4. Either value is an indicator that the physical system (in the sub region) is NOT in thermodynamic equilibrium. It is also important to recall that non-zero values of Topological Torsion imply that the Frobenius unique integrability Theorem for the Pfaffian equation, A = 0, fails. The concept of topological parity, F ˆF , has its foundations in the theory of Pfaff’s problem, with a recognizable four-dimensional formulation appearing in Forsyth [77] page 100. On a variety of 4 variables, the coefficient of the 4-form F ˆF will be defined as the topological parity (or orientation) function, K, such that Topological Parity F ˆF = Kdxˆdyˆdzˆdt = KΩ4 .

(1.12)

It is possible to ascribe the idea of entropy production (due to bulk viscosity) to the coefficient K of the Parity 4-form. The idea of Topological Torsion, AˆF , has been associated with the idea of magnetic helicity density, a concept that apparently had its electromagnetic genesis with the study of plasmas in WWII. However, the concept of helicity density is but one component of the four-dimensional Topological Torsion 4-vector.

Pfaff Topological Dimension

37

Recall that a space curve with non-zero Frenet-Serret torsion does not reside in a two-dimensional plane. Non-zero Frenet - Serret torsion of a space curve is an indicator that the geometrical dimension of the space curve is at least 3. The fact that the Pfaff topological dimension of the 1-form, A, is at least 3, when AˆF is non-zero, is the basis of why the 3-form, AˆF , was called "Topological Torsion". The idea of non-zero 3-form AˆF also appears in the theory of the Hopf Invariant [30]. The concept of AˆF has also appeared in the differential geometry of connections, where a matrix valued 3-form is known as the Chern-Simons 3-form. However, on varieties without connection or metric, the Chern-Simons concept is not well defined, but the Topological Torsion concept exists and is acceptable, for it does not depend upon the geometric features of metric and/or connection. The concepts can be extended to "pre-geometrical", and therefore topological, domains of dimension greater than 4. Pre-geometry implies that constraints of metric or connection have not been (necessarily) imposed on the base variety. It is possible to define a "curvature" dimension (at a point) in terms of the number of non-null eigenvectors of the Jacobian matrix built from the partial derivatives of the C1 functional components that define the 1-form of Action. The "Curvature" dimension is always less than the dimension of the base variety. The implication is that the determinant of the shape matrix is zero. It is possible that the Pfaff topological dimension can exceed the "curvature" dimension. The idea of the Pfaff Topological Dimension is analogous to the idea of the number of "essential parameters" in the theory of continuous groups [68]. 1.6.2 Eigendirection fields of exterior 2-forms: Vectors vs. Spinors Perhaps one of the most interesting features of the Pfaff Topological dimension is that all of the eigendirection fields of the antisymmetric matrix of 2-form coefficients, dA, are complex isotropic direction fields of zero (null) length, but with complex eigenvalues, if the Pfaff topological dimension is even. If the Pfaff topological dimension is odd, then all but one of the eigendirections is isotropic, and that unique vector direction field (in the space of dimension equal to the Pfaff topological dimension), with eigenvalue zero, has a Hamiltonian generator. The surprising realization is that these isotropic eigendirection fields are Spinors, in the sense of E. Cartan [44]. The exterior product of two spinors represents a finite area, but the inner product of a spinor with itself is zero. Spinors are, therefore, representations of "zero length" but "finite area"! Remark 8 Spinors are related to the Pfaff topological dimension, and are natural (and to be expected) occurrences when the 1-form of Work induced by a thermodynamic process is not zero. If the Pfaff topological dimension of A is one, the 2-form, dA, is null, and there are no Spinor representations. If A is the 1-form of Action representing an equilibrium

38

Non-Equilibrium Thermodynamics

thermodynamic system, then its Pfaff topological dimension is unity. It follows that processes in equilibrium systems do not have Spinor representations. If the Pfaff topological dimension of the 1-form of Action is 2, then the thermodynamic system is isolated, and it admits a Spinor representation in terms of conjugate pairs. If the Pfaff topological dimension is 3, then the system is thermodynamically closed, consists of more than one component, and admits one conjugate pair of Spinor representations. If the Pfaff topological dimension is 4, then the thermodynamic system is open, and admits two pairs of complex conjugate spinor representations. As shown below, Spinor representations of processes are required if the process is thermodynamically irreversible. 1.7

Evolutionary Invariants

Evolutionary invariants are generally those properties of physical systems that are observables in the sense of physical measurements. Invariants of continuous processes are included in the set of topological properties (invariants of homeomorphisms), and topological properties are included in the set of geometric properties (invariants of diffeomorphisms). 1.7.1 Deformation Invariants as Topological Properties Topological properties are defined as invariants with respect to homeomorphisms. A more mundane definition is that a topological property is an invariant of a continuous deformation. Certain integral properties of a thermodynamic system are deformation invariants with respect to those continuous evolutionary processes that can be described by a singly parameterized vector field. For the example of an electrodynamic thermodynamic system, the absolute deformation invariants lead to fundamental topological conservation laws, described in the physical literature of electromagnetism as the conservation of charge and the conservation of flux. Recall the definitions used to describe processes of continuous topological evolution. Definition A continuous process is defined as a map from an initial state of topology Tinitial into a final state of perhaps different topology Tf inal such that the limit points of the initial state are permuted among the limit points of the final state (see p. 97 et.seq. [141]). If the ordering of the limit points is invariant, the process is uniformly continuous. If the ordering (as in a folding of a boundary) or the number of the limit sets is changed, then the process is non-uniformly continuous. A simple description of a topological property (invariant of a homeomorphism) is an object that is a deformation invariant. Consider a rubber sheet with three holes. Stretch the rubber sheet. The holes may be deformed but the fact that there are three holes stays the same under small deformations. The concept of three holes is a topological property. It is remarkable that such topologically coherent objects (the

Evolutionary Invariants

39

holes) can be determined from those open and closed integrals which are deformation invariants. A topological deformation invariant is defined as an integral of an exterior differential p-form over a p-dimensional manifold, or cycle, zpd, such that the Lie differential of the integral of the p-form ω with respect to a singly parameterized vector field, ρV k , vanishes, for any choice of deformation parameter, ρ. Integral Deformation Invariant: L(ρV k )

Z

ω=0

any ρ.

(1.13)

p

The requirements that a given p-form becomes a deformation invariant (and therefore a topological property, invariant with respect to homeomorphisms) is expressed in terms of certain topological constraints. Those objects that remain the same under continuous deformation represent topological, not geometric, properties. However, if the topological constraints required for continuous deformation are not satisfied, then topological change takes place. Topological change would require that the number of holes in the thin rubber sheet example were to change. Topological change can occur continuously or discontinuously. The focus in this monograph is on continuous topological change, and as will be demonstrated below, topological change is a necessary requirement for thermodynamic irreversibility [192]. 1.7.2 Absolute Integral Invariants There are two types of invariant integrals, Absolute and Relative integral invariants. If the exterior p-form that forms the integrand is exact, the Absolute integral invariant places conditions only on the boundary of the domain of integration. It is these types of objects (Absolute integral invariants) that give a formality to those thermodynamic concepts whereby a physical system reaches a steady state uniformly within its interior, and yet may couple with its exterior environment via fluxes across its boundary. In such cases, only effects related to the boundary are of consequence. For example, consider physical systems that can be defined by a 1-form of Action, A, such that the derived 2-form F = dA, is exact. It follows from Stokes theorem that the two-dimensional integral of F is an absolute integral deformation invariant with respect to all continuous processes that can be defined by a singly parameterized vector field, subject to a boundary condition that the net flux, i(ρV k )F, of F, across the one-dimensional boundary of M is zero: Z Z Z Z Z Z k F = i(ρV )dF + d(i(ρV k )F ) (1.14) L(ρV k ) M M M Z = 0+ i(ρV k )F ⇒ 0. (1.15) boundary of M

This concept is at the basis of the Helmholtz theorems of vorticity conservation (or angular momentum per unit mass) in hydrodynamics, and the conservation

40

Non-Equilibrium Thermodynamics

of flux in classical electromagnetism. Herein, this concept of deformation invariance of a topologically coherent structure will be written in the form of an exterior differential system [34], F − dA = 0. The exterior differential system is to be recognized as a topological constraint. From Stokes theorem, the two-dimensional domain of finite support for F can not, in general, be compact without boundary, unless the Euler characteristic vanishes. There are two exceptional cases for absolute invariance of the integral, and they occur when the integration domain is compact without boundary. Such two-dimensional domains which have a zero Euler characteristic are the torus and the Klein-Bottle, but these situations require the additional topological constraint that F ˆF ⇒ 0. The fields in these exceptional cases must reside on these exceptional compact surfaces without boundary, which form topological coherent structures. Note that an evolutionary process could start with F ˆF 6= 0, and possibly evolve to a state with F ˆF = 0. If such residue states are compact without boundary, then they must be either tori or Klein bottles. The same integration technique can be applied to non-exact but closed pforms. 1.7.3 Relative Integral Invariants If the integration of the exact 2-form, F , is over a closed two-dimensional integration chain, designated as a two-dimensional cycle, z2d (which may or may not be a twodimensional boundary), then the integral is invariant for any deformation factor, ρ : Z Z Z Z Z Z k L(ρV k ) F = i(ρV )dF + d(i(ρV k )F ) = 0 + 0. (1.16) z2d

z2d

z2d

The two integrals on the right vanish, the first due to the fact that dF = 0, and the second due the fact that the closed integral over an exact form vanishes. Closed integrals of exact p-forms are always relative deformation integral invariants. However, the same technique can be applied to non-exact but closed p-forms. For electromagnetism, there are several exact p-forms, each producing a relative deformation integral invariant. For example, the 3-form of charge-current density is exact, J = dG. The 4-forms that define the Poincare 4-forms are exact, F ˆF = d(AˆG) and F ˆG − AˆJ = d(AˆG) (see Chapter 4.3). If the conditions of relative integral invariance are applied to an arbitrary 1-form of Action, then the relative integral invariance condition becomes, Z Z Z k L(ρV k ) A = i(ρV )dA + d(i(ρV k )A) (1.17) z1d z1d z1d Z = i(ρV k )F + 0 ⇒ 0. (1.18) z1d

It follows that i(ρV k )dA must be zero on the cycle z1d for any deformation parameter ρ. Cartan has shown that this is the condition that implies the process ρV k has a "Hamiltonian" representation [43] (see Chapter 4.5).

Evolutionary Invariants

41

1.7.4 Holder Norms, Period Integrals and Topological Quantization Besides the invariant structures considered above, the Cartan methods may be used to generate other sets of topological invariants. Realize that over a domain of Pfaff dimension n less than or equal to N, the Cartan criteria admits a submersive map to be made from N to a space of minimal dimension n. Assume the submersive map produces functions, [V 1 (x, y, z..), V 2 (x, y, z..), ...V n (x, y, z..)],

(1.19)

with a differential volume element, Ωn = dV 1 ˆdV 2ˆ...ˆ V n . Then construct the (n-1)form, C = i(V 1 , V 2 , ..., V n )Ωn .

(1.20)

Define an integrating factor ρ in terms of the Holder norm, ρ = 1/λ = 1/{a(V 1 )p + b(V 2 )p + c(V 3 )p + .....}m/p .

(1.21)

Then multiply (n-1)-form, C, by ρ to produce (n-1)-form density (current) J as: J = i(V 1 , V 2 , V 3 , ...)Ω = ρ{V 1 dV 2 ˆdV 3 ... − V 2 dV 1 ˆdV 3 ... + V 3 dV 1 ˆdV 2 ... − ...}.

(1.22)

Theorem 1. The (n-1)-form J is closed, dJ = 0, for any choice of constants a,b,c.. and for any p, if the Holder homogeneity index satisfies the equation, m = n (for proof, see Chapter 8). It is remarkable that the "current" J so defined has a vanishing exterior differential, independent of the value of p for a given m (equal to the dimension of the submersive volume element), and for all values of the constants (plus or minus a,b,c...). All such "currents" thereby define a "conservation law". As the map defining the components of the vector field in terms of the base {x,y,z..} is presumed to be differentiable, then the (n-1)-form, J, has a well defined pullback on the base space (almost everywhere), and its exterior differential on the base space also vanishes everywhere mod the defects. That is, the (n-1)-form, J, is locally exact. The number of negative coefficients in set {a, b, c, d...} determines the signature index of the Holder norm. The number m determines the homogeneity index. The Holder integrating factors are more familiar when m = 1, p = 1 which generates the barycentric coordinates {a, b, c, d...} of Moebius, [31], and for m = 1, p = 2, a = b = c..., which is known as the Gauss map. Use for both of these special Holder constructions will be developed in that which follows. The integrals of these closed currents, when integrated over closed (n-1)dimensional chains, create deformation invariants, with respect to any evolutionary process that can be described by a vector field, because Z Z Z L(ρV) J= i(ρV)dJ + d(i(ρV))J) ⇒ 0 + 0 = 0. (1.23) z(n−1)d

z(n−1)d

z(n−1)d

42

Non-Equilibrium Thermodynamics

These integral objects appear as "topological coherent" structures (which may have defects or anomalous sources, when the integrating factor 1/λ is not defined). The integration chain, z(n − 1)d, is a (n − 1)-dimensional (d) cycle z. Remark 9 The Holder norm which creates closed forms when homogeneity index m = n, is a topological idea independent from the metric signature. The compliment to the zero sets of the function λ determine the domain of support associated with the specified vector field. The closed (n-1)-form, J, that satisfies the conservation law, dJ = 0, has integrals over closed domains that have rational fraction ratios. As this (n-1)-current is closed globally, it may be deduced on a connected local domain from a (n-2)-form, G. In every case J has a well defined pullback to the base variety, {x,y,z,t}. Note that the n functions [V x (x, y, z..), V y (x, y, z..), V z (x, y, z..), ...] represent the minimum number of Clebsch variables that are equivalent to the original action, A, over the domain of support. As each of these integrals is intrinsically closed, the Lie differential with respect to any C1 vector field, ρV, is a perfect differential, such that (when integrated over closed domains that are p-1 boundaries) the evolutionary variation of these closed integrals vanishes. These n-1 integrals are relative integral invariants for any C1 evolutionary processes, or flows. The values of the integrals are zero if the closed integration domains are boundaries, or completely enclose a simply connected region. If the closed integration domains encircle the zeros of the function λ, then the values of the integrals are proportional to the integers; i.e., their ratios are rational. In general, by deRham’s theorems, the values of these period integrals, for different closed integration chains in domains where dJ = 0, have rational ratios [193]. When the evolution of a period integral is such that the integer changes, the process can describe the decay from a quantized stationary state of topological quantum number m to a state of topological quantum number n: Z Topological Quantization: L(ρV) J = n constant. (1.24) z(n−1)d

Note that each signature of λ must be investigated. For the elliptic (positive definite) signature, the singular points are the stagnation points, and the domain of support excludes those singularities. For the hyperbolic signatures, the domain of support excludes the hyperbolic singularities of lower dimension, such as the light cone. Further note that a given vector field may not generate real domains of support for all possible signatures of the quadratic form, λ. Details and applications of homogeneous constructions that give rise to period integrals are presented in Chapter 6. 1.8

Unique Continuous Evolutionary Processes

Evolutionary processes can preserve the topological properties of a physical system, or they can change them. In this monograph, those processes which can change

Unique Continuous Evolutionary Processes

43

continuously the topology of a physical system are of major interest, for topological change is a necessary requirement of thermodynamic irreversibility. Intuitively, a process applied to a physical system can be arbitrary, which implies that the process V is not necessarily dictated by the topological structure of the physical system, A. This intuitive idea is not precise. In contrast, for a given physical system, A, there are two vector direction fields that are determined uniquely from the functions that defined the topological structure of the physical system. One of these vector fields, VE , is uniquely defined on contact manifolds of odd topological dimension, 2n + 1. VE is called an "extremal" field. The second vector direction field, VT , is defined uniquely on symplectic manifolds of even topological dimension, 2n + 2. VT is called a "torsion" vector. Recall that the topological dimension can be smaller than or equal to the geometric dimension. The concept that these two vectors are uniquely determined implies that the representation is in terms of topological, not geometrical, coordinates. They are also necessary to produce topological fluctuations. The extremal vector VE on a contact manifold is proportional to the unique real eigenvector of the 2n + 1 × 2n + 1 antisymmetric matrix of functions that forms the components of the 2-form, dA. This unique eigenvector has an eigenvalue equal to zero. Such null eigenvectors exist (uniquely) only on topological domains of odd Pfaff Topological dimension, typically, in this monograph, equal to 3. The extremal vector,VE , satisfies the equations: VE : Extremal Vector i(VE )dA = 0, L(VE ) A = d(i(VE )A).

(1.25) (1.26)

It will be demonstrated in the subsequent chapters that extremal vector fields admit a Hamiltonian representation, or generator, that describes the evolutionary direction field. Extremal Hamiltonian vector fields (without fixed points) do not alter the topological properties of the physical system represented by the 1-form, A. On geometric domains of dimension greater than 2n+1, there can exist other vector direction fields that have a Hamiltonian (or better said a Bernoulli-Casimir) generator. However, these "Hamiltonian" direction fields are not uniquely defined in terms of the functions that define the topology of the physical system. For example, on a geometric domain of 2n+2 dimensions, if there exists one eigenvector of the 2-form, dA, with eigenvalue zero, then there must exist at least two such eigenvectors with eigenvalue zero. The fact that there exists even one eigenvector with eigenvalue zero, implies that the determinant of the antisymmetric matrix must be zero. The topological dimension of the matrix must be less than its geometric dimension. The differences between extremal Hamiltonian processes and the Bernoulli processes will be discussed in the subsequent chapters. On topological domains where dA is of even maximal rank, extremal vectors (eigenvectors relative to the matrix representation of dA with zero eigenvalues) do not exist. All eigendirection fields of the associated antisymmetric matrix are complex

44

Non-Equilibrium Thermodynamics

with complex eigenvalues (pure imaginary). These eigendirection fields are not zero, but have a quadratic form that vanishes. They are not strictly vectors. They are, in fact, pure Spinors, which have metric (quadratic form) properties, as do tensors (vectors), and behave as tensors with respect to transitive maps. However, Spinors do not behave as tensors (vectors) with respect to rotations [44]. Remark 10 Spinors have practical use in classical phenomena, such as minimal surface theories, propagation of topological discontinuities, the production of event horizons, and the generation of wakes. Yet, in such cases of even Pfaff topological dimension, there does exist a unique vector direction field, the Topological Torsion vector, VT , that is determined by the topological structure and the coefficient functions of the 1-form that encodes the physical system, A. Such a unique vector exists only on topological domains of evendimensional maximal rank, typically in this monograph, equal to 4. The Topological Torsion vector satisfies the equations: VT i(VT )Ω Ω L(VT ) A i(VT )A

: = = = =

Topological Torsion vector AˆdA...dA, dx1 ˆdx2 ˆ...dx2n+2 , W = i(VT )dA = σA = Q, 0.

(1.27) (1.28) (1.29) (1.30)

These equations will be developed in detail in the next section. It will also be shown that an evolutionary process which has a component in the direction of the "Topological Torsion" vector will produce a thermodynamically irreversible process on the physical system defined by the 1-form of Action, A. Evolution in the direction of the Topological Torsion vector causes topological change. Remark 11 Note that the Work 1-form, W , generated by a process in the direction of the Topological Torsion vector is not zero. Hence the Topological Torsion vector must have Spinor components. The previous paragraph contains the first introduction to the concept of Topological Torsion vector to appear in this monograph. The properties of the Topological Torsion vector will be extolled and examined again and again in the chapters that follow. The existence of a Topological Torsion direction field is a signal that the physical system is not in equilibrium. Evolution in the direction of the Topological Torsion vector is thermodynamically irreversible if the divergence of VT is not zero. The dogmatic insistence on topological invariance in many classical physical theories in effect excludes the concept of the "topological torsion". When σ = 1, the Topological Torsion vector has been called the "Liouville vector field" (see page

Unique Continuous Evolutionary Processes

45

65 [140]). If the topological structure of the physical system, A, evolves (or decays) from a domain where the rank of dA is even to a domain where the maximal rank of dA is odd, then the components of VT become proportional to a characteristic direction field, which has both extremal properties and associated properties. The extremal processes, VE , can always be put into correspondence with a Hamiltonian process, but those processes represented by a direction field component proportional to VT do not have a Hamiltonian representation, unless the divergence of VT is zero. These features apply not only to topological dimension 3 and 4, but are also valid for topological dimensions which are odd (VE for n = 2k + 1) or even (VT for n = 2k + 2). The topological refinement induced by the process forms two categories related to a Contact structure (n = 2k + 1) or to a Symplectic structure (n = 2k + 2). Note that the concept of "uniqueness" relates to the direction field V that represents a process, but such direction fields as a vector field are unique only to within an arbitrary (deformation) factor. Continuous evolution includes two equivalence classes of processes. There are those processes that preserve topological features (homeomorphisms) and those processes that do not (non-homeomorphisms). The latter class is the class of processes that describe continuous topological evolution, and it is this class which is studied extensively in this monograph. As will be demonstrated below, the topological structure of Ra physical system leads to the consideration of odd-dimensional inteR grals of the type 2k+1 AˆdA... and even-dimensional integrals of the type 2k+2 dAˆdA.... If these integrals are deformation invariants they represent a topological property that is an evolutionary invariant. Of particular interest is the set of even and odddimensional integrals where the integration chain is a closed cycle. The class of continuous processes that describe topological change can be divided into two distinct classes, A and B. • Class A. This equivalence class of non-homeomorphic continuous processes preserves the even-dimensional integrals as deformation invariants, but causes the values of the odd-dimensional integrals to change. The Helmholtz conservation of vorticity concept is a classic example of when an even-dimensional topological property is preserved. Such processes will be called Helmholtz (or Symplectic) processes, in general, when the 2-form of Action, dA, is an evolutionary invariant. The Poincare integral invariants of classical mechanics are further examples of even-dimensional integral invariants. Extremal and Hamiltonian processes are special cases of Helmholtz processes. However, it will be demonstrated that all such Helmholtz processes, which can produce topological change of the odd-dimensional integrals, are thermodynamically reversible. Topological change is a necessary, but not sufficient, condition for continuous thermodynamic irreversibility. • Class B. This equivalence class of non-homeomorphic continuous processes causes the values of both the even and the odd-dimensional integrals to change.

46

Non-Equilibrium Thermodynamics

Both the odd and the even topological features of the physical system are modified. It is this equivalence class that contains those processes which are thermodynamically irreversible. Without being too precise, both energy and angular momentum must change if a process is to be thermodynamically irreversible. Pasting together is a continuous process for which the topology of the final system state is not necessarily the same as the topology of the initial system state. Separation or cutting into parts is a discontinuous process for which the system topology of the final state is not the same as the system topology of the initial state. The obvious topological property that changes is the number of parts. Projections from higher dimensions to lower dimensions are classic examples of many to one differentiable maps that are not invertible. The obvious topological property that changes is the property of dimension. Consider a flat putty disc in the shape of an annulus. Deform the putty continuously such that the points that make up the central hole are pasted together. On the other hand make an interior cut in a disk of putty and discontinuously separate the points to make a hole. The obvious topological property that changes is the number of holes. (Discontinuous processes are more or less ignored in this presentation.) 1.9

The Arrow of Time

On a given domain, Baldwin has shown that the existence of a Cartan 1-form and its Pfaff sequence (1.8) may be used to define a "Cartan" topology over the domain (see Chapter 6 for details). If the Cartan 1-form is integrable in the sense of Frobenius, then the Cartan topology is a connected topology. If the Cartan 1-form is not integrable then the Cartan topology is a disconnected topology. Evolution from a connected topology to a disconnected topology can proceed only by means of discontinuous processes. However, evolution from a disconnected topology to a connected topology can be accomplished continuously. It is this latter class of continuous processes that will be in focus in this monograph. An important practical result of this fact is that the continuous hydrodynamic evolution from a streamline (integrable and reversible) state to a turbulent (non-integrable and irreversible) state is impossible, while the continuous evolution from a turbulent state to a streamline state is permissible [217]. The creation of irreversible turbulence is necessarily discontinuous, but the decay of turbulence can be continuous. It has been demonstrated ([192]) that for continuous but non-homeomorphic maps (C1 maps without continuous inverse) it is impossible to predict the functional form of either covariant or contravariant vector fields. That is, the functional form of tensor fields on the final state is not well defined in terms of the functional form of the field on the initial state, if the map from initial to final state is continuous but not homeomorphic. On the other hand it can be shown that covariant antisymmetric tensor fields are deterministic in a retrodictive sense, even though the

The Arrow of Time

47

continuous maps from initial to final state are not invertible. That is, the functional form of the components of differential forms defined on the final state are well defined on the initial state even if the map from the initial state to the final state is C1 smooth, but not a homeomorphism. With respect to continuous topological evolution there then exists a natural, logical, arrow of time, which is not observable with respect to diffeomorphic geometric evolution that preserves topology. Therefore, to understand irreversible phenomena, a retrodictive point of view seems to be of some value [259] and it is this non-statistical retrodictive point of view constructed on Cartan’s theory of exterior differential systems that is the point of departure in this monograph.

Figure 1.1 Continuous and discontinuous evolution The methods will be restricted at first to those evolutionary processes and systems which are C2 continuous. It is appreciated that this restriction does not cover all physical situations, where in my opinion, "true" discontinuities, not just mathematical artifacts, are possible. The continuous evolutionary processes to be considered will permit topology to change in a continuous but irreversible manner (example: the pasting together of two blobs, or the collapse of a hole). Discontinuous processes are, at present, excluded.

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Non-Equilibrium Thermodynamics

Chapter 2 TOPOLOGICAL THERMODYNAMICS 2.1

Continuous Topological Evolution and the First Law

2.1.1 Cartan’s Magic Formula The topological thermodynamic methods used herein are based upon Cartan’s theory of exterior differential forms. The thermodynamic view assumes that the physical systems to be studied can be encoded in terms of a 1-form of Action Potentials (per unit source), A, on a four-dimensional variety of ordered independent variables, {ξ 1 , ξ 2 , ξ 3 , ξ 4 }. The variety supports a volume element Ω4 = dξ 1 ˆdξ 2 ˆdξ 3 ˆdξ 4 . No metric, no connection, no constraint of gauge symmetry is imposed upon the fourdimensional variety. Topological constraints will be imposed in terms of exterior differential systems [34]. In order to make the equations more suggestive to the reader, the symbolism for the variety of independent variables will be of the format {x, y, z, t}, but be aware that no constraints of metric or connection are imposed upon this variety. For instance, it is NOT assumed that the variety is Euclidean. The 1-form of Action, A, will have components that form a covariant direction field (relative to diffeomorphisms), Ak (x, y, z, t), to within a non-zero factor. Evolutionary processes on space-time will be determined in terms of four-dimensional contravariant direction fields, V4 (x, y, z, t) , to within a non-zero factor, ρ. Continuous topological evolution (see Chapter 7) will be defined in terms of Cartan’s magic formula for the Lie differential, Cartan-Lie differential :

L(ρV4 ) A = i(ρV4 )dA + d(i(ρV4 )A).

(2.1)

The definition is to be interpreted algebraically, using the properties of the exterior differential and the inner product associated with exterior differential forms. Many derivations of the Cartan-Lie differential formula presume a dynamic constraint, such that the vector field V4 (x, y, z, t) be the generator of a single parameter group; if true, then the topological constraint of Kinematic Perfection is satisfied, Kinematic Perfection :

dxk − V k dt ⇒ 0.

(2.2)

The topological constraint in effect defines (or presumes) a limit process. The constraint on the more general Lie differential leads to the concept of the Lie derivative of the 1-form A. The evolution is represented by the infinitesimal propagation of the

50

Topological Thermodynamics

1-form, A, down the flow lines generated by the 1-parameter group. However, such a kinematic constraint is not usually imposed in the presentation in this monograph; the vector field may have multiple parameters, which leads to the important concept of topological fluctuations, discussed in section 2.6. In this series, the Lie differential (not Lie derivative) is the fundamental generator of continuous topological evolution. It is interesting to note that in Felix Klein’s discussions of the development of calculus, he says "The primary thing for him (Leibniz) was not the differential quotient (the derivative) thought of as a limit." It is important to remember that the concept of a differential form is different from the concept of a derivative, where a (topological) limit has been defined. When acting on an exterior differential 1-form of Action, A = Ak dxk , Cartan’s magic (algebraic) formula is equivalent abstractly to the first law of thermodynamics. The name "Cartan’s Magic Formula" is due to Marsden. [148]. Cartan’s Magic Formula L(ρV4 ) A First Law of Thermodynamics Inexact 1-form of Heat L(ρV4 ) A Inexact 1-form of Work W Internal Energy U

= : = = =

i(ρV4 )dA + d(i(ρV4 )A), W + dU = Q, Q, i(ρV4 )dA, i(ρV4 )A.

(2.3) (2.4) (2.5) (2.6) (2.7)

In effect, Cartan’s magic formula leads to a topological basis of thermodynamics. The First Law is a statement of continuous topological evolution in terms of deRham cohomology theory; the difference between two non-exact differential forms is equal to an exact differential. The topological methods to be described below go beyond the notion of processes which are confined to equilibrium systems. Non-equilibrium systems and processes which are thermodynamically irreversible, as well as many other classical thermodynamic ideas, can be formulated in precise mathematical terms using the topological features of the three thermodynamic 1-forms, A, W, and Q, without the use of statistics or metric constraints. 2.1.2 An electromagnetic example The thermodynamic identification of the terms in Cartan’s magic formula are not whimsical. To establish an initial level of credence in the terminology, consider the 1-form of Action, A, where the component functions are the symbols representing the vector and scalar potentials in electromagnetic theory. The coefficient functions have arguments over the independent variables {x, y, z, t}, A = Ak (x, y, z, t)dxk = A ◦ dr − φ dt.

(2.8)

Construct the 2-form of field intensities, F = dA = (∂Ak /∂xj − ∂Aj /∂xk )dxjˆ dxk = Fjk dxj ˆdxk = +Bz dxˆdy... + Ex dxˆdt... .

(2.9) (2.10)

Continuous Topological Evolution and the First Law

51

Contract the 2-form, F, with a 4-vector direction field, J (scaling parameter, ρ), J = [J, ρ] = ρ[v, 1] = ρV4 ,

(2.11)

W = i(ρV4 )dA = i(J)F, ⇒ −{ρE + J × B} ◦ dr + {J ◦ E}dt. The Lorentz force : − {fLorentz } ◦ dr component. The dissipative power : +{J ◦ E}dt component.

(2.12) (2.13) (2.14) (2.15)

to yield the expressions: The Work 1-form:

For those with experience in electromagnetism, note that the construction yields automatically the expression for the "Lorentz force" as a derivation, without further assumptions. The dot product of a force and a differential displacement defines the elementary concept of differential work. The 4-component Work 1-form, W , includes the spatial contribution related to force times differential distance, dr, and a differential time component, dt, with a coefficient which is recognized to be the dissipative power, {J ◦ E}. The Work 1-form, W , is not necessarily a perfect differential, and therefore can be path dependent. Closed cycles of Work need not be zero. Next compute the Internal Energy term, Internal Energy: U = i(J)A = J ◦ A − ρφ = ρ[v ◦ A − φ].

(2.16)

The result is to be recognized as the "interaction" energy density in electromagnetic plasma systems. It is apparent that the internal energy, U, corresponds to the interaction energy of the physical system; that is, U is the internal stress energy of system deformation. Therefore, the electromagnetic terminology can be used to demonstrate that Cartan’s magic formula is another way to state that the first law of thermodynamics makes practical sense. It is remarkable that although the symbols are different, the same basic constructions and conclusions apply to most classical physical systems. Examples given in Part II will demonstrate this feature of topological universality. The correspondence so established between the Cartan magic formula acting on a 1-form of Action, and the first law of thermodynamics is taken both literally and seriously in this monograph. The Cartan methods are not limited to applications in electromagnetic theory. As mentioned above, the thermodynamic phenomena and the associated topological results describe herein have universal qualities, and applicability to all physical theories. The topological methods permit the long sought for integration of mechanical and thermodynamic concepts, without the constraints of equilibrium systems, and statistical analysis. The methods yield explicit constructions for testing

52

Topological Thermodynamics

when a process acting on a physical system is irreversible. The methods permit irreversible adiabatic processes to be distinguished from reversible adiabatic processes analytically. Adiabatic processes need not be "slow" or quasi-static. Given any 1-form, A, W, and Q, the concept of Pfaff topological dimension permits separation of processes and systems into equivalence classes. For example, dynamical processes can be classified in terms of the topological Pfaff dimension of the Work 1-form, W . All Hamiltonian systems have a Work 1-form, W , of topological Pfaff dimension of 1, (dW = 0), and therefore cannot describe irreversible processes for which the topological Pfaff dimension is 4. A discussion of the concept of Pfaff topological dimension and its application to physical systems and physical processes appears in the next section 2.2 2.2.1

Thermodynamic Systems Applications of the Pfaff Topological Dimension

As mentioned in Chapter 1, one of the most useful topological tools is that defined as the Pfaff Topological Dimension (or class). Recall that it is possible to define many topologies on the same set of elements. For any given exterior differential 1-form of functions, say A = Ak (x, y, z, t)dxk , it is possible to construct the Pfaff sequence of terms, {A, dA, AˆdA, dAˆdA}. These elements of the Pfaff sequence may be used to construct a Cartan Topology and a Cartan Topological structure (see Chapter 6). In the Cartan topology, the exterior differential operator, d, acts as limit point generator. Hence the union of a form and its exterior differential creates the topological (Kuratowski) closure of the form. It is known that a Kuratowski closure operator defines a topology [141]. For any given 1-form, the Pfaff sequence will contain M successive non-zero terms equal to or less than N, the number of geometric dimensions of the base independent variables. The number M is defined as the "Pfaff Topological Dimension" or class of the given 1-form. The concept implies that there is a submersive map from the space of geometric dimensions, N, to a space of topological dimensions, M. The topological properties of the given 1-form are expressible in terms of this irreducible number, M, of functions and differentials. The results are invariant with respect to the submersive pullback to the geometric range space of dimension, N, from the target space of topological dimension, M ≤ N. The three important 1-forms of thermodynamics, A, W, and Q, can have different Pfaff dimensions, and can generate different topologies on the elements of the same geometric space. (It is possible to construct many topologies on the same set of base elements.) The 1-form of Action, A, is used to define the topology of the physical thermodynamic system, and the topologies induced by the 1-forms of Work, W , and Heat, Q, are described as topological refinements. The topological refinements depend on both the particular system and a particular process applied to the system. Suppose the 1-form of Work, W, is defined in terms of two functions

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53

as W = P dV. The Pfaff sequence consists of the terms {W, dW, 0, 0} as W ˆdW = 0; hence in this example, the Pfaff dimension of W is 2. From the first law, under the assumption that W = P dV, Q dQ QˆdQ dQˆdQ

= = = =

W + dU = P dV + dU, dW = dP ˆdV, W ˆdW + dUˆdW = 0 + dUˆdP ˆdV, 0.

(2.17) (2.18) (2.19) (2.20)

Hence, the Pfaff dimension of 2 for the Work 1-form, W , can be related to a Pfaff dimension of 3 for the Heat 1-form, Q, unless the pressure is a function of the internal energy and the volume. In this latter case, the Pfaff dimension of Q and W are both 2. In this monograph, attention will be focused on dissipative turbulent systems with thermodynamic irreversible processes such that the Pfaff topological dimensions of A, W, and Q will be maximal and equal to 4. (The techniques can be extended to higher-dimensional geometric spaces.) These Turbulent systems of Pfaff dimension 4 are not topologically equivalent to Equilibrium systems (for which the topological dimension is 2, at most). Topological defects in the Turbulent state will be associated with sets of space-time where the Pfaff topological dimensions of A, W, and Q are not maximal. It is remarkable that such topological defect sets can form attractors causing self organization and long lived states of Pfaff dimension 3, which are far from equilibrium. Examples will be presented below. 2.2.2 Physical Systems: Equilibrium, Isolated, Closed and Open Physical systems and processes are elements of topological categories determined by the Pfaff topological dimension (or class) of the 1-forms of Action, A, Work, W , and Heat, Q. For example, the Pfaff topological dimension of the exterior differential 1-form of Action, A, determines the various species of thermodynamic systems in terms of distinct topological categories:

dA AˆdA d(AˆdA) dAˆdA

= = = 6=

Thermodynamic Systems defined by the Pfaff Topological Dimension of A 0 Equilibrium - Pfaff dimension 1, 0 Isolated - Pfaff dimension 2, 0 Closed - Pfaff dimension 3, 0 Open - Pfaff dimension 4.

(2.21) (2.22) (2.23) (2.24)

There are two topological thermodynamic categories, one is determined by the closure (or differential ideal) of the 1-form of Action, A ∪ dA, and the other is determined by the closure of the 3-form of topological torsion, AˆdA ∪ dAˆdA. The first category is represented by a connected Cartan topology, while the second category is represented by a disconnected Cartan topology. The Cartan topology is discussed in detail in Chapter 6.

54

Topological Thermodynamics

Connected Topology AˆF = 0 1. Equilibrium physical systems are elements such that the Pfaff topological dimension of the 1-form of Action, A, is 1. 2. Isolated physical systems are elements such that the Pfaff topological dimension of the 1-form of Action, A, is 2, or less. Isolated systems of Pfaff dimension 2 need not be in equilibrium, but (in historic language) do not exchange radiation or mass with the environment. Disconnected Topology AˆF 6= 0 1. Closed physical systems are elements such that the Pfaff topological dimension of the 1-form of Action, A, is 3. Closed systems can exchange radiation, but not mass, with the environment. 2. Open physical systems are such that the Pfaff topological dimension of the 1form of Action, A, is 4. Open physical systems can exchange both radiation and mass∗ with the environment. Note that these topological specifications as given above are determined entirely from the functional properties of the physical system encoded as a 1-form of Action, A. The system topological categories do not involve a process, which is encoded (to within a factor) by some vector direction field, V4 . However, the process V4 does influence the topological properties of the Work 1-form, W , and the Heat 1-form, Q. Compare these topological definitions, whereby Equilibrium or Isolated systems are determined in terms of two independent variables at most, and Duhem’s theorem "Whatever the number of phases, components and chemical reactions, if the initial mole numbers Nk of all components are specified, the equilibrium state of a closed system is completely specified by two independent variables. (p.182 [180])" 2.2.3 Equilibrium versus Non-Equilibrium Systems The intuitive idea for an equilibrium system comes from the experimental recognition that the intensive variables of pressure, P, and temperature, T , (conjugate to volume and entropy) become domain constants in an equilibrium state, dP ⇒ 0, dT ⇒ 0. A definition made herein is that the Pfaff topological dimension in the interior of a physical system which is in the equilibrium state is at most 1 [12]. Formally, the ∗

The use of the word mass to distinguish between closed and open systems is a legacy that ought to be changed to "mole or particle" number, as it is now known that mass energy can be converted to radiation, and radiation can produce massive pairs.

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55

idea is restated such that the equilibrium state is a Lagrangian submanifold of a fourdimensional symplectic manifold, and upon this Lagrangian submanifold, the 2-form dA, that generates the symplectic structure, vanishes. Hence the equilibrium state is of Pfaff topological dimension 1, Equilibrium {A 6= 0, dA = 0, AˆdA = 0, dAˆdA = 0}.

(2.25)

The isolated physical system is of Pfaff dimension 2, Isolated {A 6= 0, dA 6= 0, AˆdA = 0, dAˆdA = 0}.

(2.26)

For both the isolated or equilibrium system, the Cartan topology generated by the elements of the Pfaff sequence for A is then a connected topology of one component, as AˆdA = 0 (see Chapter 6). Although the Pfaff topological dimension of A is at most 2 in the isolated state, processes in the isolated state are such that the Work 1-form, W , and the Heat 1-form, Q , must be of Pfaff dimension 1. For suppose W = P dV, then dW = dP ˆdV ⇒ 0 if the pressure is a domain constant. Similarly, suppose Q = T dS, then dQ = dT ˆdS ⇒ 0 if the temperature is a domain constant. Hence both W and Q are of Pfaff dimension 1 for this isolated example. If the Pfaff dimension of the 1-form of Action is 1, then dA ⇒ 0. It follows in this more stringent case that W ⇒ 0. Hence for elementary systems the pressure must vanish or the Volume is constant, and the Heat 1-form, Q, is a perfect differential, Q = d(U ). Of particular interest herein are those regions of base variables for open, nonequilibrium, turbulent physical systems, formed by the closure† of the 3-forms AˆdA, W ˆdW , and QˆdQ. For such regions, the Pfaff topological dimension of the 1-forms, A, W, and Q, are all initially of Pfaff topological dimension 4, Open dAˆdA 6= 0, dW ˆdW 6= 0, dQˆdQ 6= 0,

(2.27)

save for defect regions that are of Pfaff dimension 3 (or less). It is remarkable that evolutionary dissipative irreversible processes in such open systems can describe evolution to regions of base variables where the Pfaff topological dimension of the 1form of Action, A, changes from 4 to 3. Such processes describe topological change in the physical system. For a given 1-form of Action, A, those regions of Pfaff topological dimension 3, once created, form topological "defect structures" in the closure of the 3-form, AˆF. The defect structures of the 1-form of Action, A, (of Pfaff dimension 3) can behave as long lived (excited) states of the initial physical system, but they are far from equilibrium and are not isolated, for they are not of Pfaff topological dimension equal to 2 or less. Such excited states (of odd topological dimension) can admit extremal processes of Kinematic Perfection, and can have a Hamiltonian †

The closure of the p-form Σ is the union of Σ and dΣ, which Cartan has called a differential ideal.

56

Topological Thermodynamics

generator for the kinematics represented as a system of first order ordinary differential equations. The Hamiltonian evolution remains contained in the defect structure, unless topological fluctuations destroy the Kinematic Perfection. Such concepts can be applied to a model of cosmology (where the stars are the defect structures), to turbulent plasmas and fluids (where wakes are the defect structures), and to a better understanding of the arrow of time. Although the defects in the turbulent non-equilibrium regime are not necessarily equilibrium structures, once formed and self organized as coherent topological structures of Pfaff dimension 3, they can evolve along extremal trajectories that are not dissipative. Indeed such extremal processes have a Hamiltonian representation. These "stationary", or long lived (excited), states of Pfaff dimension 3, indeed are states "far" from the equilibrium state, which requires a Pfaff dimension of 1. Note that the word "far" does not imply a "distance". The Pfaff dimension 3 and 4 sets are not even "connected" to the equilibrium states in a topological sense. The non-equilibrium but isolated states of a physical system that are "near-by" to the equilibrium state, are "connected" to the equilibrium state, and are of Pfaff dimension 2. The descriptive words of self-organized states far from equilibrium have been abstracted from the intuition and conjectures of I. Prigogine [180]. However the topological theory presented herein presents for the first time a solid, formal, mathematical justification (with examples) for the Prigogine conjectures. Precise definitions of equilibrium and non-equilibrium systems, as well as reversible and irreversible processes can be made in terms of the topological features of Cartan’s exterior calculus. Using Cartan’s methods of exterior differential systems, thermodynamic irreversibility and the arrow of time can be well defined in a topological sense [221], a technique that goes beyond (and without) statistical analysis. Thermodynamic irreversibility and the arrow of time requires that the evolutionary process produce topological change. 2.2.4 Change of Pfaff Topological Dimension It should be noted that the closed components of the 1-form of Action do not effect the components of the 2-form of intensities, F = dA = d(Ac + A0 ) = 0 + d(A0 ) = F0 . However, these "gauge" additions of closed forms, Ac , do influence the topological dimension of the 1-form of Action. For example, let A0 be of Pfaff Topological dimension 2, representing an isolated system where A0 ˆdA0 = 0. Then by addition of a closed component to the original action, the new 1-form of Action, A = Ac + A0 could have a topological dimension of 3, as AˆdA = (Ac + A0 )ˆdA0 = Ac ˆdA0 6= 0.

(2.28)

So the addition of a closed component to the 1-form of Action could change the system from an isolated system of Pfaff dimension 2 to a closed system of Pfaff dimension 3. The 4-form dAˆdA is not influenced by the (gauge) addition to the original 1-form of Action, because dAˆdA = dA0 ˆdA0 . (2.29)

Thermodynamic Systems

57

In higher dimension, such gauge additions imply that the Pfaff dimension can change according to the rule, 2n ⇒ 2n + 1. It is also possible to change the Pfaff dimension of a 1-form by "renormalization", or better said, by "rescaling" with a multiplying function, often in the form of an integrating factor. For example, consider the 1-form A0 of Pfaff dimension 4, such that d(A0 ˆdA0 ) 6= 0. Next rescale the 1-form such that A = βA0 . Then, d(AˆdA) = d(β 2 A0 ˆdA0 ) ⇒ 0,

(2.30)

subject to the constraint that β 2 is an integrating factor for the 3-form A0 ˆdA0 . In four dimensions there exists an infinite number of such functions that serve as integrating factors for the 3-form of Topological Torsion, A0 ˆdA0 . The integrating factors (which can be formulated from Holder norms) can be interpreted as distributions of "density" which change the Pfaff topological dimension from 4 to 3, or, in general, from 2n + 2 ⇒ 2n + 1. Such distributions can be put into correspondence with "stationary" states far from equilibrium. As an example of how the Pfaff dimension of a 1-form can be modified by a gauge addition, consider a 1-form representing a Bohm-Aharanov-Abrikosov singular "vortex" string, γ = b(ydx − xdy)/(x2 + y 2 ), (2.31) to which is added a p 1/r potential for a point source [276]. The bare m/r "Coulomb" potential, A0 = m/ (x2 + y 2 + z 2 )dt exhibits no Topological Torsion but does exhibit Topological Spin. The 1/r potential term implies that A0 6= 0. Hence the 1-form of Action representing a bare "coulomb" potential, is not in equilibrium, but does represent a connected "isolated" topology of Pfaff dimension 2. The combined 1-form of Action, p A = b(ydx − xdy)/(x2 + y 2 ) + m/ (x2 + y 2 + z 2 )dt, (2.32)

even though dγ = 0, is of Pfaff dimension 3, not 2. The Topological Torsion 3-form AˆF depends on both b and m, and is zero if b = 0, or if m = 0, reducing the Pfaff dimension of the modified 1-form back to 2. 2.2.5 Systems with Multiple Components One of the most remarkable properties of the Cartan topology generated by a Pfaff sequence is due to the fact that when AˆdA = 0, (Pfaff dimension 2 or less) the physical system is reducible to a single connected topological component. This single connected topological component need not be simply connected. The Topological Torsion field vanishes on equilibrium domains. On the other hand when AˆdA 6= 0, (the Pfaff topological dimension of the 1-form, A, is 3 or more) the physical system admits more than one topological component, and the topology is a disconnected topology (see Chapter 6).

58

Topological Thermodynamics

Remark 12 When the Pfaff dimension is 3 or greater (such that conditions of the Frobenius unique integrability theorem are not satisfied), solution uniqueness to the Pfaffian differential equation, A = 0, is lost. If solutions exist, there is more than one solution. Such concepts lead to propagating discontinuities (signals), envelope solutions‡ (Huygen wavelets), an edge of regression (the Spinodal line of phase transitions) a lack of time reversal invariance, and the existence of irreducible affine torsion in the theory of connections. It is my opinion that a dogmatic insistence that a viable physical theory must give a unique prediction from a set of given initial conditions historically has hindered the understanding of irreversibility and non-equilibrium systems. Irreversibility and nonequilibrium are concepts that require non-uniqueness, and demand that the dogma of predictive uniqueness, mentioned above, has to be rejected. 2.3

Thermodynamic Processes

2.3.1 Continuous Processes All continuous processes (see Chapter 7) may be put into equivalence classes as determined by the vector direction fields, V , that locally generate a flow. For example on a domain of geometric dimension, n, and for the 1-form, A, those n-1 vector fields, Vassociated , that satisfy the transversal equation, Associated Class: i(ρVassociated )A = 0,

(2.33)

are said to be elements of the associated class of vector fields relative to the form A. If the direction field of the 1-form of Action is considered to be a fiber, then the associated vectors are also said to be "horizontal". The associated vectors will form a distribution orthogonal to the 1-form, A, but the distribution need not be a smooth foliation. That is, the fiber direction field is not necessarily the normal field to an implicit hypersurface. The requirement for a smooth foliation is that the associated 1form be of Pfaff topological dimension 2 or less. For such associated processes acting on a 1-form of Action, A, the "internal interaction energy" vanishes! As shown below, processes generated by associated vectors relative to the 1-form of Action, A, are also included in the set of thermodynamic locally adiabatic processes. Other locally adiabatic processes are generated by those processes which are associated vectors of the exterior differential of the internal energy, U. In both cases, locally adiabatic processes are null vectors of the Heat 1-form, Q, in the sense that i(ρVadiabatic )Q = 0. As defined in the previous chapter those vector fields, VE , that satisfy the equations, Extremal Class: i(ρVE )dA = 0, (2.34) are said to be elements of the extremal class of vector fields. As the matrix of functions that define the 2-form dA is antisymmetric, the extremal vector is proportional ‡

See section 7.8.

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59

to that eigenvector of the antisymmetric matrix that has a zero eigenvalue. If the matrix dA is of maximum rank, then there can be only one (unique) eigenvector with zero eigenvalue, and that null eigenvector exists only if the Pfaff topological dimension of the 1-form A is odd, (2n+1). In other words, the 2-form dA defines a Contact manifold. The extremal direction field is completely determined (to within a factor) by the component functions of the 1-form A utilized in its definition. Note that the work 1-form W = i(ρVE )dA ⇒ 0 vanishes for extremal evolutionary processes. As shown below, this constraint is the essence of Hamiltonian mechanics. It is rather remarkable (and only fully appreciated by me in February, 2005) that there is a large class of direction fields (given the symbol ρS4 ) that do not behave as vectors (with respect to rotations). They are isotropic complex vectors of zero length, defined as Spinors by E. Cartan [44], but which are eigendirection fields relative to the antisymmetric matrix generated by dA: The Spinor Class: i(ρS4 )dA 6= 0.

(2.35)

W = i(ρV4 )dA = fm dxm + P dt.

(2.36)

To understand these claims, realize that the Work 1-form, W , is a generalization of the Newtonian concept of Force times Distance:

The 2-form dA can be realized as an antisymmetric matrix of functions. The concept of Work as the 1-form W = i(ρV4 )dA focuses attention on the importance of the 2form, F = dA, and its antisymmetric matrix representation, F ' [F] = − [F]transpose . The concept of Work is (in effect) related to the matrix product of [F] and some vector direction field: W = i(ρV4 )dA ' [F] ◦ |ρV4 i .

(2.37)

i(ρV4 )dA ' [F] ◦ |ei = γ |ei .

(2.38)

i(ρV)i(ρV)dA ' he| ◦ [F] ◦ |ei = γ he| ◦ |ei .

(2.39)

i(ρV)i(ρV)dA ' he| ◦ [F] ◦ |ei = 0.

(2.40)

Suppose e =ρV4 is an eigendirection field with eigenvalue γ, such that

Then,

Due to antisymmetry, it follows that

Hence, for the antisymmetric matrix, [F], it must be true that γ he| ◦ |ei = 0.

(2.41)

For division algebras there are two choices, either γ = 0, or he| ◦ |ei = 0. The implication is that for non-zero eigenvalues γ, the quadratic form must vanish:

60

Topological Thermodynamics

he| ◦ |ei = (e1 )2 + (e2 )2 + (e3 )2 + (e4 )2 = 0.

(2.42)

|ρV4 i ⇒ |ρS4 i if γ 6= 0.

(2.43)

The null quadratic form is equal to the sum of squares of the components, if the eigenvalue is not zero. The finite eigendirection field of an antisymmetric matrix, with a non-zero eigenvalue, but a zero quadratic form, implies that the vector direction field must be composed of Spinors,

As mentioned above, Spinors have metric properties, behave as vectors with respect to transitive maps, but do not behave as vectors with respect to rotations (see p. 3, [44]). Spinors generate harmonic forms and are related to conjugate pairs of minimal surfaces. The notation that a Spinor is a complex isotropic direction field is preferred over the names "complex isotropic vector", or "null vector" that appear in the literature. As shown below, the familiar formats of Hamiltonian mechanical systems exclude the concept of Spinor process direction fields, for the processes permitted are restricted to be represented by direction fields of the extremal class, which have zero eigenvalues (see Chapter 4.5). Remark 13 Spinors are normal consequences of antisymmetric matrices, and, as topological artifacts, they are not restricted to physical microscopic or quantum constraints. According to the topological thermodynamic arguments, Spinors are implicitly involved in all processes for which the 1-form of Work is not zero. Spinors play a role in topological fluctuations and irreversible processes. Note that the 1-form of Work can be expanded in terms of a basis of Spinors and the extremal field, if it exists. As an example, consider the 1-form of Action and its associated Pfaff sequence given by the expressions, A F AˆF F ˆF

= = = =

ydx − xdy + sdz − zds, dA = 2dyˆdx + 2dsˆdz, 2{xdyˆdzˆds − ydxˆdzˆds + zdxˆdyˆds − sdxˆdtˆdz, 8dxˆdyˆdzˆds.

Note that the 4x4 antisymmetric matrix is of the ⎡ 0 1 0 ⎢ −1 0 0 [F] = ⎢ ⎣ 0 0 0 0 0 −1

with eigenvalues and eigenvectors§ , §

It would be better to say eigendirection field.

(2.44) (2.45) (2.46) (2.47)

form (neglecting a factor of 2), ⎤ 0 0 ⎥ ⎥, (2.48) 1 ⎦ 0

Thermodynamic Processes

61

Eigenvectors = [0, 0, 1, i], [1, i, 0, 0] with eigenvalue = i, Eigenvectors = [0, 0, 1, −i], [1, −i, 0, 0] with eigenvalue = −i.

(2.49) (2.50)

Each spinor eigenvector is null isotropic such that the sum of squares of the coefficients is zero. This example is a simple example generated from the 1-form, A, which is the adjoint of the three exact differentials generated by the Hopf map (see Chapters 3.5 and 5). Note that the 4x4 matrix [F] acting on a spinor eigenvector causes the eigenvector to be "rotated" by π/2 in the complex plane, and the operation is not reflexive of degree 2, but is reflexive of degree 4. That is, [F]2 ◦ |Spinori = − |Spinori and

[F]4 ◦ |Spinori = + |Spinori .

(2.51)

Note that the third power of the 4x4 matrix, [F] , is the Inverse of [F] . This works only for antisymmetric matrices in even dimensions, for there is no inverse to an antisymmetric matrix in odd dimensions (the determinant in odd dimensions is zero). If the Pfaff topological dimension of the 1-form A is even, as in the example above, then a unique extremal vector (with eigenvalue zero) does not exist. The reduced topological domain (not necessarily the entire geometric domain) is a symplectic manifold of even dimensions, (2n + 2). However, on a symplectic manifold of 4 geometric dimensions and 4 topological dimensions, it follows that there does exist a unique vector direction field, the Topological Torsion vector, VT , (or sometimes written as T4 ), completely determined (to within a factor) in terms of the functions which define the physical system. Topological Torsion Class : :

i(ρVT )dA = ρσA, i(ρVT )A = 0.

(2.52) (2.53)

Note that the internal energy generated by a process of the Torsion class is zero. For the example above, the Topological Torsion vector is proportional to the 4-vector, VT = 2[x, y, z, s], which indeed has a non-zero divergence. Moreover, i(VT )A = xy − yx + zs − sz = 0. As i(ρVT )dA is not zero, it must be composed from the 2 spinor conjugate pairs of eigenvectors of the antisymmetric matrix representation of the 2-form, dA. A related vector field is defined as the Iso-Vector, which is not associated, but satisfies the constraints, Iso-Vector Class but

: :

i(ρViso )dA = ρσA, d{i(ρViso )A} = 0.

(2.54) (2.55)

The internal energy of such Iso-vector processes is a constant. The factor ρσ is an "integrating factor" for the 1-form of Action, A, which implies that the 1-form A cannot be of Pfaff dimension greater that 2.

62

Topological Thermodynamics

In the next section it will be shown that evolution with a component in the direction of the "Topological Torsion" vector will produce an irreversible process on the physical system (as encoded by the Action 1-form), if the divergence of the "Topological Torsion" vector is not zero. This "Topological Torsion" vector, equivalent to the 3-form AˆdA = AˆF, is always an associated vector, but it is not necessarily an extremal vector, relative to the Action 1-form, A. The Torsion vector is identically zero on domains of Pfaff topological dimension 2. Hence non-zero values of the Torsion vector are an indication that the physical system, A, is not an equilibrium system. The Topological Torsion vector exists only on domains of Pfaff topological dimension 3 or greater, in the same sense that Frenet-Serret torsion exists only on domains of geometric dimension 3 or greater. With respect to evolution in the direction of the Torsion Current, the symplectic 4D volume is contracting or expanding exponentially unless σ = 0. If the divergence of VT orsion vanishes, σ ⇒ 0, and therefore such vector fields cannot represent a symplectic process (which preserves the volume element, dAˆdA). The factor, σ, is a Liapunov function and defines the stability of the process (depending on the sign of σ). When σ = 1, the Torsion vector has been called the "Liouville vector" [140]. Vector fields which are both extremal and associated are said to be elements of the characteristic class, Vcharacteristic , of vector fields [113]. They are not elements of the Spinor class, but are elements of the extremal class of processes. Characteristic Class and

: :

i(ρVcharacteristic )A = 0, i(ρVcharacteristic )dA = 0.

(2.56) (2.57)

Note that characteristic flow lines generated by Vcharacteristic of the Characteristic class preserve the Cartan topology, for each element, {A, dA, AˆdA, ..}, of the Cartan topological base is invariant with respect to the action of the Lie differential relative to characteristic flows. Characteristics are often associated with wave phenomena, and propagating discontinuities. They are locally adiabatic. The Topological Torsion vector mentioned above may have zero divergence on certain geometric subsets of space-time, but these domains are of Pfaff topological dimension 3 (although of geometric dimension 4). In such cases, the Topological Torsion vector will be a characteristic vector for the 1-form of Action, A. These and other properties of the "Topological Torsion" vector will be described in detail by examples presented below. 2.3.2 Reversible and Irreversible Processes The Pfaff topological dimension of the exterior differential 1-form of Heat, Q, determines important topological categories of processes. From classical thermodynamics "The quantity of heat in a reversible process always has an integrating factor" [86] [160]. Hence, from the Frobenius unique integrability theorem, which requires QˆdQ = 0, all reversible processes are such that the Pfaff dimension of Q is less than or equal to 2. Irreversible processes are such that the Pfaff dimension of Q is

Thermodynamic Processes

63

greater than 2, and an integrating factor does not exist. A dissipative irreversible topologically turbulent process is defined when the Pfaff dimension of Q is 4. Processes : as defined by the Pfaff dimension of Q QˆdQ = 0 Reversible - Pfaff dimension 2. d(QˆdQ) 6= 0 Turbulent - Pfaff dimension 4.

(2.58) (2.59)

Note that the Pfaff dimension of Q depends on both the choice of a process, V4 , and the system, A, upon which it acts. As reversible thermodynamic processes are such that QˆdQ = 0, and irreversible thermodynamic processes are such that QˆdQ 6= 0, Cartan’s formula of continuous topological evolution can be used to determine if a given process, V4 , acting on a physical system, A, is thermodynamically reversible or not: Processes defined by : the Lie differential of A L(ρV4 ) A = Q. Reversible Processes ρV4 : QˆdQ = 0, L(ρV4 ) AˆL(ρV4 ) dA = 0. Irreversible Processes ρV4 : QˆdQ 6= 0, L(ρV4 ) AˆL(ρV4 ) dA 6= 0.

(2.60) (2.61) (2.62) (2.63) (2.64)

Remarkably, Cartan’s magic formula can be used to describe the continuous dynamic possibilities of both reversible and irreversible processes, acting on equilibrium or non-equilibrium systems, even when the evolution induces topological change, transitions between excited states, or changes of phase, such as condensations. The idea that irreversible processes must satisfy QˆdQ 6= 0, and the idea that the Pfaff topological dimension of such irreversible domains must be related to symplectic (2n+2) domains dAˆdA 6= 0, is opposed to the presentations of Chen, who, in a set of articles of almost identical content, [49], proposed that non-equilibrium thermodynamics and irreversible processes should be studied on contact manifolds (2n+1) constrained such that QˆdQ = 0. According to the criteria that an irreversible process implies that an integrating factor does not exist for the Heat 1-form, Q, Chen’s theory is patently false. It is important to note that the direction field, V4 , need not be topologically constrained such that it is singularly parameterized. That is, the evolutionary processes described by Cartan’s magic formula are not necessarily restricted to vector fields that satisfy the topological constraints of Kinematic Perfection, dxk −V k dt = 0. A discussion of topological fluctuations, where dxk − V k dt = ∆k 6= 0, and an example fluctuation process is described in Section 2.6. In the next section it will be demonstrated that evolution in the direction of the Topological Torsion vector (or Current), T4 , defined from the components of the 3-form of topological torsion, i(T4 )dxˆdyˆdzˆdt = AˆdA,

(2.65)

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induces a process which satisfies the equations of a conformal evolutionary process, L(T4 ) A = σA

and

i(T4 )A = 0, σ 6= 0,

(2.66)

such that L(T4 ) AˆL(T4 ) dA = QˆdQ = σ2 AˆdA 6= 0.

(2.67)

Remark 14 Evolution in the direction of the Topological Torsion vector, T4 , relative to a physical system encoded by the 1-form A, is thermodynamically irreversible. A crucial idea is to recognize that irreversible processes must be on domains of Pfaff topological dimension which support Topological Torsion, AˆdA 6= 0, with its attendant properties of non-uniqueness, envelopes, regressions, and projectivized tangent bundles. Such domains are of Pfaff dimension 3 or greater. Moreover, as described below, it would appear that thermodynamic irreversibility must support a non-zero Topological Parity 4-form, dAˆdA 6= 0. Such domains are of Pfaff dimension 4 or greater. 2.3.3 Adiabatic Processes - Reversible and Irreversible The topological formulation of thermodynamics in terms of exterior differential forms permits a precise definition to be made for both reversible and irreversible adiabatic processes in terms of the topological properties of Q. On a geometrical space of N dimensions, a 1-form, Q, will admit N-1 associated vector fields, VAssociated , such that i(VAssociated )Q = 0. Processes defined by associated vector fields, VAssociated , relative to Q are defined as (locally) adiabatic processes, Vadiabatic [12]. Locally Adiabatic Processes:

i(Vadiabatic )Q = 0.

(2.68)

The N-1 associated vectors will form a distribution of adiabatic processes orthogonal to the 1-form Q. The distribution of adiabatic processes will not form a smooth hypersurface, unless the Pfaff dimension of Q is 2 or less. In other words the null curves (adiabats) form a smooth hypersurface only in the equilibrium or isolated state. Note that all adiabatic processes are defined by vector direction fields, to within an arbitrary factor, β(x, y, z, t). That is, if i(VA )Q = 0, then it is also true that i(βVA )Q = 0. The adiabatic direction fields and the 1-form of Action can be used to construct an interesting basis frame related to projective connections. The differences between the inexact 1-forms of Work and Heat become obvious in terms of the topological format. Both 1-forms, W and Q, depend on the process, V4 , and on the physical system, A. However, Work is always transversal to the process, but Heat is not, unless the process is adiabatic. Work is transversal i(V4 )W = i(V4 )i(V4 )dA = 0, Heat is NOT transversal i(V4 )Q = i(V4 )dU ; 0.

(2.69) (2.70)

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65

It is this fundamental difference between Heat, Q, and Work, W , that leads to the Carnot-like statements that it is possible to convert work into heat with 100% efficiency, but it is not possible to convert heat into work with 100% efficiency. Locally adiabatic direction fields, so defined as null curves of Q, do not imply that the Pfaff dimension of Q must be 2. That is, it is not obvious that Q can be written in the form, Q = T dS, as is possible on the manifold of equilibrium, or isolated, states. From the Cartan formulation it is apparent that if Q is not zero, then L(VA ) A = Q 6= 0, i(VA )L(VA ) A = i(VA )i(VA )dA + i(VA )d(i(VA )A) = 0(transversality) + i(VA )d(i(VA )A) = i(VA )Q.

(2.71) (2.72) (2.73) (2.74)

The necessary condition for a process to be adiabatic is given by the statement that the process is an "associated" vector relative to the exact exterior differential of the internal energy, dU . A locally adiabatic process requires that i(VA )Q = i(VA )d(i(VA )A)) ⇒ 0.

(2.75)

If Q 6= 0, a necessary adiabatic condition is given by the equation, necessary condition : i(VA )dU ⇒ 0,

(2.76)

while a sufficient condition is given by the equation, sufficient condition : d(i(VA )A) = dU ⇒ 0.

(2.77)

Note that the Topological Torsion vector is an associated vector relative to the Action 1-form, A, and therefore defines a locally adiabatic (but irreversible) process on domains of Pfaff topological dimension 4. If the Heat 1-form, Q, is zero, then the process is a globally reversible adiabatic process of a special type. A reversible process is defined such that the Pfaff dimension of Q is less than 3; or, QˆdQ = 0. Hence i(VA )(QˆdQ) = 0 for reversible processes. However, i(VA )(QˆdQ) = (i(VA )Q)ˆdQ − Qˆi(VA )dQ, (2.78)

which permits reversible and irreversible adiabatic processes to be distinguished¶ when Q 6= 0: Reversible Adiabatic Process i(VA )Q Irreversible Adiabatic Process i(VA )Q ¶

= ⇒ = ⇒

−Qˆi(VA )dQ ⇒ 0, 0, −Qˆi(VA )dQ 6= 0, 0.

(2.79) (2.80) (2.81) (2.82)

It is apparent that i(V)Q = 0 defines an adiabatic process, but not necessarily a reversible adiabatic process. This topological point clears up certain misconceptions that appear in the literature.

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It is certainly true that if L(V) A = Q = 0, identically, then all such processes are (globally) adiabatic, and reversiblek . (In the next section, it will be demonstrated how these thermodynamic ideas can be associated with the tensor processes of covariant differentiation and parallel transport.) In such special adiabatic cases, the Cartan formalism implies that W + dU = 0. Such systems are elements of the Hamiltonian-Bernoulli class of processes, where W = −dB. 2.3.4 Processes classified by connected topological constraints on the Work 1-form, W Cartan has shown that all Hamiltonian processes (systems with a generator of ordinary differential equations), ρVH , satisfy the following equations of topological constraint on the work 1-form, W :

WE WB

A Hamiltonian process VH is either VE or VB , Extremal Hamiltonian VE = i(ρVE )dA = 0 Pfaff dimension of W = 0 . Bernoulli-Casimir Hamiltonian VB = i(ρVB )dA = −dB Pfaff dimension of W = 1.

(2.83) (2.84)

More details about Cartan’s development of Hamiltonian systems appear in section 4.5. A special case occurs when the Bernoulli function is equal to the negative of the internal energy, for then the Heat 1-form, Q, produced by this special Hamiltonian process vanishes. For Helmholtz processes (which are not strictly Hamiltonian) the situation is a bit more intricate, but in all cases the Pfaff dimension of the Work 1-form, W , is at most 1. Hamiltonian processes are subsets of Helmholtz processes. Helmholtz (Symplectic) Process VS WS = i(ρVS )dA = −dB + γ Pfaff dimension of W = 1 , dWS = 0 as γ is closed but not exact. ρVS is Symplectic when dAˆdA 6= 0, WS 6= 0, dWS = 0.

(2.85) (2.86) (2.87)

Helmholtz-Symplectic processes satisfy the following equation which is known as the Helmholtz conservation of vorticity theorem: Helmholtz : Conservation of Vorticity L(ρVS ) dA = dWS + ddU = 0 + 0 = dQ ⇒ 0.

(2.88)

Note that a symplectic process preserves the 2-form dA but does not generate a symplectic manifold relative to the Action 1-form, A. The symplectic physical system is different from the symplectic physical process. k

Some authors do not distinguish between globally adiabatic and locally adiabatic.

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67

However, the closed but not exact component of work can have finite period integrals, so the evolutionary Helmholtz process can involve changing topology. The closed integrals of Action are not invariant with respect to ρVS unless γ = 0, as Z Z Z A= γ= Q 6= 0. (2.89) L(ρVS ) z1d

z1d

z1d

The Helmholtz class of processes (2.89) can be split into two types: Type HA are those processes for which the connectivity of the domain of support for the 1-form A is invariant, Z Z Z Helmholtz type A : L(ρV) A ⇒ 0, any ρ 6= 0, W = Q = 0. (2.90) z1

z1

z1

Type HB are those processes for which the connectivity of the domain of support for the 1-form A can change (the number of holes and handles can change), Helmholtz type B : L(ρV)

Z

z1

A 6= 0, any ρ 6= 0,

Z

z1

W =

Z

z1

Q 6= 0.

(2.91)

Cartan proved [43] that if the 1-form of Action is taken to be of the classic "Hamiltonian" format, (2.92) A = pk dqk − H(pk , q k , t)dt, on a (2n+1)-dimensional domain of variables {pk , q k , t}, there exists a unique extremal vector field, ρVE , that satisfies the conditions of Helmholtz type A processes. The closed but not exact 1-forms, γ, introduce non-uniqueness into the definition of the Work 1-form, W , for Helmholtz type B processes. As dQ = d(−dB + γ + dU) = 0 for all three processes defined above, all three processes are thermodynamically reversible (see (2.58)). Remark 15 Processes of the Helmholtz class, Type B, demonstrate that uniform continuity, dQ = 0, can cause topological change, but non-uniform continuity, dQ 6= 0, is required to produce thermodynamic irreversibility. 2.3.5

Planck’s Harmonic Oscillator and Type B processes - How does energy get quantized ? Consider a symplectic Harmonic Oscillator system with a Lagrange function, Lagrange function, L(t, x, v) = −1/2kx2 + 1/2mv 2 + m0 c2 , and a 1-form of Action, A = pdx − (pv − L(t, x, v))dt, . where }k = p − ∂L/∂v = p − mv 6= 0.

(2.93) (2.94) (2.95)

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Topological Thermodynamics

Then search for evolutionary vector fields such that the symplectic non-zero virtual work is of the form, W = i(W)dA, = [−(}k)(dv − adt) + F diss (dx − vdt)], = (F diss )dx − (}k)dv + {(}k)a − F diss v}dt.

(2.96)

Consider those cases where the functions, F diss v = βΓωv 2 , }ka = βΓωxa,

(2.97) (2.98)

are constrained to yield the Virial Equation {(}k)a − F diss v) ⇒ 0,

βΓω(xa − v2 ) ⇒ 0.

(2.99)

The Work 1-form, W , then becomes, W = Γβ{vd(ωx) − (ωx)dv}.

(2.100)

If β is chosen to be a polynomial distribution of Holder norms, where each term is of the form, β(p) = 1/{(ωx)p ± (v)p }2/p , (2.101)

then each term contributes an integer to the integral, I X W = Γ2π = (integers).

(2.102)

In other words, the Virial constraint causes the 1-form of work to be of Pfaff dimension 1 (dW = 0). However, the 1-form of Work, W , is closed, but not exact. Remark 16 An "Open Question" remains: Does a Planck Distribution have a relationship to the polynomial of Holder norms? Such processes on a thermodynamic system are examples of Helmholtz type B processes on symplectic manifolds. Topological fluctuations (see Chapter 2.6) in both kinematic position and velocity are permitted, but are tamed by the constraint of the Virial condition to yield energy quantization. However, such processes that are uniformly continuous can describe the change in certain topological quantum numbers. Remark 17 The radiation pressure (fluctuation in (dx − vdt)) or temperature (fluctuation in (dv − adt)) prevents change in the orbit. Angular momentum is constant, but interaction with the environment gives a Bohr-like picture. It would appear that the application of the Virial theorem can have both statistical and topological significance. From a statistical average point of view, the Virial theorem leads to Boyle’s ideal gas Law, P V = nRT . From a topological point of view the Virial theorem appears to be related to the discrete oscillation frequencies associated with quantum mechanics.

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69

2.3.6 Locally Adiabatic Processes Each of the reversible processes must satisfy an additional topological constraint if the process is to be locally adiabatic: Locally Adiabatic Processes Adiabatic process i(VA )Q = i(VA )d(i(VA )A)) ⇒ 0, Q 6= 0, with a sufficient condition = i(VA )A ⇒ 0.

(2.103) (2.104)

If −dB = 0, then ρVE is a characteristic process relative to the 2-form F. If the Work 1-form, W , is of Pfaff topological dimension 0, then the process is an extremal process relative to A (see (2.56)). Extremal processes cannot exist on a non-singular symplectic domain, because a non-degenerate antisymmetric matrix (formed by the coefficients of the 2-form dA) does not have eigenvectors with zero eigenvalues on spaces of even dimensions. Although unique extremal stationary states do not exist on the domain of Pfaff topological dimension 4, there can exist evolutionary invariant Bernoulli-Casimir functions, B, that generate non-extremal, "stationary"states. Such Bernoulli processes can correspond to energy dissipative Helmholtz processes, but they, as well as all Helmholtz processes, are reversible in the thermodynamic sense described in section 2.3.3. The mechanical energy need not be constant, but the Bernoulli-Casimir function(s), B, are evolutionary invariant(s), and may be used to describe non-unique stationary state(s). The equations given above define several familiar categories of processes, which are in effect constraints on the Work 1-form, W . The Work 1-form, W , is generated by a process describing the topological evolution of any physical system encoded by an Action 1-form, A. The Pfaff dimension of the 1-form of virtual work, W = i(V)dA is 1 or less for all three sub categories of Helmholtz processes described above. The Extremal constraint of equation (2.83) can be used to generate the Euler equations of hydrodynamics for a incompressible fluid. The Bernoulli-Casimir constraint of equation (2.84) can be used to generate the equations for a barotropic compressible fluid. The Helmholtz constraint of equation (2.86) can be used to generate the equations for a Stokes flow. All such processes are thermodynamically reversible as dQ = 0. None of these constraints on the Work 1-form, W , above will generate the dissipative Navier-Stokes equations, which require that the topological dimension of the 1-form of virtual work must be greater than 2. Remark 18 Dissipative Navier-Stokes processes must contain spinor components. Note that for a given 1-form of Action, A, it is possible to construct a matrix of N-1 associated vectors, and then to compute the adjoint matrix of cofactors transposed to create the unique direction field (to within a factor), VAdjoint . Evolution in the direction of VAdjoint does not represent an adiabatic process path, as

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i(VAdjoint )A 6= 0. However, for a given A, the N-1 associated vectors represent locally adiabatic processes, but they need not span a smooth hypersurface whose surface normal is proportional to a gradient field. In fact, the components of the 1-form of Action, A, may be viewed as the normal vector to an implicit hypersurface, but the implicit hypersurface is not necessarily defined as the zero set of some smooth function.

2.3.7

Reversible processes when the Pfaff topological dimension of Work is 2 or 3

Before studying irreversible processes, it is of some importance to study those reversible processes for which the Pfaff dimension is 2 or 3. In the process examples above, the Work 1-form, W , was of Pfaff dimension 1 at most. As such, the Helmholtz conservation of vorticity theorem is valid, and the differential 1-form of Heat is closed, dQ = 0. It follows that all such processes are thermodynamically reversible as QˆdQ = 0. However, there are processes where the work 1-form W is of Pfaff dimension >1, and yet the process involved is reversible. Remember that all processes for which the Work 1-form, W , is not zero, can involve spinor representations, singly or in combination. First consider Stokes processes where the Pfaff dimension of W is 2:

Stokes Processes dW Q dQ QˆdQ

: = = = =

If W = −βdU = d(βU) + Udβ, −dβˆdU, (1 − β)dU, −dβdU, −(1 − β)dUˆdβˆdU ⇒ 0.

(2.105) (2.106) (2.107) (2.108) (2.109)

Although the Pfaff topological dimension of the work 1-form is 2, as QˆdQ = 0, the Stokes process is a reversible process. Next consider Chaotic reversible processes where the Work 1-form, W , is of Pfaff dimension 3. The topology induced by the Work 1-form, W , is a disconnected topology. The functions φ and χ are completely arbitrary in this example, and can be associated with the classical thermodynamic potentials. The contact structure (as the Pfaff topological dimension of Work, W , is equal to 3) can be of two types, Tight and Overtwisted [2]. Tight contact structures have a global Pfaff dimension equal to 3, Overtwisted contact structures also have a 3-form which is not zero, except at

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71

certain singular subsets. The 3-form is not global. If W dW Q dQ QˆdQ

= = = = = =

Tight Contact Structures φdχ − dU = d(φχ − U) − χdφ, dφˆdχ, (W + dU ), dW, (W + dU )ˆdW, −dUˆdW + dUˆdW ⇒ 0.

(2.110) (2.111) (2.112) (2.113) (2.114) (2.115)

As QˆdQ ⇒ 0, these specialized processes which induce a Work 1-form, W , of Pfaff topological dimension 3 are thermodynamically reversible. The Work 1-form, W , generates a contact 3-manifold which has no limit cycles [69]. It will be shown below how such processes are related to the classical thermodynamic potentials, for specific choices of the function (φχ − U). Neither of these last two processes conserve vorticity (think angular momentum). Yet they are candidates for investigating reconnection processes [99]. Of particular interest are those processes for which the Work 1-form, W , generates an "Over-twisted Contact structure". Such structures are important for they are the domain of limit cycles. As an example, define the Holder function as a quadratic form in terms of two independent functions, φ and χ, as: Remark 19 A special "Holder" function can be defined as, h2 = φ2 ± χ2 . A constant value for the square Holder norm is elliptic or hyperbolic depending upon the ± sign. Next define the closed 1-form of Pfaff dimension 1, as Definition The closed but not exact 1-form, γ, is defined as, γ = (φdχ − χdφ)/(φ2 ± χ2 ) = (φdχ − χdφ)/h2 , dγ ⇒ 0.

(2.116) (2.117)

The closed form γ plays the role of a "differential angle variable, δθ" in the elliptic case. Now study those processes where the Work 1-form, W , is of Pfaff dimension 3, but not globally such that it admits defect structures.

If W dW W ˆdW Q QˆdQ

= = = = =

Over-twisted Contact Structures (Limit cycles) f (h)γ − dU, ∂f/∂hdhˆγ, {−∂f /∂h }dUˆdhˆγ, (W + dU), dQ = dW, −dU ˆdW + dUˆdW ⇒ 0.

(2.118) (2.119) (2.120) (2.121) (2.122)

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Topological Thermodynamics

Although the Work 3-form W ˆdW is not zero almost everywhere, the heat 3-form QˆdQ = 0 is zero globally, Hence the process is thermodynamically reversible. However, the 3-form volume element created by the Work 1-form, W , is not global and will admit defect structures. In the example above, the work 3—form, W ˆdW, considered as a 3D volume element, has singularities which occur at the zeros of the function −∂f /∂h. If, for example, f (h) = (b + h − h3 /3a2 ),

(2.123)

then the circle, −∂f/∂h = 0, defines a limit cycle in the elliptic case, and the cycle resides in the two-dimensional plane defined by φ and χ: ¡ ¢ −∂f/∂h = − 1 − h2 /a2 ⇒ 0, (2.124) h2 = φ2 + χ2 = a2 .

(2.125)

The limit cycle is stable (attracting) if the volume element has a negative orientation (contracting), and is unstable otherwise. To cement the ideas, rewrite the Work 1form, W , in terms of a more suggestive set of symbols, and observe that the rotational term has the format of the component of angular momentum orthogonal to the plane of rotation. If W

= i(ρV4 )dA = mΓ(h)(xdy − ydx) − dU, ⇒ Γ(h){m(xV y − yV x )}dt − dU, = Γ(h)Lz dt − dU.

(2.126) (2.127) (2.128)

As dγ = 0, except at the fixed point of the "rotation", the Pfaff dimension of W has evolved from Pfaff dimension 3 to Pfaff dimension 1, as ∂Γ/∂h ⇒ 0. In the Pfaff dimension 1 state, Helmholtz theorem becomes valid and "vorticity" is preserved. In the Pfaff dimension 3 mode, the process does not conserve vorticity. When the system decays (or is attracted) to the Pfaff dimension 1 state, the subsequent work done by a cyclic process is not necessarily zero. The closed, but not exact 1-form γ, can contribute to a period integral. Upon reflection, what has been described is the approach (Pfaff dimension 3) to a limit cycle (Pfaff dimension 1). The entire process has been done reversibly. Other forms of both the tight and the overtwisted contact structures defined by the Work 1-form, W „ can occur and such C2 processes can be thermodynamically irreversible. However, it will be demonstrated below that sequential C1 processes exist for all contact structures that are thermodynamically reversible. Remark 20 Limit cycles in thermodynamic systems are artifacts of Spinors. 2.4

A Physical System with Topological Torsion

For maximal, non-equilibrium, turbulent systems in space-time, the maximal element in the Pfaff sequence generated by A, W, or Q, is a 4-form. On the geometric space of

A Physical System with Topological Torsion

73

4 independent variables, every 4-form is globally closed, in the sense that its exterior differential vanishes everywhere. It follows that every 4-form is exact and can be generated by the exterior differential of a 3-form. The exterior differential of the 3form is related to the concept of a divergence of a contravariant vector field. A large fraction of the development in this monograph will be devoted to the study of such 3-forms, and their kernels, for it is 3-forms that form indicators of non-equilibrium systems and processes. It is a remarkable fact that all 3-forms (in general, (N-1)forms) admit integrating denominators, such that the exterior differential of a rescaled 3-form is zero almost everywhere. Space-time points upon which the integrating denominator has a zero value produce singularities defined as elements of the class of topological defect structures. As mentioned earlier, when the Action for a physical system is of Pfaff dimension 4, there exists a unique direction field, T4 , defined as the Topological Torsion 4-vector, that can be evaluated entirely in terms of those component functions of the 1-form of Action which define the physical system. To within a factor, this direction field∗∗ has the four coefficients of the 3-form AˆdA, with the following properties: i(T4 )Ω4 W U L(T4 ) A QˆdQ dAˆdA

= = = = = =

Properties of the Topological Torsion vector T4 AˆdA, (2.129) (2.130) i(T4 )dA = σ A, (2.131) i(T4 )A = 0, σ A, (2.132) 2 (2.133) L(T4 ) AˆL(T4 ) dA = σ AˆdA 6= 0, (2.134) (2!) σ Ω4 .

Note that a T4 process is locally adiabatic. Hence, by equation (2.133) evolution in the direction of T4 is thermodynamically irreversible, when σ 6= 0 and A is of Pfaff topological dimension 4. The kernel of this vector field is defined as the zero set under the mapping induced by exterior differentiation. In engineering language, the kernel of this vector field are those point sets upon which the divergence of the vector field vanishes. The Pfaff topological dimension of the Action 1-form is 3 in the defect regions defined by the kernel of T4 . The coefficient σ can be interpreted as a measure of space-time volumetric expansion or contraction. It follows that both expansion and contraction processes (of spacetime) are related to irreversible processes. It is here that contact is made with the phenomenological concept of "bulk" viscosity = (2!)σ. For symplectic systems of higher Pfaff dimension m = 2n + 2 ≥ 4, the numeric factor becomes (m/2)!). It is important to note that the concept of an irreversible process depends on the square ∗∗

A direction field is defined by the components of a vector field which establish the "line of action" of the vector in a projective sense. An arbitrary factor times the direction field defines the same projective line of action, just reparameterized. In metric based situations, the arbitrary factor can be interpreted as a renormalization factor.

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of the coefficient, σ. It follows that both expansion and contraction processes (of space-time) are related to irreversible processes. It is tempting to identify σ 2 with the concept of entropy production. The Topological Torsion vector vanishes when the Pfaff topological dimension of A is 2 or less. Note that the Frenet-Serret geometric torsion of a space curve vanishes when the geometric dimension is 2 or less. It is this analog dependence on dimension 3 or more that led to the name "Topological Torsion" for the 3-form AˆdA. Solution uniqueness is lost when the Topological Torsion vector is not zero. In 4D, the 3-form Aˆ(dA) has been defined as the Topological Torsion 3-form. The Torsion current depends only on the system (the Action) and not upon a process. The divergence of this Torsion current is proportional to the measure of the 4D volume, that defines the symplectic space, and cannot be zero on the symplectic domain. The components of the Topological Torsion vector T4 generate what is called the "subsidiary Pfaffian system" by Forsyth [77]. For purposes of more rapid comprehension, consider a 1-form of Action, A, with an exterior differential, dA, and a notation that admits an electromagnetic interpretation (E = −∂A/∂t − ∇φ, and B = ∇ × A)†† . The explicit format of the Electromagnetic Topological Torsion 4-vector, T4 becomes, T4 = −[E × A + Bφ, A ◦ B] , AˆdA = i(T4 )Ω4 , = T4x dyˆdzˆdt − T4y dxˆdzˆdt +T4z dxˆdyˆdt − T4t dxˆdyˆdz, dAˆdA = 2(E ◦ B) Ω4 = KΩ4 , = {∂T4x /∂x + ∂T4y /∂y + ∂T4z /∂z + ∂T4t /∂t} Ω4 .

(2.135) (2.136) (2.137) (2.138) (2.139)

When the divergence of the topological torsion vector is not zero, σ = (E◦B) 6= 0, and A is of Pfaff dimension 4, W is of Pfaff dimension 4, and Q is of Pfaff dimension 4. The process generated by T4 is thermodynamically irreversible, as QˆdQ = L(T4 ) AˆL(T4 ) dA = σ2 AˆdA 6= 0.

(2.140)

The evolution of the volume element relative to the irreversible process T4 is given by the expression, L(T4 ) Ω4 = i(T4 )dΩ4 + d(i(T4 )Ω4 ) = 0 + d(AˆdA) = 2(E ◦ B) Ω4 .

(2.141) (2.142)

Hence, the differential volume element Ω4 is expanding or contracting depending on the sign and magnitude of E ◦ B, a useful fact when topological thermodynamics is ††

The bold letter A represents the first 3 components of the 4-vector of potentials, with the order in agreement with the ordering of the independent variables. The letter A represents the 1-form of Action.

A Physical System with Topological Torsion

75

applied to cosmology. The irreversible dissipation induced by a T4 process can be compared to a bulk viscosity coefficient. A cosmology on 4D can have an expanding volume element, Ω4 , but with embedded 3D defect structures (the galaxies) which are not "expanding". If A is (or becomes) of Pfaff dimension 3, then dAˆdA ⇒ 0 which implies that σ 2 ⇒ 0, but AˆdA 6= 0. The differential geometric volume element Ω4 is subsequently an evolutionary invariant, and evolution in the direction of the topological torsion vector is thermodynamically reversible. The physical system is not in equilibrium, but the divergence free T4 evolutionary process forces the Pfaff dimension of W to be zero, and the Pfaff dimension of Q to be at most 1. Indeed, a divergence free T4 evolutionary process has a Hamiltonian representation, and belongs to the characteristic class of vector fields. In the domain of Pfaff dimension 3 for the Action, A, the subsequent continuous evolution of the system, A, relative to the process T4 , proceeds in an energy conserving manner, representing a "stationary" or "excited" state far from equilibrium. These excited states can be interpreted as the evolutionary topological defects in the Turbulent dissipative system of Pfaff dimension 4. The Topological Torsion vector becomes an adiabatic, extremal, characteristic direction field in the space of geometric dimension 4, but where the Pfaff dimension of the physical system, A, is of Pfaff topological dimension 3.

Figure 2.1 The emergence of non-equilibrium and equilibrium states by continuous topological evolution On a geometric domain of four dimensions, assume that the evolutionary process generated by T4 starts from an initial condition (or state) where the Pfaff

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topological dimension of A is also 4. Depending on the sign of the divergence of T4 , the process follows an irreversible path for which the divergence represents an expansion or a contraction. If the irreversible evolutionary path is attracted to a region (or state) where the Pfaff topological dimension of the 1-form of Action is 3, then E ◦ B becomes (or has decayed to) zero. The zero set of the function E ◦ B defines a hypersurface in the four-dimensional space. If the process remains trapped on this hypersurface of Pfaff dimension 3, E ◦ B remains zero, and the T4 process becomes an extremal, adiabatic, characteristic direction field. Such extremal fields are such that the virtual Work 1-form, W , vanishes, W = i(T4 )dA = 0. The direction field that represented an irreversible process, in domains where the divergence goes to zero, becomes a representation for a reversible conservative extremal Hamiltonian process. Although the extremal process is conservative in a Hamiltonian sense, the physical system can be in a "excited" state on the hypersurface that is far from equilibrium, for the Pfaff dimension of the 1-form of Action is 3, and not 2. (If the path is attracted to a region where the function E ◦ B is oscillatory, the system evolutionary path defines a limit cycle, or what has been called a "breather".) The fundamental claim made in this monograph is that it is these topological defects that self organize from the dissipative irreversible evolution of the Turbulent State into "stationary" states far from equilibrium. These long lived stationary states form the stars and the galaxies of the cosmos at a cosmological level. They represent the long lived remnants or wakes generated from irreversible processes in a dissipative non-equilibrium macroscopic turbulent fluid. On another scale, these topological defects form the excited quantum states at the microscopic level. 2.5

The Lie differential L(V ) and the Covariant differential ∇(V )

The covariant derivative of tensor analysis, as used in General Relativity, is often defined in terms of isometric diffeomorphic processes (that preserve the differential line element) and can be used to describe rigid body motions and isometric bendings, but not deformations and shear processes associated with convective fluid flow. Another definition of the covariant derivative is based on the concept of a connection, such that the differential process acting on a tensor produces a tensor. The definition of the covariant derivative usually depends upon the additional structure (or constraint) of a metric, or of a connection, placed on a given variety, while the Lie differential does not. As the Lie differential is not so constrained, it may be used to describe non-diffeomorphic processes for which the topology changes continuously. The covariant derivative is avoided in this monograph. Koszul (see p. 262 in [96]) has given a set of axioms that can be used to define a linear affine connection and a covariant derivative. The covariant derivative axioms require that, ∇(f V ) ω = f ∇(V ) ω, and ∇(V ) f ω = (∇(V ) f )ω + f ∇(V ) ω.

(2.143) (2.144)

The Lie differential L(V ) and the Covariant differential ∇(V )

77

This axiomatic representation of a covariant derivative and an affine connection should be compared to the Lie differential, L(f V ) A = f L(V ) A + df (i(V )A), and L(V ) fA = (L(V ) f )A + f L(V ) A.

(2.145) (2.146)

Only if the last term in the expansion of the Lie differential, df (i(V )A), is zero does the formula for the Lie differential have an equivalent representation as a covariant derivative in terms of a connection. From a thermodynamic point of view, the difference between the Lie differential and the Covariant differential is related to the internal energy i(V )A and a scale variation, df , L(f V ) A − ∇(f V ) A = d(ln f )(i(f V )A).

(2.147)

If the scale variation of the process is a constant over the domain, then df = 0 and the difference between the Lie and Covariant differential can be made to vanish. If the scaled process is not constant over the domain, then the difference between the Lie and the covariant differential is zero only when the "internal" energy, defined as U = (i(f V )A) ⇒ 0. The process in that case is an associated vector relative to the Action 1-form. Even more remarkable in a thermodynamic sense is the comment made by Mason and Woodhouse (see p. 49 [161] and also [9]): Remark 21 "... then there is a Higgs field.... which measures the difference between the Covariant derivative (differential) along V and the Lie derivative (differential) along V ." The implication is that the concept of a Higgs field represents the difference between a process that is not dependent upon the constraint of a gauge group (the Lie differential), and a process that is restricted to a specific choice of a connection defined by some gauge group, (the Covariant differential). For the cases where (i(f V )A) = 0 the two differentials are equivalent, then L(f V ) A = = But i(V )Q = where i(V )Q =

f f f 0

L(V ) A + d(ln f ) (i(f V )A), L(V ) A = f · i(V )dA = f · ∇(V ) A = f Q. i(V )i(V )dA ⇒ 0, defines an adiabatic process.

(2.148) (2.149) (2.150) (2.151)

Theorem 2. Hence, all covariant derivatives with respect to an affine connection have an equivalent representation as an adiabatic process!!! (Such covariant adiabatic processes need not be thermodynamically reversible.)

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Topological Thermodynamics

Suppose that the covariant process is such that L(V ) A = ∇(V ) A = Q ⇒ 0.

(2.152)

d(L(V ) A) = L(V ) dA = dQ ⇒ 0,

(2.153)

Then, and it follows that the covariant (adiabatic) process is reversible. However, the covariant condition that Q ⇒ 0 zero is the equivalent to the condition of parallel transport: L(V ) ω ⇒ ∇(V ) ω = 0. (2.154) Theorem 3. The remarkable conclusion is that the concept of parallel transport in tensor analysis is - in effect - an adiabatic, reversible process!!! As it is a matter of experience that not all evolutionary processes are adiabatic, much less reversible, it seems sensible to conclude that theories (such as general relativity) that invoke the use of a covariant derivative, and or parallel transport, to describe evolutionary processes have allowed irreversible phenomena, in the words of Sir Arthur Eddington, "to slip through the net". Moreover, the suggestion of Mason and Woodhouse leads to the idea that this thermodynamic defect can be related to the appearance of Higgs fields. It is extraordinary that a theoretical concept having its source in the theory of elementary particles has a place in a topological perspective of thermodynamics. 2.6

Topological Fluctuations

2.6.1 The Cartan-Hilbert Action 1-form This subsection considers those physical systems that can be described by a Lagrange function L(q, v,t) and a 1-form of Action given by Cartan-Hilbert format, A = L(qk , vk ,t)dt + pk ·(dqk − vk dt).

(2.155)

The classic Lagrange function, L(qk , vk ,t)dt, is extended to include fluctuations in the kinematic variables. It is no longer assumed that the equation of Kinematic Perfection is satisfied. Fluctuations of the topological constraint of Kinematic Perfection are permitted; Topological Fluctuations in position: ∆q = (dqk − vk dt) 6= 0.

(2.156)

When dealing with fluctuations, the geometric dimension will not be constrained to 4 independent variables. At first glance it appears that the domain of definition is a (3n+1)-dimensional variety of independent base variables, {pk , qk , vk ,t}. Do not assume that p is constrained to be a jet; e.g., pk 6= ∂L/∂vk . Instead, consider pk to be a (set of) Lagrange multiplier(s) to be determined later. Note that the Action

Topological Fluctuations

79

1-form has the format used in the Cartan-Hilbert invariant integral [52], except that it is not assumed that pk is canonical; pk 6= ∂L/∂vk necessarily. Also, do not assume at this stage that v is a kinematic velocity function, such that (dqk − vk dt) ⇒ 0. The classical idea is to assert that topological fluctuations in position are related to pressure. For the given Action, construct the Pfaff sequence (1.8) in order to determine the Pfaff dimension or class [139] of the Cartan-Hilbert 1-form of Action. The top Pfaffian is defined as the non-zero p-form of largest degree p in the sequence. The top Pfaffian for the Cartan-Hilbert Action is given by the formula,

n+1

(dA) Ω2n+1

Top Pfaffian 2n+2 = (n + 1)!{Σnk=1 (∂L/∂v k − pk )dv k }ˆΩ2n+1 , = dp1 ˆ...dpn ˆdq 1 ˆ..dq n ˆdt.

(2.157) (2.158)

The formula is a bit surprising in that it indicates that the Pfaff topological dimension of the Cartan-Hilbert 1-form is 2n+2, and not the geometrical dimension 3n+1. For n = 3 degrees of freedom, the top Pfaffian indicates that the Pfaff topological dimension of the 2-form, dA is 2n + 2 = 8. The value 3n + 1 = 10 might be expected as the 1-form was defined initially on a space of 3n + 1 "independent" base variables. The implication is that there exists an irreducible number of independent variables equal to 2n + 2 = 8 which completely characterize the differential topology of the first order system described by the Cartan-Hilbert Action. It follows that the exact 2-form, dA, satisfies the equations (dA)n+1 6= 0, but Aˆ(dA)n+1 = 0.

(2.159)

Remark 22 The idea that the 2-form, dA, is a symplectic generator of even maximal rank, 2n+2, implies that ALL eigendirection fields are complex isotropic Spinors, and all processes on such domains have Spinor components. The format of the top Pfaffian requires that the bracketed factor in the expression above, {Σnk=1 (∂L/∂vk − pk )dv k }, can be represented (to within a factor) by a perfect differential, dS. dS = (n + 1)!{Σnk=1 (∂L/∂vk − pk )dv k }.

(2.160)

The result is also true for any closed addition γ added to A; e.g., the result is "gauge invariant". Addition of a closed 1-form does not change the Pfaff dimension from even to odd. On the other hand the result is not renormalizable, for multiplication of the Action 1-form by a function can change the algebraic Pfaff dimension from even to odd. On the 2n+2 domain, the components of (2n+1)-form T = Aˆ(dA)n generate what has been defined herein as the Topological Torsion vector, to within a factor

80

Topological Thermodynamics

equal to the Torsion Current. The coefficients of the (2n+1)-form are components of a contravariant vector density Tm defined as the Topological Torsion vector, the same concept as defined previously, but now extended to (2n+2)-dimensions. This vector is orthogonal (transversal) to the 2n+2 components of the covector, Am . In other words, AˆT = Aˆ(Aˆ(dA)n ) = 0 ⇒ i(T)(A) =

P

Tm Am = 0.

(2.161)

This result demonstrates that the extended Topological Torsion vector represents an adiabatic process. This topological result does not depend upon geometric ideas such as metric. It was demonstrated above that, on a space of 4 independent variables, evolution in the direction of the Topological Torsion vector is irreversible in a thermodynamic sense, subject to the symplectic condition of non-zero divergence, d(AˆdA) 6= 0. The same result holds on dimension 2n+2. The 2n+2 symplectic domain so constructed can not be compact without boundary for it has a volume element which is exact. By Stokes theorem, if the boundary is empty, then the surface integral is zero, which would require that the volume element vanishes; but that is in contradiction to the assumption that the volume element is finite. For the 2n+2 domain to be symplectic, the top Pfaffian can never vanish. The domain is therefore orientable, but has two components, of opposite orientation. Examination of the constraint that the symplectic space be of dimension 2n+2 implies that the Lagrange multipliers, pk , cannot be used to define momenta in the classical "conjugate or canonical" manner. Define the non-canonical components of the momentum, }kj , as, non-canonical momentum: }kj = (pj − ∂L/∂v j ),

(2.162)

such that the top Pfaffian can be written as, (dA)n+1 = (n + 1)!{Σnj=1 }kj dv j }ˆΩ2n+1 , 1

n

Ω2n+1 = dp1 ˆ...dpn ˆdq ˆ..dq ˆdt.

(2.163) (2.164)

For the Cartan-Hilbert Action to be of Pfaff topological dimension 2n+2, the factor {Σnj=1 }kj dv j } 6= 0. It is important to note, however, that as (dA)n+1 is a volume element of geometric dimension 2n+2, the 1-form Σnj=1 }kj dv j is exact (to within a factor, say T (qk , t, pk ,Sv )); hence, Σnj=1 }kj dv j = T dSv .

(2.165)

Tentatively, this 1-form, dSv , will be defined as the Topological Entropy production relative to fluctuations of differential position. The concept of entropy with respect to continuous topological evolution will be discussed in more detail in Section 2.7.

Topological Fluctuations

81

2.6.2 Thermodynamics and Topological Fluctuations of Work Topological fluctuations are admitted when the evolutionary vector direction fields are not singly parametrized. It is historical to consider the interpretations of equilibrium statistical fluctuations in terms of pressure and temperature. These concepts are assumed to be transported to topological fluctuations: Fluctuations in position : (pressure) dq − vdt = ∆q 6= 0. Fluctuations in velocity : (temperature) dv − adt = ∆v 6= 0.

(2.166) (2.167)

These "failures" of Kinematic Perfection undo the topological refinements imposed by a "kinematic particle" point of view, and place emphasis on the continuum methods inherent in fluids and plasmas. For the maximal non-canonical symplectic physical system of Pfaff dimension 2n+2, consider evolutionary processes to be representable by vector fields of the form, γV3n+1 = γ{v, a, f, 1},

(2.168)

relative to the independent variables, {q, v, p, t}. Use the Cartan magic formula definition of the "virtual work" 1-form, W , to compute W = i(γV3n+1 )dA. The Work 1-form must vanish for the case of extremal evolution, and be non-zero, but closed, for the case of symplectic evolution. First compute the 2-form, dA from the Cartan-Hilbert Action (eq. 2.155); dA = {∂L/∂vk − pk }(∆vk )ˆdt + {dpk − ∂L/∂xk dt}ˆ(∆q k ).

(2.169)

Then compute the Work 1-form, W = (γV3n+1 )dA = {p − ∂L/∂v} • ∆v + {f − ∂L/∂q} • ∆q.

(2.170)

The resulting formula is correct in the sense that the work is transversal, but not necessarily adiabatic, with respect to the process, (γV3n+1 ), as i(γV3n+1 )W = (γV3n+1 )(γV3n+1 )dA ⇒ 0.

(2.171)

Note that {p − ∂L/∂v} is the definition of the non-canonical momentum, }kj , and {f − ∂L/∂q} represents those components of the force that are not conservative. When the fluctuations in velocity are zero (temperature) and the fluctuations in position are zero (pressure), then the Work 1-form, W , will vanish, and the process and physical system admits an extremal Hamiltonian representation. On the other hand if the fluctuations in velocity are not zero and the fluctuations in position are not zero, then the Work 1-form vanishes only if the momenta (the Lagrange multipliers, p) are canonically defined, {p − ∂L/∂v} ⇒ 0, and the Newtonian force is a gradient,

82

Topological Thermodynamics

{f −∂L/∂x} ⇒ 0. These topological constraints are ubiquitously assumed in classical conservative Hamiltonian mechanics. Note that the condition that momenta are canonical is given by the equation, {∂L/∂vk − pk } = 0,

(2.172)

which reduces the top Pfaffian to topological dimension equal to 2n+1. The manifold is therefore a contact manifold of Pfaff topological dimension 2n+1, and the 2-form, dA, admits one eigenvector with eigenvalue zero. This eigenvector defines the extremal field, but then the Work 1-form must vanish, W = i(V )dA = 0. Hence the second bracket must vanish, yielding the Lagrange Euler equations: {dpk − ∂L/∂xk dt} ⇒ 0 = d(∂L/∂vk ) − ∂L/∂xk dt, or d(∂L/∂vk )/dt = ∂L/∂xk .

(2.173) (2.174)

It is important to note that the Lagrange-Euler equations for the extremal field do NOT require that the fluctuations in position and velocity must vanish. Remark 23 The result indicates that canonical momentum and a gradient force yield the Lagrange-Euler equations, of variational calculus fame, for the Cartan-Hilbert 1form of Action. It is not necessary to assume that the velocity and acceleration are without fluctuations - a case known as Kinematic Perfection. When all topological fluctuations vanish, then the Pfaff dimension of the Work 1-form, W , is also zero. This is a sufficient but not necessary condition for equilibrium.‡‡ It is possible that when the momenta are canonical, and the force is conservative, the equilibrium state can admit fluctuations, and yet the Work 1-form vanishes and the Heat 1-form, Q, is exact. This result can be used as a starting point for a statistical analysis of the equilibrium state (statistical methods are more or less ignored in this monograph). Fluctuations in Pfaff topological dimension 2n+2 and 2n+1 When the 2-form dA is non-zero, all processes acting on the Cartan-Hilbert Action, generate a Work 1-form, W , of the form given in equation (2.170). The maximum topological dimension for the Cartan-Hilbert Action is 2n+2. Suppose that the 2form dA is constructed in terms of these "2n+2 topological coordinates". The 2-form dA is said to be non-degenerate, or of maximal rank, on the (2n+2)-dimensional space in regions where the antisymmetric matrix representing dA has no zero eigenvalues. (Recall that closed non-degenerate 2-forms of even rank define a symplectic structure [139].) However, there may exist singularities in the space of topological coordinates ‡‡

Inanimate is perhaps a better description of the state with zero fluctuations.

Topological Fluctuations

83

2n+2 where the 2-form dA becomes "singular". In such 2n + 2 regions the 2-form dA on the 2n + 2 space becomes degenerate and admits zero eigenvalues. Such regions, where dA is of Pfaff dimension 2n + 1, can be considered to be topological defects or subspaces in the 2n+2 topological domain. In such subspaces, the 2-form dA, expressed in terms of the 2n+2 topological coordinates, admits two eigenvectors with eigenvalue zero. As the eigenvalues of an antisymmetric matrix of (2n + 2) × (2n + 2) functional elements come in pairs, vectors representing topological defects of the symplectic domain are not unique, a well known result of the calculus of variations having envelope solutions (see section 9.10). One of these eigenvectors with a zero eigenvalue (of 2n + 2 components) is the unique Hamiltonian-extremal field, and the other is related to the topological torsion vector (of 2n + 2 components) with zero divergence in the subspace of topological defects with Pfaff dimension, 2n+1. This latter vector is in effect reduced to a characteristic vector relative to the Action. Caution: the characteristic vector is equivalent to the Topological Torsion vector, only if the divergence of the Topological Torsion vector is zero. This second solution vector with a zero eigenvalue (and which defines an adiabatic process in a thermodynamic sense) can be related to the Hamilton-Jacobi theory in classical mechanics. Processes defined by the extremal field or the characteristic field (degenerate topological torsion vector) are thermodynamically reversible. In contrast, the process generated by the topological torsion vector with non-zero divergence is thermodynamically irreversible. These facts can now be combined with the expression for the Work 1-form, W , given in equation (2.170). In the regions where dA is non-degenerate, the Work cannot vanish (as this would imply a null eigenvector). If the Work 1-form does not vanish, the process must have components composed from the Spinor (see [277]) eigendirection fields of the 2-form, dA. It follows that the following 4 situations are NOT allowed when dA is of maximal rank. Case 1. Canonical momentum and gradient forces {p − ∂L/∂v} = 0 and {f − ∂L/∂q} = 0.

(2.175)

Case 2. Canonical momentum and zero kinematic fluctuations in position {p − ∂L/∂v} = 0 and ∆q = 0.

(2.176)

Case 3. Zero kinematic fluctuations in velocity and gradient forces ∆v = 0 and {f − ∂L/∂q} = 0.

(2.177)

Case 4. Zero kinematic fluctuations in velocity and Zero kinematic fluctuations in position ∆v = 0 and ∆q = 0.

(2.178)

84

Topological Thermodynamics

Conversely, when dA generates a contact manifold of Pfaff topological dimension 2n+1, one of the four cases above must be true. In the contact 2n+1 domain, however, there exists a unique vector field with a null eigenvalue, such that the virtual Work 1-form, W , indeed vanishes, W = i(X)dA = 0. This result serves as the basis of the d’Alembert principle. An elementary case is based upon the assumption that Case 4 is valid. That is, there exists a kinematic description of the process at both the first and the second order (velocities and accelerations are singly parameterized). Another case that is common is based on the assumption that the momentum is canonically defined. Then, for the Contact extremal case to exist, and as {p − ∂L/∂v} = 0, it is necessary that the Work 1-form, W , reduces to vanishing expression W = {f − ∂L/∂q} ◦ ∆q ⇒ 0 in the extremal case. (2.179) The extremal constraint is satisfied when the bracket factor vanishes, which is then the equivalent of the Lagrange-Euler equations of classical mechanics. However, the Contact constraints are also satisfied when the force is a gradient field, or there exist zero fluctuations in position, or the non-zero components of the force (the otherwise dissipative components) are orthogonal to the kinematic fluctuations in position.

Bernoulli-Hamiltonian Processes and fluctuations in Work A Bernoulli-Hamiltonian process is not uniquely defined by the 1-form of Action representing the physical process. Recall that the extremal direction field in the domain of Pfaff dimension 2n+1 and the topological torsion direction field in the domain of Pfaff dimension 2n+2 are uniquely defined by the functional format of the 1-form of Action representing the physical system. Further recall that a BernoulliHamiltonian process is defined by the Work 1-form being non-zero and exact, W = i(X)dA = −dB 6= 0, where B is an arbitrary function, often called a "Casimir" - or somewhat inappropriately, a "Hamiltonian". In non-singular regions where the 1form A is of Pfaff dimension 2n+2, and is non-degenerate, the functions B are never constant and never without a gradient. Although they are not constants over the domain, these "potential" or "energy" functions B are evolutionary invariants of the Bernoulli-Hamiltonian process, X. That is, a Bernoulli function, B, is an invariant along a given path, but can have different values for B on neighboring paths. Most engineers and applied scientists have a greater appreciation for these functions when it is pointed out that they are equivalent to the Bernoulli invariants in hydrodynamics and the thermodynamic potentials in classical thermodynamics. The engineer would call B a Bernoulli "constant", a function invariant along a streamline, but which has different values for different neighboring streamlines, B = (P + ρgh + ρv 2 /2). To prove that the Bernoulli-Casimirs are always evolutionary invariants with respect to the vector fields, X, construct the Lie differential of B with respect to X: L(X) B = i(X)dB + d(i(X)B) = i(X)i(X)dA + d(i(X)B) = 0 + 0.

(2.180)

Topological Fluctuations

85

Both the first and second terms vanish algebraically. However, for the classic "Hamiltonian" defined above in terms of the Legendre transformation, H(t, qk , v k , pk ) = {pk vk − L(t, q k , v k )},

(2.181)

a direct computation indicates that the Hamiltonian need not be an invariant of a symplectic process - even if the Hamiltonian is explicitly time independent. For consider the evolutionary equation, L(X) H = i(X)dH = {(∂H/∂q) • v + (∂H/∂p) · f + (∂H/∂v) · a + (∂H/∂t)}, (2.182) or equivalently L(X) H = {(p − ∂L/∂v) • a + (f − ∂L/∂q) • v − (∂L/∂t)}.

(2.183)

For the domain of the Cartan-Hilbert Action which is of Pfaff topological dimension 2n+2, the first factor of the first term cannot vanish. The first factor of the second term, when set to zero, is equivalent to the classical Lagrange-Euler equations, and the forces are conservative gradient fields. Suppose that (∂L/∂t) = −(∂H/∂t) = 0, and the non-conservative forces are orthogonal to the velocities, then, even in this case, if the accelerations a are such that (p−∂L/∂v)·a 6= 0, the "Hamiltonian energy" H, is not an evolutionary invariant relative to X. Yet the Bernoulli-Casimir energies are evolutionary invariants relative to X. A simple example of this situation is where the mechanical (Hamiltonian) energy of a system decays to perhaps some nonzero value at a singular point of the 2n+2 domain, but the angular momentum stays constant during the process. Numerical simulations of such evolutionary possibilities in fluids have been studied numerically by Carnevale [39]. Thermodynamic Potentials and Reversible Processes From the topological version of the first law in terms of Cartan’s magic formula, and from the concept that thermodynamic reversibility requires that QˆdQ = 0, it follows that for reversible processes, QˆdQ = (W + dU)ˆdW = W ˆdW + dUˆdW ⇒ 0.

(2.184)

Suppose that the Work 1-form is restricted to the format of Pfaff dimension 3 such that, Pfaff dimension 3, W = −dU + φˆdχ = d{−U + φχ} − χdφ, dW = dφˆdχ, W ˆdW = −dU ˆdW.

(2.185) (2.186) (2.187)

Subject to these constraints it follows that the process that created the Work 1-form, W , is thermodynamically reversible, as QˆdQ = (W + dU)ˆdW = W ˆdW + dUˆdW = −dUˆdW + dU ˆdW = 0. (2.188)

86

Topological Thermodynamics

The functions φ and χ are completely arbitrary. The quantities {−U + φχ} are defined as the thermodynamic potentials for specific choices of φ and χ. For example, the classic choices for potential energy are: {−U {−U {−U {−U

+ φχ} + φχ} + φχ} + φχ}

−U (Internal Potential) φ = 0, χ = 0, T S − U (Helmholtz) φ = T, χ = S, −(P V + U) (Enthalpy) φ = −P, χ = V, −(U − T S + P V ) (Gibbs).

(2.189) (2.190) (2.191) (2.192)

for the Internal Potential. − T S) − SdT, Helmholtz Potential. + P V ) + V dP, Enthalpy Potential. − T S + P V ) + V dP − SdT, Gibbs Potential.

(2.193) (2.194) (2.195) (2.196)

= = = =

In each case the (reversible) Work 1-form, W , is given by the formula: W W W W

= = = =

−dU, −d(U −d(U −d(U

The Helmholtz potential is useful for reversible processes for which the temperature is constant, dT = 0. The Enthalpy potential is useful for reversible processes for which the pressure is constant, dP = 0. The Gibbs potential is useful for reversible processes which involve constant pressure and temperature, dP = 0 and dT = 0. However, note that the function pair φ and χ is completely arbitrary. Remark 24 The importance of the thermodynamic potentials is their relationship to reversible processes, where the Work 1-form, W , is of Pfaff dimension 3, but the Pfaff dimension of the Heat 1-form, Q, is 2. It is important to realize that the thermodynamic potentials so constructed above imply that the contact (2n+1=3)-dimensional structures generated by the Work 1-form, W , are "tight" and without limit cycles. 2.6.3 Thermodynamic Potentials as Bernoulli evolutionary invariants Under the appropriate conditions of constant pressure, constant temperature, or both, each of the thermodynamic potentials above have the format of Bernoulli functions, W = i(ρV4 )dA = −dB. Under the constraints of constant temperature or pressure, each of the Potentials is an Bernoulli invariant of the path generated by, ρV4 , but each potential is not necessarily a global invariant. The proof is easy: L(ρV4 ) B = i(ρV4 )dB = i(ρV4 )(L(ρV4 ) (−W )) = −i(ρV4 )(i(ρV4 )dA) = 0.

(2.197)

In other words, depending on the choice of the Bernoulli function, B, representing the Work 1-form in terms of constrained topological fluctuations, the following evolutionary invariants are determined. L(ρV4 ) BGibbs L(ρV4 ) BEnthalpy L(ρV4 ) BHelmholtz L(ρV4 ) Binternal

= = = =

L(ρV4 ) (U − T S + P V ) = 0, L(ρV4 ) (U + P V ) = 0, L(ρV4 ) (U − T S) = 0, L(ρV4 ) (U) = 0 (extremal).

(2.198) (2.199) (2.200) (2.201)

Topological Fluctuations

87

Hence, the empirical thermodynamic potentials, more than 100 years old in concept, are to be recognized as the Bernoulli-Casimir evolutionary invariants of reversible processes that admit topological fluctuations. These reversible processes can exist on symplectic spaces of topological dimension 2n+2, where the Work 1-form does not vanish, and is Pfaff dimension 3. The need for recognizing the differences between mechanical energy and the thermodynamic energies was discussed by Stuke [252], where, without mention of symplectic evolution, he deduces the need for "acceleration" potentials in certain dissipative systems. These acceleration potentials, which can be shown to be the equivalent of Bernoulli-Casimir functions, were used by Stuke to construct the Enthalpy and Gibbs free energy in certain hydrodynamic examples. The thermodynamic concepts of pressure and temperature are explicitly absent from that version of classical mechanics which has focused attention on the extremal contact manifolds of dimension 2n+1, and which has ignored the concept of topological differential fluctuations on symplectic spaces of dimension 2n+2. It is suggested that the occurrence of a pressure gradient, or a temperature gradient should be taken as the signature of a symplectic process. On a symplectic domain of Pfaff topological dimension 2n+2, unique ubiquitous extremal fields of classical Hamiltonian mechanics do not exist. There are no solutions to the extremal equation, i(V)dA = 0, on the symplectic domain, but there do exist non-unique vector fields V that satisfy the Helmholtz constraint equation, d(i(V)dA) = 0. In the subset of exact cases, where i(V)dA = −dB, these vector fields generate "Hamiltonian-like" dynamical systems, or processes, (on the (2n+1)submanifold transversal to dB), similar to the dynamical systems that are associated with the 2n+1 contact manifolds of classical State Space. The Action integral is a relative (stationary) integral invariant with respect to such Hamiltonian dynamical processes. The function B is a Bernoulli-Casimir evolutionary invariant, but these evolutionary invariants (stationary states) are not unique, not independent of gauge conditions, not global constants over the domain, and are strongly dependent upon boundary conditions. The somewhat larger class of vector fields that satisfy the Helmholtz condition d(i(V)dA) = 0 are defined as symplectic vector fields, and as dynamical systems they define symplectic processes. However, all such symplectic processes, exact or not, on symplectic domains of dimension 2n+2, still represent reversible thermodynamic processes. Note that the symplectic processes must be constructed in terms of the Spinor eigendirection fields of the 2-form, dA when the Work 1-form, i(V)dA, is closed, but not zero. Remarkably, and repeated here again for emphasis, on the 2n+2 symplectic domain there exists a unique non-Hamiltonian vector field which leaves the Action integral a conformal, not stationary, invariant [189]. This unique vector field, defined as the Topological Torsion current, T, although defined on a symplectic domain of the Action 1-form, does not satisfy the condition to be a symplectic "process" (dA is not an evolutionary invariant with respect to T). Instead the Topological Torsion current

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satisfies the equation, i(T)dA = ΓA, as suggested in the 1974 article [185]. Moreover, it now can be demonstrated that this unique vector field generates dynamical systems that represent irreversible processes in a thermodynamic sense. This unique vector field (to within a factor) is generated by the formulas, Aˆ(dA)n = i(T)Ω2n+2 vol .

(2.202)

The symplectic space of dimension 2n+2 on which the Torsion current exists is defined as Thermodynamic Space, in order to distinguish it from the classic State Space of dimension 2n+1. The divergence of this Torsion vector field defines a density function on the 2n+2 space. The zero sets of this density function define smooth attractors (inertial manifolds) of dimension 2n+1 on the (2n+2)-dimensional domain. The irreversible dynamical system generated by the Torsion vector irreversibly decays to these sets of measure zero which form the "stationary" states of a 2n+1 contact manifold. Once in the stationary state, the evolution can take place by a reversible Hamiltonian process. 2.7

Entropy of Continuous Topological Evolution and Equilibrium Submanifolds

A remarkable achievement of a non-equilibrium thermodynamics, expressed in terms of continuous topological evolution, is the ability to formulate a concept of entropy production in an analytic, non-phenomenological, way - and without the use of statistics. The classic concept of entropy has been extremely hard to define in nonstatistical terms, for, like potential energy, classical mechanics does not yield a clear visual intuitive picture of "just what is" entropy. Numerous phenomenological constructions have been suggested (such as entropy is a measure of disorder, entropy is the inverse of information, entropy is proportional to area, entropy is related to the logarithm of the ratio of the linear mean to the geometric mean ....), but encoding such concepts is difficult. Associated with the concept of entropy is the idea of a system in equilibrium, which at least in approximation is recognized from experience. Cold water poured into a hot bath comes to equilibrium within a perceptibly short time span. On the other hand, interchangeability of kinetic energy and potential energy, ultimately yields a visual perception of "energy" related to dynamics, but there seems to be no visual equivalent for "entropy". Moreover, the currently accepted dogma is that entropy always increases on a global scale. These concepts are hard to formulate analytically using physical techniques that have been based upon geometric concepts. It is the purpose of that which follows to demonstrate how a topological, not geometric, point of view enables an analytic coding of the concept of Entropy - without the use of statistics or a phenomenological assumption. The topological view also gives a mathematical definition of what is meant by an equilibrium physical system. The topological difference between a connected and a disconnected topology is a sufficient topological property which can be used

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to distinguish an equilibrium system from a non-equilibrium physical system. This concept is based on the Frobenius unique integrability theorem, which is valid for both an equilibrium system and an isolated system (of Pfaff topological dimension ≤ 2), but not for non-equilibrium systems (of Pfaff topological dimension ≥ 3). However, the concept of equilibrium is more subtle. Bamberg and Sternberg [12] suggest that a thermodynamic equilibrium state corresponds to a solution of a Lagrangian submanifold structure to an exterior differential system (in 4D). In 4D, the Lagrangian submanifold of a symplectic manifold generated by a 2-form, dA, is a two-dimensional submanifold upon which the 2-form dA vanishes. Of more interest to this monograph is how such a submanifold structure may be viewed in terms of the limit set of topological fluctuations in arbitrary dimension. The basic idea is that: Proposition Topological Fluctuations lead to a concept of an entropy relative to continuous topological evolution. 2.7.1 Extensions of the Cartan-Hilbert Action 1-form This subsection considers in more detail those physical systems that can be described by a Lagrange function L(q, v,t) and a 1-form of Action given by the expression, A = L(qk , vk ,t)dt + pk ·(dqk − vk dt).

(2.203)

The classic Action, L(qk , vk ,t)dt, is extended to include topological fluctuations in all of the kinematic variables. It is no longer assumed that the equation of Kinematic Perfection is satisfied. That is, fluctuations of the topological constraint of Kinematic Perfection are permitted: Topological Fluctuations : in position ∆q = (dqk − vk dt) 6= 0, in velocity ∆v = (dvk − ak dt) 6= 0, in momenta ∆p = (dpk − fk dt) 6= 0.

(2.204) (2.205) (2.206)

These topological fluctuations are not merely functions of time, but can be fluctuations in space and perhaps other parametric variables. Note that the topological fluctuations are not derivatives, but are differentials - the limit process has not been explicitly stated. One particular fluctuation problem is related to the choice of an "observer’s" origin. For example, in mechanics it is often assumed that the origin is located at the center of mass. Such an approach can lead to imprecision and fluctuations of parameters, such as mass. The only origin that is free from such defects is a singular barycentric system, but that cannot be defined with parameters that are positive definite (such as mass). In the singular barycentric system of projective systems, any point can be used as the "origin" for all other points in an equivalent manner.

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When dealing with topological fluctuations, the pre-geometric dimension will not be constrained to only 4 independent variables. At first glance it appears that the domain of definition for the Cartan-Hilbert Action 1-form, above, is a (3n+1)dimensional variety of independent base variables, {pk , qk , vk ,t}. The reader is warned that p is NOT constrained to be a jet; e.g., pk 6= ∂L/∂vk . Instead, the pk are considered to be (a set of) Lagrange multiplier(s) to be determined later. Note that the Action 1-form has the format used in the Cartan-Hilbert invariant integral [51], except that herein it is not assumed that the pk are canonical; that is, pk 6= ∂L/∂vk necessarily. Also, it is NOT assumed at this stage that the vector field, v, is a kinematic velocity function, such that (dqk − vk dt) ⇒ 0. A classical inference is to assert that topological fluctuations in kinematic velocity, ∆q, are related to pressure, and topological fluctuations in kinematic acceleration, ∆v, are related to temperature. As explained in the previous section, the top Pfaffian for the Cartan-Hilbert Action can be evaluated, and is given by the formula, n+1

(dA) Ω2n+1

Top Pfaffian 2n+2 = (n + 1)!{Σnk=1 (∂L/∂v k − pk )dv k }ˆΩ2n+1 , = dp1 ˆ...dpn ˆdq 1 ˆ..dq n ˆdt.

(2.207) (2.208)

This formula emphasizes the fact the topological Pfaff dimension of the CartanHilbert 1-form is 2n+2, and not the "geometrical" dimension 3n+1. From the fact that the top Pfaffian represents a 2n+2 volume element, (dA)n+1 ⇒ Ω2n+2 = dSˆdp1 ˆ...dpn ˆdq 1 ˆ..dq n ˆdt,

(2.209)

such that the bracketed expression in the formula for the top Pfaffian must reduce to an exact differential, dS, (n + 1)!{Σnk=1 (∂L/∂v k − pk )dv k } = (n + 1)!Σnj=1 }kj dv j ⇒ dS.

(2.210) (2.211)

Remark 25 As the (2n+2)-form represents a volume element, the coefficient of the top Pfaffian has a representation (to within a factor) as a perfect differential of a function, S, which is independent from the {pk , q k , t}. The differential change of the function S is explicitly dependent upon the differentials of velocity dv k and the non-canonical components of momentum (∂L/∂vk − pk ). Definition The change in entropy, due to continuous topological evolution, is defined as dS, and is given by the expression, dS = (n + 1)!{Σnk=1 (∂L/∂vk − pk )dv k }.

(2.212)

Definition The function S whose differential is the 1-form given in equation (2.212) is defined as the entropy of continuous topological evolution.

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The even-dimensional (2n+2)-form represents an orientable volume element, and once an orientation has been fixed (say +1), then as continuous evolution is constrained to maintain the volume element and its sign, the change in the entropy, dS, must be of one sign. So entropy of topological evolution, if it changes globally, can be only of one sign (chosen to be positive in the historic literature). Also, as dA is presumed to be non-degenerate, then the differential, dS, can not change sign by continuous topological evolution on the (2n+2)-dimensional space. Remark 26 The fact that global changes in entropy of continuous topological evolution must be of one sign ≥ 0 is an artifact of topological orientability, hence dS represents entropy production. Next consider subspaces of the Symplectic 2n+2 space. In particular consider a Lagrangian submanifold, which must be dimension n+1. By definition, on the Lagrangian submanifold (of dimension n+1) of the Symplectic space (of dimension 2n+2), the 2-form dA must vanish. The 2-form can be written as dA = dSˆdt + {dpk − ∂L/∂xk dt}ˆ(∆q k ) ⇒ 0.

(2.213)

Observe that the immersion ψ of the configuration space with differentials {dq ˆ..dq n ˆdt} into the top Pfaffian space {dSˆΩ2n+1 }, defines a Lagrangian submanifold when the pullback of the 2-form dA vanishes. The 2-form dA has an expression given by the equation above. Consider the case where the immersion, ψ, into the 3n+1 space is such that the pullback of (∆q k ) ⇒ 0. If the immersion mapping is given by the expression, 1

ψ : (q 1 , .., q n , t) ⇒ (S(q, p, t, v), p1 , ..., pn , q 1 , .., q n , v 1 , .., v n , t),

(2.214)

then the 2-form has a pullback realization such that, ψ∗ (dA) = dSˆdt ⇒ 0 for a Lagrange submanifold.

(2.215)

The Pfaff topological dimension of the constrained 1-form of Action is then 2 on configuration space, and induces a connected Cartan Topology. The 2-form vanishes when the entropy is a constant: Remark 27 Equilibrium implies dSequil (q, p, t, v) ⇒ dSequil (t) ⇒ 0. It is also remarkable to note that if the momenta are canonically defined, such that {∂L/∂vk − pk } ⇒ 0 ⊃ dS = {Σnk=1 (∂L/∂v k − pk )dv k } ⇒ 0,

(2.216)

then the entropy production, dS, vanishes. The concept of an entropy of continuous topological evolution is explicitly dependent upon the existence of non-canonical momenta.

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It is remarkable that the symplectic systems of irreducible topological dimension 2n+2 seem to resolve the Boltzmann-Loschmidt-Zermelo paradox of why canonical Hamiltonian mechanics is not able to describe the decay to an equilibrium state, and why the usual (extremal) methods of Hamiltonian mechanics do not give any insight into the concept of pressure, temperature, Entropy or the Gibbs free energy. It is extraordinary that answers to these 150 year old paradoxes of physics seem to follow without recourse to statistics if one utilizes a topological perspective. Remark 28 The interpretation of the fact that the top Pfaffian (for a physical system that can be encoded by a Cartan-Hilbert 1-form of Action) is of dimension 2n + 2 and not 3n + 1 is, at present, not complete. The implication is that there must exist (3n + 1) − (2n + 2) = n − 1 topological invariants in these systems. Consider a process starting with some initial conditions in the turbulent domain of Pfaff dimension 4 for A, W, and Q. If the process proceeds by evolution such that the process path enters a region of the geometric domain where either T ⇒ 0, or the Topological Entropy production rate vanishes by orthogonality, Orthogonality: {Σnj=1 }kj dv j } = dSv ⇒ 0,

(2.217)

or, if a domain is reached such that the momenta become canonical Canonical Momenta pk = ∂L/∂v k ,

(2.218)

it follows that the Cartan-Hilbert Action, A, decreases its topological dimension from 2n + 2 to 2n + 1. This 2n + 1 Contact manifold is the state space of classical mechanics. When that Action 1-form generates a Contact manifold, there is always a unique extremal vector field which generates a system of first order ODE’s known as Hamilton’s equations describing the extremal process. If at subsequent steps of the evolutionary path all of the differentials dpk become zero, then the dimensionality of the 2n+2 manifold becomes the configuration space manifold of dimension n+1, a LaGrangian submanifold. If the Pfaff dimension of A is equal to 1 when A is restricted to the submanifold, the equilibrium state has been defined in which the entropy function, S, is a constant; e.g., dSv = 0. The important facts are that there are two classes of processes that can represent the topological change from a Pfaff topological dimension 2n+2 to a Pfaff topological dimension of 2n+1. The 2n+2 system supports thermodynamically irreversible dissipative processes. The 2n+1 system supports stationary reversible Hamiltonian processes. The two classes of processes are distinguished by the property that the velocity field is either orthogonal to the non-canonical momenta, or the process causes the non-canonical momenta to vanish. If the domain of definition is constrained such that the momenta are defined canonically, ∂L/∂v k −pk = 0, then the 2-form dA does not define a symplectic manifold

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93

of Pfaff topological dimension 2n+2, but the 2-form does define Contact structure on 2n+1 with the formula for the Top Pfaffian given by the expression, n

Aˆ(dA)

Top Pfaffian 2n+1 = n!{(pk vk − L(t, q k , v k )}Ω2n+1 .

(2.219)

The coefficient in brackets is recognized as the Legendre transform of the Lagrangian producing the format of the classic Hamiltonian energy. It is this (2n+1)-dimensional contact manifold that served as the arena for most of classical mechanics prior to 1955, especially for those theories which were built from the calculus of variations. The (2n+1)-dimensional contact manifold, or state space, admits a unique "extremal" evolutionary field, i(V)dA = 0, that satisfies "Hamilton’s equations". The coefficient of the state space volume is to be recognized as the Legendre transform of the physicist’s Hamiltonian energy function. The Legendre transformation is defined by the equation, Legendre transformation L(t, q k , v k ) = pk vk − H(t, qk , v k , pk ).

(2.220)

When the constraints of canonical momenta are valid, it follows that ∂H(t, q k , v k , pk )/∂vk = 0.

(2.221)

This result is interpreted by the statement that the "Hamiltonian" is to be expressed in terms of the variables {t, qk , pk } only. The Top Pfaffian becomes, Aˆ(dA)n = n!{H(t, q k , pk )}Ω2n+1 .

(2.222)

The 2n+1 space maintains its contact structure as long as the "total Hamiltonian energy" is never zero, and the momenta are canonically defined. If further topological evolution causes the Pfaff topological dimension to change from 2n+2 to 2n, then it follows that the Hamiltonian energy must vanish. That is (using the canonical constraint), reduction of the Pfaff dimension from 2n+1 to 2n (state space to phase space) requires that the Lagrange function be homogeneous of degree 1 in the velocities, v k . For then, (dA)n

Top Pfaffian 2n = {vk ∂L(t, q k , v k )/∂vk − L(t, q k , v k )} ⇒ 0.

(2.223)

The result is remarkable in that the definition of a Finsler space is precisely that constraint situation where the coefficient {pk v k −L(t, q, v} of the 2n+1 manifold vanishes, and the momenta are canonical. These two constraints, first that the momenta be canonical, and second that the Lagrangian function be homogeneous of degree 1 in the velocities, are precisely Chern’s constraints used to formulate a Finsler geometry [52]. Finsler spaces include non-Riemannian geometries (when the Lagrange function contains more than quadratic powers of v) and spaces with torsion [32]. Remark 29 Note that the processes of topological reduction of the Cartan-Hilbert 1-form of Action, {2n + 2 → 2n + 1 → 2n}, as described above are not equivalent to forming an arbitrary section(s) in the form of holonomic constraints.

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2.8 2.8.1

Topological Thermodynamics

An Irreversible Example: The Sliding Bowling Ball An Overview

To give substance to the methods described above, an example of an irreversible process, decaying from a set of initial conditions in a domain of Pfaff dimension 2n+2, will be given. The irreversible process continues until the system reaches a point where the Pfaff topological dimension becomes (or decays to) the value 2n+1. Once the evolutionary orbit enters the domain of Pfaff topological dimension 2n+1, further motion can be described by a Hamiltonian process without further dissipation. The physical system is not an equilibrium system, for the Pfaff topological dimension is 2n+1, yet the evolution is stationary in the sense of a Hamiltonian extremal field.

Figure 2.2 Irreversible decay to a state far from equilibrium

2.8.2

The Observation

Consider a bowling ball given an initial amount of translational energy and rotational energy. Assume the angular momentum and the linear momentum are orthogonal to themselves and also orthogonal to the ambient gravitational field. Then place the bowling ball, subject to these initial conditions, in contact with the bowling alley. Initially, it is observed that the ball slips or skids, dissipating its linear and angular momentum, until the No-Slip condition is achieved. Note that it is possible for the angular momentum or the linear momentum to change sign during the irreversible phase of the evolution. The dynamical system representing the evolutionary process is irreversible until the No-Slip condition is reached. Thereafter, the dynamical system is reversible, and momentum is conserved.

An Irreversible Example: The Sliding Bowling Ball

95

2.8.3 The Analysis Assume that the physical system may be represented by a 1-form of Action constructed from a Lagrange function, L = L(x, θ, v, ω, t) = {βm(λω)2 /2 + mv 2 /2}.

(2.224)

The constants are: m=mass, β = moment of inertial factor (2/5 for a sphere), λ = effective "radius" of the object, the moment of inertia = βmλ2 . Let the topological constraints be defined anholonomically by the Pfaffian system, {dx − vdt} ⇒ 0, {dθ − ωdt} ⇒ 0, {dx − λdθ} ⇒ 0. (2.225) Define the constrained 1-form of Action as, A = L(x, θ, v, ω, t)dt + p{dx − vdt} + l{dθ − ωdt} + s{λdθ − dx},

(2.226)

where {p, l, s} are Lagrange multipliers. Rearrange the variables to give (in the language of optimal control theory) a pre-Hamiltonian action, A = (+p − ms)dx + (l + λs)dθ − {pv + lω − L}dt.

(2.227)

It is apparent that the Pfaff dimension of this Action 1-form is 2n+2 = 6. The Action defines a symplectic manifold of dimension 6. For simplicity, assume, initially, that two of the Lagrange multipliers (momenta) are defined canonically; e.g., p = ∂L/∂v ⇒ mv,

l = ∂L/∂ω ⇒ βmλ2 ω,

(2.228)

which implies that, A = (mv − s)dx + (βmλ2 ω + λs)dθ − {mv 2 /2 + βm(λω)2 /2}dt.

(2.229)

The volume element of the symplectic manifold is given by the expression, 6V ol = 6m2 βλ2 {v − λω}dxˆdθˆdvˆdωˆdsˆdt = dAˆdAˆdA.

(2.230)

The 6D volume element is either expanding or contracting (irreversibly) with a coefficient 6m3 βλ2 {v − λω}. This dissipative coefficient is related to the concept of "bulk" viscosity. The symplectic manifold has a singular subset upon which the Pfaff dimension of the Action 1-form is 2n+1 = 5. The constraint for such a contact manifold is precisely the no-slip condition (when the "Bulk viscosity" goes to zero). No Slip Condition

{v − λω} ⇒ 0.

(2.231)

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Topological Thermodynamics

This condition is the analogue of the zero divergence condition in incompressible hydrodynamics, only the divergence is that associated with the topological torsion vector, d(AˆdAˆdA) in a six-dimensional (2n+2)-space. On the five-dimensional contact manifold there exists a unique extremal (Hamiltonian) field which (to within a projective factor) defines the conservative reversible part of the evolutionary process. As this unique extremal vector satisfies the equation, (2.232)

i(V)dA = 0,

it is easy to show that dynamical systems defined by such vector fields must be reversible in the thermodynamic sense. (As dQ = d(i(V)dA) = 0 for all Hamiltonian or symplectic processes, it follows that QˆdQ = 0.) However, on the six-dimensional symplectic manifold, there does not exist a unique extremal field, nor a unique stationary field, that can be used to define the dynamical system. The symplectic manifold does support vector fields, S, that leave the Action integral invariant, but these vector fields are not unique in the sense that that they depend on an arbitrary gauge addition to the 1-form of Action that may be required to satisfy initial conditions. There does exist a unique torsion field (or current) defined (to within a projective factor, σ) by the 6 components of the 5-form, Topological Torsion = AˆdAˆdA.

(2.233)

Relative to the topological coordinates [dx, dθ, dt, dv, dω, ds], the Topological Torsion vector has the components T6 AˆdAˆdA Tv Tω Ts

= = = = =

[0, 0, 0, Tv , Tω , Ts ], i(T6 )Ω6 , m2 βλ2 {+βλ2 ω2 + 2λvω − v 2 }, m2 λ{+βλ2 ω2 − 2βλvω − v2 }, m2 βλ2 {+βmλ2 ω 2 + mv 2 + 2(λω − v)s}.

(2.234) (2.235) (2.236) (2.237) (2.238)

If the three non-zero components of the Topological Torsion vector are treated as a dynamical system, then it is to be noted that the dynamical system is a Volterra system generated on a Finsler space (see p. 205 [5]). This unique Topological Torsion vector, T6 , independent of gauge additions, has the properties such that, L(T) A = Γ · A

and

i(T)A = 0.

(2.239)

This "Torsion" vector field satisfies the equation, L(T) AˆL(T) dA = QˆdQ = (6m2 βλ2 {v − λω})2 AˆdA 6= 0.

(2.240)

Second order versus first order ODE’s

97

Hence a dynamical system having a component constructed from this unique Torsion vector field becomes a candidate to describe the initial irreversible decay of angular momentum and kinetic energy. It is to be noted that the non-canonical "symplectic momentum" variables, defined by inspection from the constrained 1-form of Action lead to the momentum map: . . Px = mv − s, Pθ = mβλ2 ω + sλ. (2.241) Substitution in terms of the momentum variables leads to the generic form (p. 31 [299], also see [147]) for the 1-form of Action, A = Px dx + Pθ dθ − Hdt,

(2.242)

where H is an independent variable on the six-dimensional manifold. The H map is given by the expression for energy where v and ω are eliminated in terms of the Px and the Pθ , H = (mv 2 /2 + βm(λω)2 /2) ⇒ (1/2m)[(Px + s)2 + (1/β)(Pθ /λ − s)2 ].

(2.243)

Note that v = ∂H/∂Px and ω = ∂H/∂Pθ . Each component of "canonical momenta" decays with the same rate in the canonical domain. 2.9

Second order versus first order ODE’s

It is also possible to consider those physical systems that can be described by a Lagrange function L(q, v, a,t) and a 1-form of Action given by the extended CartanHilbert Action to include possible fluctuations in differential velocities directly in the 1-form of Action. Consider the Action 1-form of the type, A = L(qk , vk ,t)dt + pk ·(dqk − vk dt) + λk ·(dvk − ak dt),

(2.244)

with Topological Fluctuations in differential velocity, ∆v = (dvk − ak dt) 6= 0.

(2.245)

At first glance it appears that the domain of definition is a (5n+1)-dimensional variety of independent variables, {qk , vk , ak , pk , λk , t}. Do not assume that pk or λk are constrained to be jets; e.g., pk 6= ∂L/∂vk and λk 6= ∂L/∂ak . Instead, consider pk and λk to be a (set of) Lagrange multiplier(s) to be determined later. Do not assume at this stage that vk or ak are kinematic velocity functions, such that (dqk − vk dt) ⇒ 0 or (dvk − ak dt) ⇒ 0. The classical idea is to assert that topological fluctuations in kinematic velocity (dq − vdt) = ∆q are related to pressure, and that topological fluctuations in kinematic acceleration (dv − adt) = ∆v are related to temperature.

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For the given Action, construct the Pfaff sequence (see (1.8)) in order to determine the Pfaff topological dimension or class of the 1-form [238]. The top (nonzero) Pfaffian of this sequence is given by the formula, (dA)n+1 = (2n + 1)!{Σnk=1 (∂L/∂ak − λk ) • dak }ˆΩqpvt , Ωqpvt = dv 1 ˆ..dv n ˆdp1 ˆ...dpn ˆ dq 1 ˆ..dq n ˆdt.

(2.246) (2.247)

The equation indicates that the Pfaff topological dimension is 4n + 2 and not the geometrical dimension 5n + 1, which might be expected as the 1-form was defined initially on a space of 5n + 1 "independent" base variables. The implication is that there exists an irreducible number of independent variables equal to 4n + 2 which completely characterize the differential topology of the second order system. It follows that the exact 2-form dA satisfies the equations, (dA)2n+1 6= 0, but Aˆ(dA)2n+1 = 0.

(2.248)

For n = 3 degrees of freedom, the top Pfaffian indicates that the Pfaff topological dimension of the 2-form, dA is 4n + 2 = 14. The algebra for these systems becomes formidable. Again, there are two evolutionary routes that lead to a reduction of the even Pfaff dimension to the odd Pfaff dimension, which admits stationary Hamiltonian evolutionary processes. Topological reduction by dissipative irreversible processes on the topological domain of Pfaff dimension 4n+2 reduces to a topological domain of Pfaff dimension 4n+1, if 1. the process induces orthogonality such that, {Σnj=1 (∂L/∂aj − λj )daj } = 0,

(2.249)

2. or, the process induces the topological constraint such that the Lagrange multipliers are canonical, λk = ∂L/∂ak . (2.250) The format of the top Pfaffian of dimension 4n+2 indicates that the bracketed factor involving the non-canonical components of the Lagrange multipliers, λk , must be exact, dSa = {Σnk=1 (∂L/∂ak − λk ) • dak }. (2.251) The procedure permits a definition of entropy (to within a factor), Sa , based upon accelerations. The restriction to a Lagrangian submanifold of the (4n+2)-dimensional space implies that dSa ⇒ 0 on the (2n+1)-dimensional subspace. The ubiquitous (in classical mechanics) assumption that the Lagrange multipliers are "canonical" eliminates thermodynamic irreversible processes in favor of Hamiltonian extremal processes.

Summary

99

Note that the Lagrangian submanifold of the symplectic space of dimension 4n+2 is a space of dimension 2n+1. Such a space is a configuration space in {q k , v k , t} which always has a process with a Hamiltonian representation in terms of a unique extremal vector field. When the fluctuations ∆v ⇒ 0 and ∆q ⇒ 0, the 2-form dA must vanish for the submanifold to be Lagrangian. However, zero fluctuations are not the only way in which the submanifold can have a 2-form which vanishes. 2.10

Summary

In this chapter, the basic tools of non-equilibrium, irreversible thermodynamics from the perspective of continuous topological evolutions have been introduced and defined. The two most useful ideas are the concept of Pfaff topological dimension and its evolution (for a physical system encoded by a 1-form of Action, A). In the next chapter, the idea of a universal phase function (generated from the Jacobian matrix of the coefficient functions of the 1-form of Action, A) will be explored.

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Part II Applications to Physical Systems

101

Chapter 3 THE UBIQUITOUS TOPOLOGICAL VAN DER WAALS GAS 3.1

Introduction

In this section, a topological perspective is applied to the development of a universal thermodynamic Phase function on a four-dimensional domain. The method refines the Cartan topological structure, encoded by a 1-form of Action, A, to recognize the projective similarity invariants encoded by the Cayley-Hamilton theorem applied to the Jacobian matrix constructed from the coefficients of the 1-form, A. The theory starts with a review of the classic Ideal Gas and the van der Waals Gas. Of particular interest is the ultimate understanding of the (topologically continuous but irreversible) condensation processes that create topological defects, of topological dimension 3 or less, in dissipative domains of topological dimension 4. The success of the Landau-Ginzburg theories indicates that the fundamental Phase function is a fourth order polynomial. Herein it is theorized that this fourth order polynomial is generated by the Cayley-Hamilton theorem as applied to the Jacobian matrix constructed from the coefficients of that 1-form of Action, A, of Pfaff topological dimension 4, used to encode the thermodynamic system. But first, consider the classic theories, and especially van der Waals contributions. The simplistic equation of state for an ideal (perfect) gas, Ideal Gas:

P/RT = ρ = n/V,

(3.1)

does not encode certain thermodynamic features (phase transitions and critical point behavior) which are observable in "real" gases. It has been argued that the "real" gas consists of n geometric "parts" that interact with one another, in contrast to an ideal gas, where geometric features (size and shape) of such "parts" and their interactions had been ignored. The classic interpretation is that n represents the number of moles, where moles is interpreted in terms of microscopic "molecules". Motivated by such ideas, van der Waals created, phenomenologically, an equation of state for "real" gases in terms of two parameters, a and b, which were introduced to encode the interaction and geometric size features of the "molecular" components. The resulting formula for an equation of state was cubic in the molar density, ρ = n/V. ρRT van der Waals: P = − aρ2 , (3.2) 1 − bρ or: abρ3 − aρ2 + {RT + bP }ρ − P = 0. (3.3)

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The Ubiquitous Topological van der Waals gas

The formula has enjoyed remarkable success for qualitatively explaining the thermodynamic features of real gases. The formula represents an implicit surface in the space of variables, {P, T, ρ}. However, the development was phenomenological, and although motivated by the concept of microscopic "molecules", the fundamental properties were independent from the geometric size of its parts. From a topological point of view, size is not of primary concern. Topological properties are independent from size and shape. What is important is the number of parts. As Sommerfeld has said (without explicit reference to topology, but inferring that microscopic molecules is not of thermodynamic importance): "The atomistic, microscopic point of view is alien to thermodynamics. Consequently, as suggested by Ostwald, it is better to use moles rather than molecules." p. 11 [245]. The ideal gas approximation has been found to be of utility to the study of agglomerates of parts that range from the geometric size of nuclei to the geometric size of stars. A major purpose of this section is to demonstrate the universality of the van der Waals gas to the study of condensates of all types of "parts" in non-equilibrium configurations based upon topological issues. By differentiating the Van der Waals equation of state with respect to the molar density, it can be determined that there exists a "critical point" on the hypersurface at which the values of the pressure, temperature and molar density take on values such that at the critical point, Critical Point Pc /Tc ρc = constant.

(3.4)

When the thermodynamic variables are expressed in terms of dimensionless (reduced) variables, scaled in terms of their values at the critical point, the values of those parameters, a and b, which were used to model the geometric-interaction-shape and geometric size features cancel out. In this sense, the "renormalized" or "reduced" van der Waals equation of state became an element of topological equivalent class with universal topological properties (independent from scales), and independent from the size and interaction magnitudes of its component parts. In terms of these dimensionless variables, Pe = P/Pc , Ve = V /Vc , Te = T /Tc , ρ = n/V,

(3.5)

the classic van der Waals equation may be considered as a cubic constraint on the space of variables {n; Pe, Te, e ρ} where e ρ = n/Ve is defined as the dimensionless molar density. The reduced universal van der Waals equation of state is given by the classic cubic expression, Classic van der Waals equation (in reduced coordinates) 0 = e ρ3 − 3e ρ2 + {(8Te + Pe)/3}e ρ − Pe.

(3.6)

Introduction

105

It is this scale independent polynomial equation that promotes the correspondence between topological ideas and thermodynamics. This formula should be memorized, for it yields a direct connection of the van der Waals gas and a cubic polynomial. In the development that follows the formula will be related to the CayleyHamilton equation for a 4 × 4 Jacobian matrix, [J] constructed from the coefficients of the 1-form of Action, A, used to encode the thermodynamic system∗ . The CayleyHamilton formula is of the type, Cayley-Hamilton polynomial = ξ 4 − XM ξ 3 + YG ξ 2 − ZA ξ + TK = 0.

(3.7)

If TK = 0, the determinant of the Jacobian correlation matrix vanishes, and then the Cayley-Hamilton equation for the singular correlation matrix becomes, Singular Cayley-Hamilton polynomial = (ξ 3 − XM ξ 2 + YG ξ 1 − ZA )ξ = 0.

(3.8)

The similarity coefficients become related to the "Curvatures" of the implicit surface induced by the molar density. The first (cubic) factor can be put into direct correspondence with the Classic van der Waals equation ξ = Linear : ξ 1 +ξ 2 +ξ 3 +ξ 4 = Quadratic : ξ 1 ξ 2 +ξ 2 ξ 3 +ξ 3 ξ 1 +ξ 1 ξ 4 +ξ 2 ξ 4 +ξ 3 ξ 4 = Cubic : = ξ 1 ξ 2 ξ 3 +ξ 1 ξ 2 ξ 4 +ξ 2 ξ 3 ξ 4 +ξ 3 ξ 1 ξ 4 Quartic : ξ1ξ2ξ3ξ4 =

e ρ,

XM ⇒ 3,

(3.9) (3.10)

YG ⇒ (8Te + Pe)/3,

(3.12)

TK .

(3.13)

ZA ⇒ Pe,

(3.11)

In the above formulas the ξ 1 , ξ 2 , ξ 3 , ξ 4 are the local eigenvalues of the matrix when evaluated at the point {x, y, z, t}. The Projective dimension of the correlation matrix can be determined by the rank of the Jacobian matrix, or by the zero sets of the similarity invariants. If only one eigenvalue is zero, then TK ⇒ 0, and there is a projection from 4D to 3D. If in addition, ZA ⇒ 0, then two eigenvalues are zero, and there is a projection from 4D to 2D. Recall that non-equilibrium systems are determined by domains of Pfaff topological dimension equal to 3 or 4. The remarkable result, demonstrated by examples at the end of this chapter, are there are situations where the Pfaff dimension is 3, and the Projective dimension is 4, and other situations where the Pfaff dimension is 4, but the Projective dimension is 3. These results are entwined with the idea that there are correlations constructed from 1-forms with integrating factors such that the determinant of the correlation can be ∗

It is apparent that this Jacobian matrix is a correlation in the sense of projective geometry.

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The Ubiquitous Topological van der Waals gas

made to vanish. On the other hand their are (N-1)-forms with integrating factors that can make the divergence (trace of the collineation) vanish. Forces and energies associated with the Linear curvature are typical of surface tension effects. It becomes apparent that forces and energies associated with the Cubic curvature represent the pressures of interactions. The Gauss quadratic curvatures are dominated by temperature, with a pressure contribution. A Pe, e ρ projection of the implicit universal van der Waals surface is given in Figure 3.1.

Figure 3.1 Negative Pressure in a van der Waal’s gas The diagram displays a critical isotherm that separates a single phase (the gas) from the different topological domains that can be interpreted as liquids and vapor. Note that the word "vapor" is used for the phase between the critical isotherm and the Spinodal line, for molar densities below the critical point value. The word "gas" is used to describe the phase above the critical isotherm. Note that the word "liquid" is used for the phase between the critical isotherm and the Spinodal line, for molar densities above the critical point value. The difference between vapor and gas is not appreciated in the historical literature. The molal roots to the characteristic polynomial represented by a van der Waals gas are complex above the critical isotherm and real below the critical isotherm. This result can be attributed to the eigendirection fields associated with Jacobian matrix of the Action 1-form, A, used to encode the thermodynamic system. Above the critical isotherm, the eigendirection fields consist of one real vector, and two complex spinors. Below the critical

Introduction

107

isotherm, the eigendirection fields consist of three real vectors. These concepts will be studied in detail in that which follows. The shape of the critical isotherm should be remembered, for above the critical isotherm, there exists a unique value for the pressure, and below the critical isotherm there is more than one value for the pressure. This feature represents a topological property of the van der Waals gas, and will have importance in the study of nonequilibrium systems. Of interest to cosmologists who are interested in dark energy and negative pressure, note that the pressure for the van der Waals gas, for values below the critical isotherm, can take on negative values. As will be shown below, the Phase function below the critical isotherm has the shape of a quartic Higgs potential (see section 3.4). There exists a dual surface to the equation of state as defined by a Legendre transformation to the Gibbs function, g = u − T s + P v. The implicit surface defined by the Gibbs function (for a van der Waals gas) is not single valued, and appears as a deformation of a swallowtail bifurcation set. The actual Gibbs surface for the van der Waals gas can be numerically computed and is presented in Figure 3.2. Remark 30 The topological features of the van der Waals gas are universal features (deformation invariants) for all physical systems that admit a realization over 4D space-time variety. The van der Waals gas is one element of a topological equivalence class.

Figure 3.2 The Gibbs surface with topological defects An accurate drawing of the 3D Gibbs surface appears only occasionally in thermodynamic text books. Most presentations, if in 3D, are given by sketches, and not by actual computations. For example, in [3] p.196, the Gibbs surface misses

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The Ubiquitous Topological van der Waals gas

the fact that the Spinodal line forms a cusp at the critical point. In Figure 3.2, the salient features are displayed by numeric computation of the Gibbs surface for the van der Waals gas. Remarkably, the dual Gibbs surface displays the envelope features of the Universal Phase function. The cuspoidal critical point singularity, the winged cusp representing the Spinodal line, and the Binodal self intersection are universal topological features of the discriminant (envelope) hypersurface. In Figure 3.2, the white region is where the temperature is above the critical isotherm and represents the pure gas. The other sectors are below the critical isotherm, and are influenced by the "Higgs" features of the Phase potential. The dark gray sector represents the fluid phase, and the light gray sector represents the vapor phase. The light blue sector represents the unstable mixed phase region. Historically the two implicit surfaces defined by the reduced van der Waals equation became quite useful to chemical engineers and led to the law of corresponding states. If properties of a gas near its critical point could be measured, then the law of corresponding states permitted estimates to be made for the properties of the gas by comparison to the universal van der Waals model. The topological results were independent of the geometric parameters of size, b, and interaction length, a. In this and following sections, the universal topological features of the Phase function and the Gibbs surface of the generalized† van der Waals gas will be developed and applied to non-equilibrium systems. The "generalization" consists of adding a contribution to the reciprocal volume use in the interaction term. Recall that non-equilibrium requires that the Pfaff topological dimension of the Action 1-form is 3 or greater in certain regions. Non-equilibrium systems can exist in "stationary" states where their topological coherence properties are evolutionary invariants. The principle (universal) topological defect structure of a van der Waals gas is the existence of a non-zero critical point. When expressed in terms of reduced coordinates, {Pe, Te, e ρ}, the critical point of the implicit surface representing the equation of state, is where the reduced (dimensionless) functions all have the common value unity. The topological significance of the critical isotherm, which passes through the critical point, has already been mentioned above. Another important topological defect structure is the existence of a Spinodal line, of ultimate phase stability, consisting of two parts that meet in a cusp at the critical point. The Spinodal line will be established by an edge of regression in the Gibbs surface. Yet another topological defect structure is exhibited by the Binodal line, defining portions of a ruled surface representing the region of mixed phases. The Binodal line can be described by a deformation of a pitchfork bifurcation emanating from the critical point, and line which outlines the domain of mixed phases. The domain of mixed phase is related to regions where the Pfaff topological dimension of the encoded physical system (the 1-form of Action) is at least 3. The domains of isolated single phase are related to regions where the Pfaff topological dimension is 2 or less. †

The "generalization" consists of adding a contribution to the reciprocal volume use in the interaction term.

Introduction

109

A lot can be learned from the van der Waals example, for its features are experimentally verifiable. The universal qualities are obtained in terms of variables that represent deformations and non-equilibrium extensions of the van der Waals properties. The van der Waals internal energy is a Lagrangian (phase) function in terms of extensive variables. In the language of classical mechanics, the Lagrange function is a function of the base variables, q k , and their first derivatives, vk , or velocity "extensive" functions. A Legendre transformation leads to a Hamiltonian function in terms of intensive variables, the momenta, pk . The classic van der Waals phase function defines a hypersurface in the space of extensive variables of entropy, S, volume, V, and energy, U. A Legendre transformation produces a "GibbsHamiltonian" function of intensive variables, temperature T, pressure, P, and Gibbs free energy, Gibbs. The zero sets of certain algebraic combinations of the similarity curvature invariants of these hypersurfaces define universal topological features of the physical system, which are of value to the study of both equilibrium and non-equilibrium systems. Rather than formulating the non-equilibrium universal phase equation in a phenomenological manner, it will be demonstrated that such a universal phase function can be generated as the Cayley-Hamilton polynomial equation of the Jacobian matrix for the 1-form of Action, A, that represents the physical system. The topological Pfaff dimension of A permits the delineation between those phase functions that represent non-equilibrium systems and those that do not. The following subsections first will discuss the ideas associated with Extensive and Intensive variables. Then the classic van der Waals expression for a Phase equation will be used to define an internal energy surface in terms of intensive variables. A dual construction will be used to create the Gibbs energy in terms of intensive variables. The Gibbs surface is deformably (topologically) equivalent to the swallowtail discriminant, or envelope of the classic phase equation. After this review of classical theory in the language of topological evolution, the theory will be extended to include non-equilibrium systems of the closed and open types. 3.1.1 Extensive and Intensive variables Experiment has indicated that there are two species of variables in thermodynamic systems, Extensive variables such as Volume and Entropy {V and S} - which are additive quantities, and intensive variables, such as pressure and temperature {P and T } - which are not additive intensities. As Tisza points out [259], commenting on the geometrical approaches to thermodynamics, "It is remarkable that intrinsic subspace curvature properties can have any thermodynamic meaning, as metrical based geometries can not be used to distinguish between the two classes of intensive and extensive thermodynamic variables." Be aware of the fact that physical theories that do not distinguish between extensive and intensive variables can have only a limited application to the problem

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The Ubiquitous Topological van der Waals gas

of understanding thermodynamics. In electromagnetism, the electric and magnetic field intensities (E and B) are examples of intensive variables, and the electric and magnetic field excitations (D and H) are examples of extensive variables. It will be demonstrated below that such thermodynamic differences are related to the concepts of scalar exterior differential forms and exterior differential form densities. Intensities will be related to covariant tensor field coefficients (scalar exterior differential forms and wave phenomena), and Extensive quantities will be related to contravariant tensor coefficients (exterior differential form densities and particle phenomena). In the calculus of variations it is known that those extremal principles that are independent of parametric scales lead to projective geometries and Finsler spaces. Indeed, Chern [51] has shown that the key assumption of a Finsler geometry is that the variational integrand be homogeneous of degree 1 in the variables of the tangent space, thereby forming, in his words, a “projectivized" tangent bundle. Finsler spaces are not well known, but have had a modest number of physical applications [5]. Remark 31 The physical science of thermodynamics, based upon functions which are homogeneous of degree 1, is where the theory of projectivized Finsler spaces can be of practical application. A function, Θ, that is homogenous of degree 1 in the extensive variables, (S, V, U ), satisfies the Euler equation, V k ∂Θ/∂V j = S∂Θ/∂S + V ∂Θ/∂V + U∂Θ/∂U... = Θ,

(3.14)

and the scaling equation Θ(λS, λV, λU ) = λΘ(S, V, U).

(3.15)

The partial derivative coefficients in the Euler equation define the intensive variables. The test for homogeneity can be put into correspondence with Cartan’s magic formula operating on functions. Let R = [S, V, U...] be a position vector in a space (of extensive variables), and let Θ(S, V, ...n) be a function on that space. Then, L(R) F = i(R)dΘ = S∂Θ/∂S + V ∂Θ/∂V + U∂Θ/∂U... .

(3.16)

So if Θ is homogeneous of degree 1, then the evolution in the direction of the expansion (position) vector, R, yields the result, Homogeneous of degree 1 : L(R) Θ(R) = 1 · Θ(R).

(3.17)

The application of integer and fractal homogeneous concepts are discussed in more detail in Chapter 8. It is also of some importance to note that most textbook treatments of thermodynamics agree with the idea that the phase function, Θ(U, S, V, n...), must be

Introduction

111

homogenous of degree 1, but the formulas often presented for the ideal or van der Waals gas do not satisfy the Euler criteria of homogeneity. A correct formulation is presented below. It is a fact that any function can be made homogeneous of degree 1 by merely adding a new variable, and dividing, or renormalizing, each of the old variables by the new variable, and then multiplying the new function by the new variable. For example, consider the function, f (x, y, z). Then F (x, y, z, s) = sf (x/s, y/s, z/s) is homogeneous of degree 1 in the variables, {x, y, z, s}. This same idea can be extended to vectors; for example, let v = [V x , V y , V z ] ⇒ V = [V x , V y , V z , λ(V x , V y , V z )] , ⇒ J = λ [V x /λ, V y /λ, V z /λ].

(3.18) (3.19)

The key feature of the phase function of thermodynamics is that n, “the number of parts, molecules, or phases" plays the role of the renormalization variable (designated as s in the example of (3.15)). For thermodynamics the expression n · L(S/n, V /n, U/n..) = n · L(s, v, u..) = Θ(S, V, U, n..)

(3.20)

is to be recognized as a function which is homogeneous of degree 1. The phase function, Θ(S, V, U, n..), is a projection from a space of 1 higher (possibly complex) dimension (the coordinate n) to include the concept of multiple components or phases of disconnected parts. The "Lagrangian" function, L, is composed of functions, S/n, V /n, U/n, per mole and the composite function, n·L(s, v, u..) = Θ, is homogeneous of degree 1. In classical mechanics it is also appreciated that there are two classes of variables, contravariant vectors such as the components of particle velocities, and covariant wave vectors, such as the components of a gradient field. Lagrangians are often functions of contravariant vector components (velocities, V k ), while Hamiltonians are functions of covariant variables (momenta, Pk ). 3.1.2 Lagrangian-Hamiltonian features In Hamiltonian-Lagrange formalism, the Legendre transformation is given as an equation constraining the Hamiltonian function to the Lagrange function in terms of a hyperbolic product of dual variables, H + L = Pk V k .

(3.21)

Pk ∂H/∂Pk − H = −(Pk ∂L/∂Pk − L), V k ∂H/∂V k − H = −(V k ∂L/∂V k − L).

(3.22) (3.23)

It follows that

If L is homogeneous of degree 1 in the V k then the last equation implies that either H is also Homogeneous of degree 1 in the V k , or H is homogeneous of degree zero in the V k ,

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The Ubiquitous Topological van der Waals gas

with H = 0 defining an energy hypersurface. It is to be noted that the Euler criteria for homogeneity of degree 1 in the V k is given by the expression, (V k ∂L/∂V k −L) = 0. The variational problem when the variational integrand, L(v), is homogeneous of degree 1 in v is known as the Homogeneous Problem, v∂L/∂v−L(v) ⇒ 0, which is precisely that constraint used in the theory of special relativity, the theory of minimal surfaces, and in Chern’s version of Finsler geometries built on projective connections. Based on the concept that different thermodynamic phases represent topological properties, and that a phase change is to be recognized as a signature of a topological evolution, the basic ideas of projective differential geometry mentioned above were utilized [202] to define certain topological properties of hydrodynamic flows (see section 4.5.1). Different domains of initial conditions for a given hydrodynamic flow could be associated with different phase regions of a thermodynamic substance. A specific example was given in [202] for a dynamical system in which the three-dimensional flow explicitly induced the Gibbs free energy surface typical for the Van der Waals gas. It then was possible to determine that there were domains of initial conditions for which the system could be put into correspondence with the pure liquid, pure gas, or mixed phase regions of a two phase system. An unstable region would be in the domain which is interior to the Spinodal line on the surface representing the equation of state. Although intuition implied that this correspondence was a universal result, no satisfactory argument was known at that time to substantiate the idea of universality, except in specific examples. It will be demonstrated below that the observations described above are indeed universal concepts. Any dynamical system that can be described in terms of a non-linear C1 vector field in three variables can be associated with the thermodynamics of a Van der Waals gas. This universal behavior not only justifies the law of corresponding states in chemistry, but also yields explicit universal formulas (in terms of cross ratios of similarity invariants) to describe the limits of phase stability that are equivalent to the Spinodal Line and the Binodal line of two phase thermodynamic systems. In addition, the ideas lead to a well defined procedure for treating non-equilibrium thermodynamic systems as complex deviations from the isolated, or equilibrium, systems. This claim of universality is not to be treated lightly. For example, it should be remarked that historically many authors, including Thom, have recognized that the cusp catastrophe generated by the cubic fold has many qualitative features of a Van der Waals gas. In 1977 Sewell [241] noted a relationship between Legendre transformations and bifurcation theory, and clearly defined the relationship between the Gibbs free energy surface of a van der Waals gas and its relationship to the swallowtail catastrophe. However, the claims that catastrophe concepts have universal significance have been criticized sharply, both on method and style of presentation and specifically on the grounds that not all dynamical systems have a gradient representation. However, the analysis herein gives credence to some of Thom’s claims of universality by demonstrating how the cusp singularity can be constructed in terms of

Introduction

113

any C1 three-dimensional vector field, and the similarity invariants of its Jacobian matrix. Annulling individual similarity invariants (with respect to that special subset of projective transformations (equi-affine transformations) that preserve parallelism and perpendicularity) leads to local bifurcations, and constraining dimensionless cross ratios of similarity invariants leads to global bifurcation diagrams. Then using Sewell’s result that the bifurcation set of the swallowtail singularity is related to the Legendre dual of the Cusp singularity completes the universal correspondence. Others have attempted to use differential geometric methods to analyze thermodynamic systems, but almost always these attempts have tried to construct a suitable metric formalism. For example, Tisza mentions that Blaschke attempted to deduce a (an affine) differential geometry that would apply to the metric free Gibbs space, but with only limited success. In Blaschke’s geometry, the projective space was confined to the equi-affine group, which forces the shape matrix to be symmetric. Such equi-affine systems admit only real eigenvalues for the shape matrix, where the richness of non-equilibrium thermodynamics, and its possible application to the theory of dynamical systems, requires the existence of domains of both real and complex eigenvalues. In this section, a projective geometry without metric is presumed to be the natural basis for non-equilibrium, but reversible thermodynamics. In the earlier work reported at the 1977 Aspen Conference on "New Frontiers in Thermodynamics", the Gibbs space used in deriving the shape matrix of the equilibrium “surface" was assumed to be a projective geometry of three dimensions, (U/n, S/n, V /n}, on which the projective constraint was that given by the first law of thermodynamics, ω = dU − Q + W ⇒ 0. (3.24)

The presumption of classical thermodynamics is that the first law is locally equivalent to a Darboux representation, ω = dU − T dS + P dV + µdn ⇒ 0.

(3.25)

The seven-dimensional space of variables, {U, T, S, P, V, µ, n}, is topologically constrained by an exterior differential system that consists of a function homogeneous of degree 1 in the variables, {U, S, V, n}, Θ(U, S, V, n) = n · L(U/n, S/n, V /n) ⇒ 0,

(3.26)

and the differential 1-form, ω = dU − T dS + P dV + µdn ⇒ 0.

(3.27)

The vanishing of the 3-form, ωˆdω = 0, insures that the 1-form, ω, is integrable in the sense of Frobenius, and the condition is at the foundation if Caratheodory’s theory of equilibrium. The existence of this 1-form defines a non-standard, or Cartan surface, for which the shape matrix is not necessarily symmetric, and therefore

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can have complex eigenvalues (see Chapter 8). In equilibrium thermodynamics the additional constraint that ω is integrable implies the existence of a unique solution function, n·L(U/n, S/n, V /n), which is homogeneous of degree 1 in the extensive variables {U, S, V, n}. The partial derivatives of the solution function with respect to the extensive variables, yield the thermodynamic intensities. Whether the 1-form ω is integrable or not, the vanishing of the 1-form constrains the projective shape matrix to be symmetric. A point of departure is realized when the projective constraint is chosen such that the shape matrix admits complex eigenvalues. In all projective geometries, the fundamental invariants are constructed from six primitive cross ratios, two of which are bounded by negative infinity and zero, two of which are bounded by zero and one (the probability domain) and two of which are bounded by one and infinity. It will be demonstrated below how this signature of the three projective equivalence classes appear in the relationships that relate envelopes to bifurcations in projective space. When the Gibbs primitive phase surface of the van der Waals gas is mapped to its dual by means of a Legendre transformation, the Spinodal line can be interpreted as an edge of regression in the dual surface of “Gibbs free energy". It is this clue that focuses attention on the theory of envelopes, for the edge of regression is a singularity in an enveloping surface [251]. It is apparent in the dual surface of Gibbs free energy that, in addition to the edge of regression, there exists another topological feature of singularity, a line of self intersection (which is not an intrinsic property that can be determined locally). This non-metrical feature of self intersection was interpreted as the Binodal line in the earlier work mentioned. Usually, the Binodal line is defined through a heuristic Maxwell construction on the PVT surface representing the equation of state. As Tisza states [259] in reference to the Maxwell procedure, "...a van der Waals gas (referring to the equation of state) does not constitute a fully defined thermodynamic system. A complete definition would include the specific heat as a function of say temperature and volume.... In the concept of a van der Waals gas a spurious interpolation (the Maxwell construction) through the instable range (of the equation of state) is substituted for the missing (specific heat) information." In differential geometry, the line of self intersection is a locus of singularities, and as such would offer a projective geometric definition of the Binodal line, without the heuristic Maxwell assumption. Although visually apparent in the equilibrium surface representing the Gibbs free energy, the differential geometry of the extrinsic Binodal line eluded algebraic formulation. 3.2

The Phase function for a van der Waals Gas

In the classical development of thermodynamics, the van der Waals gas is often used as a cornerstone example. However, the phase function, Θ, given in many textbook

The Phase function for a van der Waals Gas

115

treatments is not explicitly homogeneous of degree 1 in the extensive variables. A homogeneously correct formulation, to within a constant, is given by the relation, S

Θ{...S, V, n; U} = n[e nCv (

R V a U − b)− Cv − V − ] ⇒ 0. n n ( n + cb)

(3.28)

The constant, b, is a representative size of the "particles" that make up molar quantities of the gas. Currently, it is usual to consider the "molar" quantities to be microscopic molecules, but the molar quantities from a topological perspective can be any size, ranging from nuclei to stars. To repeat Sommerfeld’s statement: "The atomistic, microscopic point of view is alien to thermodynamics. Consequently, as suggested by Ostwald, it is better to use moles rather than molecules." p. 11 [245]. The constant a is representative of the interaction forces between the molar quantities. The term a/( Vn )2 has been described by Sommerfeld as representing the "forces (or energy) of cohesion" p. 58 [245]. Note that a correction factor, c, has been used such that, b ⇒ cb, in the historical collision term, a/(V /n) → a/(V /n+cb). The correction factor is used in order to account for the finite interaction size (or an effective scattering wavelength, or coherence length, cb) of the interacting molar particles. In addition, the coefficient c can be adjusted to give a better fit of the van der Waals gas equation to the experimental data of Ωc = (nRTc /Pc Vc ) at the critical point. This equation for Θ{S, V, n; U} satisfies the Euler condition for homogeneity of degree 1, with respect to the extensive variables, {S, V, n; U }, U∂Θ/∂U + V ∂Θ/∂V + S∂Θ/∂S + n∂Θ/∂n − Θ = 0.

(3.29)

The partial derivatives of the phase function, Θ, with respect to the extensive variables may be used to define intensive variables, (P, T, µ, β), (P = −∂Θ/∂V, T = ∂Θ/∂S, µ = −∂Θ/∂n, β = −∂Θ/∂U ).

(3.30)

From the phase function (3.28), partial differentiation yields: S R ∂ V (Θ) = (e nCv ( − b)− Cv )/Cv , ∂S n nRT n2 ∂ . (Θ) = −a P = − ∂V V − bn (V + cbn)2

T =

(3.31) (3.32)

Differentiating P with with respect to V yields, n2 nRT + 2a , ∂P/∂V = − (−V + bn)2 (V + cbn)3

(3.33)

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and differentiation again leads to, ∂ 2 P/∂V 2 = −2

nRT n2 − 6a . (−V + bn)3 (V + cbn)4

(3.34)

The classic argument to determine the critical point sets these differential relations to zero. The values of the thermodynamic variables at the critical point are given by the expressions: Vc = bn(2c + 3),

Tc =

8a/27 , bR(c + 1)

Pc =

a/27 . + 1)2

b2 (c

(3.35)

Note that if the critical molar density is defined as ρc = n/Vc , the previous equations which leads to the universal constant, Ωc , independent from the geometrical parameters {a, b}: Ωc = R(ρc Tc /Pc ) = nRTc /Pc Vc = 8

c+1 . 2c + 3

(3.36)

The reciprocal of Ωc is classically defined as the compressibility, Z = 1/Ωc . For the van der Waals gas (c = 0), Ωc =1/.375, but for many real gases, the experimental value is closer to Ωc =1/.27. This result is in good agreement with the value of c = 4. The value of 1/Ωc versus the coefficient c is displayed in Figure 3.3:

Figure 3.3 Compressibility versus coherence length Other equations of state (such as the Redlich-Kwong) have been used yielding a value for the compressibility Z = 0.333, which, though better that the van der

The Phase function for a van der Waals Gas

117

Waals gas (c = 0) Z = 0.375, does not match the Z = .27 value of experiment. Much more complicated expressions for the equation of state have evolved, guided by the study of critical exponents [112]. For the classic van der Waals gas (c = 0), a rescaled equation of state can be obtained in terms of the dimensionless variables, scaled by their values at the critical point: Te = T /Tc Pe = P/Pc , ρ2 + {(8Te + Pe)/3}e ρ − Pe. 0 = e ρ3 − 3e

e ρ = ρ/ρc

(3.37) (3.38)

At the critical point, e ρ = 1, Te = 1, Pe = 1. What is remarkable is that the coefficients a and b introduced to better account for the properties of the "particles" cancel out in the rescaled formulas. It is this feature that makes the van der Waals gas formulas have a universal appeal, and leads to the idea of "corresponding states". The classic rescaled van der Waals formula leads to a critical isotherm that topologically separates the pure gas phase from those regions that admit liquid, or vapor, coexistent mixed phases. The "universal shape" of the critical isotherm is given in Figure 3.4. It is a topological invariant and is to be recognized by its distinctive shape. Above the critical isotherm, the roots of the phase function can be complex; below the critical isotherm, the roots of the phase function are real.

Figure 3.4 The universal critical isotherm For arbitrary coefficient c, the cubic formula for the reduced equation of state is also independent from the van der Waals parameters a and b, but is of a somewhat

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The Ubiquitous Topological van der Waals gas

more complicated format: Θ = (8Tec3 + (27 + 8Te + Pe)c2 + 54c + 27)ρ3c +((−27 − Pe + 16Te)c2 + (−54 + 16Te + 2Pe)c − 27)ρ2c

ρc

+((−2Pe + 8Te)c + 8Te + Pe)ρc ) − Pe ⇒ 0. = e ρ/(2c + 3).

(3.39) (3.40)

For the special case c = 1/2 the cubic formula mimics the classical Virial expansion where the pressure is expanded in powers of the density: Pe = +3Tee ρ + 1/16(−243/4 + 12Te + 3/4Pe)e ρ2 +1/64(3Te + 1/4Pe + 243/4)e ρ3 .

(3.41)

The (Pe, e ρ) projection of the implicit surface generated for value c = 4 is given in Figure 3.5. This figure should be compared to the case presented above for c = 0. It is obvious that the two figures are deformationally equivalent and belong to the same topological component.

Figure 3.5 Van der Waals gas for c = 4 The reader is encouraged to pick out the critical isotherm, which is a deformation of the critical isotherm for the van der Waals gas with c = 0 (see Figure 3.1 for the reduced pressure at c = 0). It is also true that the Gibbs function for c=4 is a deformation equivalent to Figure 3.2 that describes the swallowtail surface of the van der Waals (c=0) gas.

The Jacobian Matrix of the Action 1-form

3.3

119

The Jacobian Matrix of the Action 1-form

3.3.1 Collineations and Coordinate Diffeomorphisms In Part I, the Cartan topological, metric free, methods of exterior differential forms have emphasized the antisymmetric features of a physical system, especially through the antisymmetric matrix that encodes the coefficients of the 2-form, F = dA. In this Part II, this topological information will be refined by using projective geometric methods based upon correlations and collineations, (see p. 136 in [151]). Symmetric and antisymmetric features will be admissible features to study. Guided by tensor analysis two species of linear arrays will be considered: 1. Contravariant vectors with component functions arranged as a column of k ¯ k® ¯ rows, X . 2. Covariant vectors with component functions arranged as a row of k columns, hAk |.

The concept of a general collineation can be constructed in terms of a possibly non-linear map, α, from a set of coordinates, {xm }, to a contravariant vector field, V k (x). The mapping functions, if C1 differentiable, induce a linear relationship, dα, between the differentials, dxk and dV k . (3.42) α : xm ⇒ V k (xm ), ¯ k® £ k m £ k ¤ ¤ m n n n ¯ dα : |dx i ⇒ dV = ∂V (x )/∂x ◦ |dx i = Kn (x) coll ◦ |dx i . (3.43) ¤ £ ¤ £ The Jacobian matrix of the mapping, Kkn (x) coll = ∂V k (xm )/∂xn , is a classic model for a collineation. A diffeomorphic (possibly non-linear) map that defines an "allowed" coordinate transformation, φ, has a Jacobian matrix, [Qkn (x)], that is also a collineation. φ : xm ⇒ qk (xm ), Diffeomorphism ¯ ® £ ¤ dφ : |dxm i ⇒ ¯dqk = ∂q k (xm )/∂xn ◦ |dxn i = [Qkn (x)] ◦ |dxn i .

(3.44) (3.45)

The diffeomorphism implies that the inverse of the collineation, [Qkn (x)], exists, and that the inverse mapping also exists. The collineations generated by coordinate transformations are a highly specialized and constrained subset of the general collineations. It is possible to conceive of a collineation matrix that is not integrable and not invertible. £ ¤ A general collineation, Kkn (x) coll , maps contravariant vectors into contravariant vectors: ¯ k ® ¯ ® £ ¤ ¯X (x) ⇒ ¯Y j (x) = Kkn (x)

coll

¯ ® ◦ ¯X k (x) .

(3.46)

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The Ubiquitous Topological van der Waals gas

The general collineation also maps covariant vectors into covariant vectors: £ ¤ hAj | ◦ Kkn (x) coll = hAk | ⇐ hBj |

(3.47)

Relative to coordinate transformations, φ, the general collineation is equivalent under transformations generated by the Jacobian of the coordinate ¤ £ similarity k mapping, Qn . If

£ ¤ £ k ¤ Kn (q) coll = [Qkn (x)] ◦ Kkn (x) coll ◦ [Qkn (x)]−1 . £ ¤ £ k ¤ Kn (q) coll = f (q) Kkn (x) coll ,

(3.48)

(3.49)

then the collineation is invariant in a projective sense. In projective geometry, similarity transformations are special projective transformations that preserve parallelism and orthogonality (or better said, preserve points at infinity and the point that defines the origin). 3.3.2 Correlations and the 1-form of Action The concept of a correlation can be modeled in terms of a possibly non-linear map, β, from a set of coordinates, {xm }, to a covariant vector field, Ak . The mapping functions, if C1 differentiable, induce a linear relationship, dβ, between the differentials, dxk and dAk (m). β : xm ⇒ Ak (xm ), (3.50) m m n n n dβ : |dx i ⇒ |dAk i = [∂Ak (x )/∂x ] ◦ |dx i = [Jkn (x)]corr ◦ |dx i . (3.51) The Jacobian matrix of the mapping, [Jkn (x)]corr , is a classic correlation. The starting point for topological thermodynamics is the formulation of a 1form of Action, A = Ak dxk . The Jacobian matrix, [Jkn (x)]corr = [∂Ak (x)/∂xj ] , of partial derivative functions, created from the coefficients of the 1-form of Action, Ak (x), is a correlation, not a collineation. Correlations are very different from collineations. Correlations map contravariant vectors into covariant vectors; associated correlations map covariant vectors into contravariant vectors. In projective geometry of the plane, it would be said that correlations map points into lines, and collineations map points into points. Relative to diffeomorphic "coordinate" transformations, correlations are equivalent under congruent transformations (to within a factor), where collineations are equivalent under similarity transformations (to within a factor). Correlations are not invariant under similarity transformations induced by coordinate diffeomorphisms. Collineations can be invariant under coordinate diffeomorphisms. However, the similarity invariants of a correlation are invariants of coordinate transformations.

The Jacobian Matrix of the Action 1-form

121

Remark 32 The importance of ubiquitous orthogonal transformations resides with the fact that then the congruent transformation and the similarity transformation are the same. The Jacobian correlation matrix, [Jkn (x)]corr , contains both symmetric and antisymmetric components. The matrix representation of the 2-form, dA, emphasizes the antisymmetric features of the Jacobian correlation matrix, [Jkn (x)]corr . When dA = 0, the Jacobian matrix is symmetric, and the correlation defines (in projective geometry) that which is called a "polarity". If the correlation, [Jkn (x)]corr , is symmetric, then all its eigenvalues are real. It is the antisymmetric parts of the correlation matrix that can lead to complex eigenvalues, and the possibility of Spinor eigendirection fields. These situations related to complex eigenvalues have interesting correspondence to universal (topological) equivalence classes of thermodynamic systems. 3.3.3 The Thermodynamic Phase function Every n × n matrix satisfies a polynomial Cayley-Hamilton matrix equation, which leads to an algebraic Cayley-Hamilton (characteristic) polynomial equation of degree n. It is the characteristic polynomial based on the correlation matrix, [Jkn (x)]corr , that will be used to define a "universal" thermodynamic Phase function, Θ(x, y, z, t; ξ). On a variety of four dimensions, the 4 × 4 Cayley-Hamilton characteristic polynomial can be written in terms of the (possibly complex) eigenvalues, ξ, of the Jacobian matrix, [Jkn (x)]corr as, Cayley-Hamilton polynomial = ξ 4 − XM ξ 3 + YG ξ 2 − ZA ξ 1 + TK = 0.

(3.52)

It is a notable result that the coefficients of the Cayley-Hamilton polynomial are invariant with respect to similarity transformations of the correlation matrix. That is, the coefficients are invariants with respect to coordinate similarity transformations generated by [Qkn ]. It is assumed that this characteristic equation, as a polynomial of 4th degree, is in effect a Universal Thermodynamic Phase function, Θ(x, y, z, t; ξ) = 0, representing a family of implicit surface functions in a four-dimensional space, with "coordinates" defined in terms of the symmetry invariants, {XM (xm ), YG (xm ), ZA (xm ), TK (xm )}. The family parameter is the variable, ξ, which represents the eigenvalues of the Jacobian matrix, Phase Function Θ(XM , YG , ZA , TK ; ξ) ⇒ 0. (3.53)

The family parameter can be real (single parameter) or complex (two parameter‡ ). The Phase function is distinct for different choices of the 1-form of Action, A, but all such Phase functions can be related to the deformation equivalence classes that include the classic van der Waals gas. It will be demonstrated below how and when the similarity invariants can be related to "curvatures" of the universal implicit ‡

Classical thermodynamic string theory

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The Ubiquitous Topological van der Waals gas

hypersurface. However, no metric is used explicitly to define the "curvatures". The rank of the determinant of the Jacobian matrix can be interpreted as a "projective dimension". If the rank of [Jkn (x)]corr is 4, then the projective dimension is 4. If the rank is 3, then the projective dimension is 3. If the rank is 3, then one of the 4 possible eigenvalues, ξ, must be zero. Rank 3 requires that TK ⇒ 0. If the rank is 2, then two eigenvalues must be zero, and therefore TK ⇒ 0, and ZA ⇒ 0. The non-equilibrium extensions of the van der Waals gas (of Pfaff topological dimension 3 or 4) are to a certain extent encoded in the third and fourth order similarity invariants, ZA and TK , and the possibilities that the characteristic polynomial can have complex roots. If the Pfaff topological dimension of the 1-form of Action, A, is 1, then the Jacobian matrix, [Jkn (x)]corr , is a symmetric correlation, and has only real eigenvalues. If the Pfaff dimension is 2, then an integrating factor exists, and the isolated system (ignoring defects) is not a non-equilibrium system. The antisymmetric parts of the Jacobian matrix constructed from the product of the 1-form, A, and the integrating factor, go to zero. However, if the Pfaff topological dimension of the 1-form is 3 or 4, then the thermodynamic system is not in equilibrium. The antisymmetric part of the Jacobian matrix cannot be removed by algebraic rescaling, for an integrating factor does not exist. The antisymmetric portion of the Jacobian, equivalent to dA, will have 1 pair of complex conjugate spinor eigenvalues if the Pfaff dimension is 3, and will have 2 pairs of complex conjugate spinor eigenvalues if the Pfaff topological dimension is 4. From a study of the van der Waals gas, it becomes apparent that the pure gas region above the critical isotherm is the region of complex eigenvalues. In the classical theory of a van der Waals gas, the concept of a universal thermodynamic critical point, and a Spinodal line of ultimate phase stability have visual exhibition in terms of the non-smooth (topological) defect structures of the Gibb’s swallowtail surface. For the equilibrium projection 3D to 2D (see Figure 3.2) the Spinodal line is recognized as the edge of regression of the swallowtail. The edge of regression corresponds to the case where the quadratic similarity invariant (the Gauss curvature) vanishes. The thermodynamic critical point occurs when both the Mean curvature and the Gauss curvature of the equilibrium surface vanish. Extending these universal ideas to 4D leads to the suggestion that a Cosmological universe might have a representation as a non-equilibrium van der Waals gas near its critical point, where fluctuations in density are the condensates called stars and galaxies [223]. 3.3.4 The details of the Universal Characteristic Phase Function The 1-form of Action, used to encode a physical system, contains other useful topological information, as well as geometric information. Reconsider the details of an open thermodynamic system generated by a 1-form of Action, A, of Pfaff topological dimension 4. The component functions of the Action 1-form can be used to construct a 4 × 4 Jacobian matrix of partial derivatives, [Jkn (x)]corr = [∂Ak /∂xn ]. This matrix can be considered as a correlation in a projective geometry. In general, this

The Jacobian Matrix of the Action 1-form

123

Jacobian matrix will be a 4×4 matrix that satisfies a 4th order Cayley-Hamilton characteristic polynomial equation, Θ(x, y, z, t; ξ) = 0, with four, perhaps complex, roots representing the four, perhaps complex, eigenvalues, ξ k , of the Jacobian matrix. Θ(x, y, z, t; ξ) = ξ 4 − XM ξ 3 + YG ξ 2 − ZA ξ 1 + TK ⇒ 0.

(3.54)

It should be noted that the characteristic polynomial emphasizes the symmetries of the Jacobian matrix, as the coefficients, {XM , YG , ZA , TK } are invariant under symmetrical permutations and similarity transformations. The antisymmetric components of the Jacobian are not ignored, and exhibit artifacts in the form of complex eigenvalues. If the Jacobian matrix is symmetric, its eigenvalues are real. The topological 2-form, F = dA, encodes the antisymmetric components of the Jacobian matrix. Hence, if the 1-form of Action, A, is of Pfaff dimension 1 (such that F = dA = 0), the antisymmetric components of the Jacobian matrix do not exist; all eigenvalues are real. Such is the case for an equilibrium thermodynamic system (see Chapter 2.2.2). However, there exist 1-forms of Action that are of Pfaff topological dimension 4, yet the fourth order symmetric similarity invariant TK ⇒ 0. Such is the case when the projective dimension is 3 instead of 4. Similarly, there are examples where ZA and TK both go to zero, but the system defined by the 1-form of Action, A, is of Pfaff dimension 3, and therefore the Pfaff topological dimension defines a non-equilibrium system. The relationships between Pfaff dimension and the projective dimension of the root structure of the characteristic polynomial can be quite complex. The theory of these relationships is not complete but will appear in [277]. The Cayley-Hamilton polynomial equation defines a family of implicit functions in the space of the real similarity invariant variables, XM (x, y, z, t), YG (x, y, z, t), ZA (x, y, z, t), TK (x, y, z, t), even though the eigenvalues can be complex. As first stated in the first section of this chapter, the similarity invariants are given by the expressions: XM YG ZA TK

= = = =

ξ 1 + ξ 2 + ξ 3 + ξ 4 = T race [Jjk ] , ξ 1ξ 2 + ξ 2ξ 3 + ξ 3ξ1 + ξ4ξ 1 + ξ 4ξ 2 + ξ 4ξ 3, ξ 1ξ 2ξ 3 + ξ 4ξ1ξ 2 + ξ 4ξ 2ξ 3 + ξ 4ξ 3ξ 1, ξ 1 ξ 2 ξ 3 ξ 4 = det [Jjk ] .

(3.55) (3.56) (3.57) (3.58)

The 4-fold degeneracy that defines the critical point has roots ξ = [1, 1, 1, 1], such that XM = 4, YG = 6, ZA = 4, TK = 1. The critical point is not a fixed point (with zero eigenvalues). These results facilitate the law of corresponding states. From the theory of strings and surface tension, the XM term can be interpreted as a linear deformation contribution to the "energy" of the system. The coefficient YG can be related to the Gauss (quadratic) curvature of the system, and is related to an area deformation contribution. The coefficient ZA can be related to the Interaction (Cubic) curvature of the system, and is related to a volume deformation contribution

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The Ubiquitous Topological van der Waals gas

(a pressure) to the "energy". The last term TK is a quartic contribution and can be related to an expansion or contraction of the four-dimensional volume element. Symbolically, multiply the phase function by u/ξ 4 and and consider u/ξ to be a length deformation, δ Length , u/ξ 2 to be an area deformation, δArea , u/ξ 3 to be a volume deformation, δ V ol , and u/ξ 4 to be a space-time expansion deformation, δ Exp_xyzt . The suggestive formula becomes, Θ = u − XM · δ Length + YG · δ Area −ZA · δ V ol + TK · δ Exp_xyzt . By comparison with a van der Waals gas, XM ≈ "String or Surface Tension", YG ≈ ”Temperature - Entropy", ZA ≈ ”Pressure - Interaction", TK ≈ xyzt- "Higgs" Expansion.

(3.59) (3.60) (3.61) (3.62) (3.63) (3.64)

Automatically, the phase function incorporates string or surface tension effects through XM , where XM can be related to a mean four-dimensional curvature expression. Gravity effects, related to the 4D Gauss curvature, G = YG /6 are "area" related. Remark 33 From the idea that the entropy of a gravitational black hole is related to an area, and the fact that the phase formula for a van der Waals gas implies that YG is dominated by the temperature (see (3.9)), the universal phase formula suggests that the idea of gravity (and the Gauss curvature) is a temperature - entropic concept, contributing energy of the type T S. The phase formula for a van der Waals gas implies that the ZA coefficient is related to pressure (which can be both negative or positive), and the "energy" contribution is of the type P V. The last term represents a xyzt expansion, which from the topological theory of thermodynamics presented above can be related to irreversible dissipation caused by "bulk viscosity". It is sometimes more convenient to express the similarity invariants in terms of their averages, where the average is determined by dividing by the number of non-zero eigenvalues. This leads to a sequence of maps from the original variety of independent variables, {x, y, z, t} ⇒ {XM , YG , ZA , TK } ⇒ {M, G, A, K}. (Do not confuse the A ↔ ZA with the symbol used to denote the 1-form of Action, A.) When the similarity invariants are treated as generalized "coordinates"§ , then the characteristic polynomial becomes a Universal Phase function, and will be used to encode universal thermodynamic properties. A similar procedure can be applied to domains of lesser dimension. For example, suppose the dimension of the domain is reduced from projective dimension §

This concept is similar to the concept of "natural" equations in the theory of space curves, (see p.26 [251]).

The Jacobian Matrix of the Action 1-form

125

4 to projective dimension 3 by the constraint that the determinant of the Jacobian matrix vanishes, TK ⇒ 0. If the Pfaff topological dimension is also 3, the thermodynamic system is a non-equilibrium system, far from equilibrium, which can support "steady" states. The Phase equation must have one null eigenvalue, that represents a null eigenvector, or fixed point of the Jacobian matrix. The Phase equation with one eigenvalue equal to zero (say ξ 4 = 0) reduces to Θ(x, y, z, t; ξ) with XM YG ZA

= = = =

(ξ 3 − XM ξ 2 + YG ξ − ZA )ξ ⇒ 0, (ξ 1 + ξ 2 + ξ 3 ), (ξ 1 ξ 2 + ξ 2 ξ 3 + ξ 3 ξ 1 ), ξ 1 ξ 2 ξ 3 = ZA .

(3.65) (3.66) (3.67) (3.68)

Conjecture An objective herein is to exploit the striking similarity between the cubic factor of the 3D phase equation (3.65), and the cubic equation of the rescaled van der Waals gas given by equation (3.38 ). The fundamental assumption is that the eigenvalue of the Cayley-Hamilton characteristic polynomial for the Jacobian matrix, [Jkn (x)]corr = [∂Ak /∂xn ], plays the role of the rescaled molar density ρ/ρc in thermodynamics. The critical point in 3D occurs for the set {XM = 3, YG = 3, ZA = 1, TK = 0} with ξ ⇔ e ρ = [1, 1, 1, 0]. A comparison of the universal equation and the van der Waals gas equation yields, 3 = XM ,

(3.69)

{8Te + Pe}/3 = YG , Pe = ZA .

(3.70)

Θ(x, y; z, t ξ) = (ξ 2 − XM ξ 1 + XG )ξ 2 ⇒ 0, with XM = (ξ 1 + ξ 2 ), YG = (ξ 1 ξ 2 ).

(3.72) (3.73) (3.74)

(3.71)

For the classic van der Waals gas, it is apparent that the (linear) similarity invariant (which is composed of the sum of molar density eigenvalues) is at its critical point value, XM = 3. The (quadratic) similarity invariant is equal to YG = ({8Te + Pe}/3) and is composed of both temperature and pressure terms. The Adjoint (cubic) interaction similarity invariant is equal to ZA = Pe, the rescaled pressure. It is remarkable that for the van der Waals gas, the linear terms (representing string or surface tension effects, have been fixed at their "Critical Point" values. These concepts will be presented in more detail in Chapter 5. If a further reduction in dimension occurs to Pfaff dimension 2, (with 2 null eigenvalues) the Phase equation with {M, G} reduces to

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The Ubiquitous Topological van der Waals gas

The critical point in 2D occurs for the set {XM = 2, YG = 1, ZA = 0, TK = 0} with ξ k = [1, 1, 0, 0]. 3.4

The Reduced Phase function in 4D

3.4.1 The Higgs potential There exists a well known transformation of a complex variable which will reformulate the characteristic polynomial. Substitute ξ = s + M/4 into the formula (3.54) for Θ(x, y, z, t; ξ). The result is a new "reduced" Phase polynomial, Φ(x, y, z, t; s) = Θ(x, y, z, t; ξ)reduced of the form, s4 + gs2 − as + k = 0. 2 /8 + YG ), (−3XM (XM /2)3 − YG XM /2 + ZA ), TK − ZA (XM /4) + YG (XM /4)2 −3(XM /4)4 , s = ξ − XM /4.

Φ(x, y, z, t; s) g a k

= = = =

(3.75) (3.76) (3.77) (3.78) (3.79)

The "reduced" Phase function is not the same as the "rescaled" Phase function. The roots of the cubic have been shifted by the amount M/4. For a van der Waals gas (XM = 3, TK ⇒ 0), the reduced coefficients become, Van der Waals gas - a special case g = −27/8 + YG = −27/8 + {8Te + Pe}/3, a = −27/8 + {8Te − Pe}/2, k = −243/256 + TK − 9/16Pe + 3/2Te, s = ξ − 3/4.

(3.80) (3.81) (3.82) (3.83)

The critical point has been moved to s = 1/4 for the van der Waals gas, as one of the eigenvalues is presumed to be zero. The reduced formula has eliminated the cubic term in the universal phase function by displacing the critical point to the origin in terms of the variable s, if all eigenvalues are not zero. Consider the reduced Phase formula, Φ, and its derivatives with respect to the family parameter, s: Φ = ∴ Φs = ∴ Φss = ∴

s4 + gs2 − as + k = 0, k = −(s4 + gs2 − as), ∂Φ/∂s = 4s3 + 2gs − a = 0, a = 4s3 + 2gs, Φs = ∂ 2 Φ/∂s2 = 12s2 + 2g = 0, g = −6s2 .

(3.84) (3.85) (3.86) (3.87) (3.88) (3.89)

The Reduced Phase function in 4D

127

Replacing the parameter a (from the envelope condition, Φs = 0) in the equation for k yields k = s2 (3s2 + g). (3.90) A plot of the equation for k is given below for various g and s. The plot yields the quartic function shape, hence the name, the "Higgs potential". This is a second clue to the idea that there is a fundamental thermodynamic difference between the Covariant differential (which depends upon a gauge constraint) and the Lie differential (which does not).

Figure 3.6 The vertical axis represents the function k(s,g) = s2 (3s2 +g) In the general case with TK 6= 0, then g = 0, s = 0, k = 0, represents the "critical point". Then for g (think reduced temperature) values below the critical point, the function k is a polynomial of 4th degree, but above the "critical temperature" the function k is quadratic. It is evident that below the critical isotherm, the "expansion" term k can have both negative and positive values. The formula for the 4D expansion coefficient therefore can also have positive or negative values. The quartic "potential" is reminiscent of the "Higgs" potential in relativistic field theories and the "Landau" potential in mean field theories. Note that these properties have been obtained without explicit use of a metric or connection, nor quantum mechanics. Also note that the surface, k = 0, intersects the Higgs potential in a pitchfork bifurcation. From the van der Waals theory, the first partial derivative of the classic phase function yields the pressure. For the universal Phase polynomial the "reduced pressure a" is determined by the equation Φs = 0. Indeed, the formula a = 4s3 + 2gs yields the universal cubic equation for a (the pressure) in terms of the molar density ”s”. A plot of a (pressure) versus s (molar density) at fixed values of g (temperature) gives the familiar cubic shape, deformably equivalent to the van der Waals gas.

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The Ubiquitous Topological van der Waals gas

For the critical temperature, (g = 0), the shape of the critical isotherm is exactly the same as for the critical isotherm of the van der Waals gas. Both k (Higgs Expansion in solid black) and a (Pressure in dashed blue) are presented in the following diagrams as constant g (∼temperature) slices above and below the critical point.

Figure 3.7

Figure 3.8

The Reduced Phase function in 4D

129

Figure 3.9

Figure 3.10 Higgs parameter, k, and reduced pressure, a, at constant reduced temperature, g. The Critical Isotherm The topology of the quartic phase (potential) function is separated by a critical isotherm into two sectors. For temperatures below the critical temperature, the

130

The Ubiquitous Topological van der Waals gas

quartic formula yields a Higgs-like sector where expansion properties k can be negative, and where "liquid and vapor" phases can coexist. Above the critical temperature the 4th order expansion properties k are positive, and the sector has lost its Higgs-like quartic properties. The critical isotherm, a = 4s3 (when g = 0) defines a line of singularities separating the two sectors. The shape of the curve mimics the dividing center of hysteresis phenomena. Note that the reduced pressure function, a, has maxima and minima at those points where the partial derivative of k vanishes, ∂k/∂s = 0. 3.4.2

The Binodal line

Guided by experience with the van der Waals gas the coefficient a has been defined as the reduced pressure. The zero (real) sets of the "reduced pressure a" occur only for temperatures below the critical point and are described by the solutions to the formula, 2(2s2 + g)s ⇒ 0. The Binodal line (along with the solution, s= 0) is generated by the formula (in reduced variables),

Binodal Line

g = −2s2 .

(3.91)

Note that the curve for reduced pressure satisfies the Maxwell "equal area" construction for defining the mixed phase region. The two real solution curves (s = 0, s2 = −g/2) define a pitchfork bifurcation in terms of the zeros of the "reduced" pressure, a, consistent with the constraint that that the reduced phase function admits an envelope, Φs = 0. This result appears to be the first non-phenomenological derivation of the Binodal line. 3.4.3

The Spinodal Line

A second piece of topological information can be obtained from those points where the partial derivative of the "reduced pressure a" vanishes. In terms of an envelope, these points are given by solutions to the equation, Φss = 12s2 + 2g = 0, which is the necessary condition that the envelope has an edge of regression. The spinodal line determines an edge of regression (also see Figure 3.2 and Figure 5.3).

Spinodal Line

g = −6s2 .

(3.92)

Again only for temperatures g below the critical isotherm will the formula give a set of points that describes classically what has been called the Spinodal line. In van der Waals theory the Spinodal line defines the "limit" of single phase stability and can only be realized transiently, in the absence of fluctuations. Both Spinodal line (black dashed) and the Binodal line (blue solid) are plotted in Figure 3.11.

Oscillations, Spinors and the Hopf bifurcation

131

Figure 3.11 Higgs surface, Binodal and Spinodal lines The Binodal line and the Spinodal line can be related to homology invariants of projective transformations (see Section 8.2.1). 3.5

Oscillations, Spinors and the Hopf bifurcation

Bifurcation and singularity theory involves the zero sets of the similarity invariants, and the algebraic intersections of the implicit hypersurfaces so generated by these zero sets. Recall that the theory of linear (local) stability requires that the eigenvalues of the Jacobian matrix have real parts which are not greater than zero. For a 4th order polynomial, either all four eigenvalues are real; or, two eigenvalues are real, and two eigenvalues are complex conjugate pairs; or there are two distinct complex conjugate pairs. Local stability therefore requires: Local Stability XM YG ZA TK

≤ ≥ ≤ ≥

0, 0, 0, 0.

(3.93) (3.94) (3.95) (3.96)

In that which follows, the similarity coefficients of the thermodynamic 1-form, A, will be studied for physical systems of various Pfaff topological dimensions. In the next chapter, 1-forms representing turbulent non-equilibrium systems will be studied as candidates for Falaco Solitons and cosmological models.

132

The Ubiquitous Topological van der Waals gas

3.5.1 The Hopf Map and Hopf vectors A particularly useful and interesting 1-form of Action, A, that exhibits local stability can be deduced from the Hopf map. The Hopf map¶ is a rather remarkable projective map from 4 to 3 (real or complex) dimensions that has topological properties related to links and braids and other forms of entanglement. As will be demonstrated, the adjoint 1-form to the Hopf map satisfies the criteria of Local Stability, and yet is not an integrable system. The 1-form deduced from the Hopf map is of Pfaff Topological dimension 4, and admits irreversible dissipation for processes in the direction of the Topological Torsion vector. Consider the map from R4[X,Y,Z,S] to R3[u,v,w] given by the formulas, H1 = [u1, v1, w1] = [2(XZ + Y S), 2(XS − Y Z), (X 2 + Y 2 ) − (Z 2 + S 2 )].

(3.97)

The components [u1, v1, w1] can be considered as the velocity components of a dynamical system. These formulas define the format of a Hopf map. The 3-component Hopf vector H1 is real, and has the property, H1 · H1 = (u1)2 + (v1)2 + (w1)2 = (X 2 + Y 2 + Z 2 + S 2 )2 .

(3.98)

Hence a real (and imaginary) four-dimensional sphere maps to a real three-dimensional sphere. If the functions [u1, v1, w1] are defined as [x/ct, y/ct, z/ct], then the 4D sphere (X 2 + Y 2 + Z 2 + S 2 )2 = 1, implies that the Hopf map formulas are equivalent to the 4D light cone. The Hopf map and its associated Hopf vector can also be represented in terms of complex functions by a map from C2 to R3, as given by the formulas, H1 = [u1, v1, w1] = [α · β ∗ + β · α∗ , i(α · β ∗ − β · α∗ ), α · α∗ − β · β ∗ .

(3.99)

The variables α and β also can be viewed as two distinct complex variables defining ordered pairs of the four variables [X, Y, Z, S]. For example, the classic format given above for H1 can be obtained from the expansion, α = X + iY , β = Z + iS. Other selections for the ordered pairs of (X, Y, Z, S) (along with permutations of the 3 vector components) give distinctly different Hopf vectors. For example, the ordered pairs, α = X + iZ , β = Y + iS, give H2 = [2(Y X − SZ), X 2 + Z 2 − Y 2 − S 2 , −2(ZY + SX)],

(3.100)

which is another Hopf vector, a map from R4 to R3, but with the property that H2 is orthogonal to H1, ¶

Also see Chapter 12 in [276]

Oscillations, Spinors and the Hopf bifurcation

H2 · H1 = 0.

133

(3.101)

Similarly, a third linearly independent orthogonal Hopf vector H3 can be found, H3 = [X 2 + Y 2 − Z 2 − S 2 , −2(Y X + SZ), 2(−ZX + SY )],

(3.102)

such that

H2 · H1 = H3 · H2 = H2 · H3 = 0, H1 · H1 = H2 · H2 = H3 · H3 = (X 2 + Y 2 + Z 2 + S 2 )2 .

(3.103) (3.104)

The three linearly independent Hopf vectors can be used as a basis of R3 excluding those points where the quartic form vanishes. The mapping functions (u, v, w) of the Hopf vector can be differentiated with respect to (X, Y, Z, S) to produce a set of three exact 1-forms whose coefficients form three independent, 4component, vectors on R4. The matrix of these three 4D vectors can be constructed, and its adjoint determined to produce a 4th linearly independent vector. This adjoint field defines, to within a factor, the Hopf 1-form, AHopf : AHopf = {−Y d(X) + Xd(Y ) − Sd(Z) + Zd(S)}.

(3.105)

The formula for the Hopf 1-form, AHopf , can be generalized to include constant coefficients of chirality, R = ±1 and L = ±1, to read AHopf = {R(−Y d(X) + Xd(Y )) + L(−Sd(Z) + Zd(S))}/Λ.

(3.106)

Each chiral pair, (−Y d(X) + Xd(Y ) and (−Sd(Z) + Zd(S)), can have the same or opposite chirality. For Λ = 1, it follows that the Hopf 1-form is of Pfaff dimension 4 with the topological torsion 4-vector, T4 = −RL[X, Y, Z, S],

(3.107)

and with a dissipation coefficient, dAHopf ˆdAHopf = −RL{8dXˆdY ˆdZˆdS} ≶ 0. (3.108) As the sign of the Topological Parity 4-form determines whether of not the 4D volume element is expanding or contracting, it follows that the relative chirality of the two links is physically measurable. Hopf Topological Parity

134

The Ubiquitous Topological van der Waals gas

The Jacobian matrix of the coefficients of the Hopf 1-form (for Λ = 1) becomes, ⎡ ⎤ 0 R 0 0 ⎢ −R 0 0 0 ⎥ ⎥ (3.109) JACHopf := ⎢ ⎣ 0 0 0 −L ⎦ , 0 0 L 0

with eigenvalues e1 = iR, e2 = −iR, e3 = iL, e4 = −iL, and with similarity invariants, Hopf XM YG ZA TK

: = = = =

Similarity Invariants 0 ≤ 0, R2 + L2 ≥ 0, 0 ≤ 0, R2 L2 ≥ 0.

(3.110) (3.111) (3.112) (3.113)

Hence the canonical Hopf 1-form is locally stable. Note that both the Hopf Topological Parity and the Topological Torsion vector depend upon the signs of the two chirality coefficients, R and L, but the similarity invariants do not. The topological properties are chiral sensitive to antisymmetries, where the similarity coefficients are not. When the eigenvalues of the characteristic polynomial are pure imaginary, Hopf oscillations can occur. Hence the criteria for a double Hopf oscillation frequency requires that the algebraically odd similarity invariants vanish (hence there exists a minimal surface) and the algebraically even similarity invariants are positive definite. For a single Hopf oscillation frequency (L = 0), the Hopf conditions are, XM = 0, YG = R2 > 0, ZA = 0, TK = 0. These conditions can be computed relatively easily, and will be demonstrated in the examples below. Note that the minimal hypersurface condition XM = 0 may be also be satisfied by states with YG < 0, in accord with the examples of soap films. Such conditions are related to non-oscillatory solitons which form "stationary states", but are globally stabilized far from equilibrium (see Chapter 5.8 on Falaco Solitons, with more detail in [274]). 3.5.2 Minimal surfaces The Universal Phase function, Θ, may be considered as a family of hypersurfaces in the four-dimensional space, {XM , YG , ZA , TK } with a complex family (order) parameter, ξ. Moreover, it should be realized that the Universal Phase Function is a holomorphic function, Θ = φ + iχ in the complex variable ξ = u + iv. That is, Θ(XM , YG , ZA , TK ; ξ) ⇒ φ + iχ,

(3.114)

φ = u4 − 6u2 v 2 + v4 − XM (u2 − 3v 2 )u +YG (u2 − v2 ) − ZA u + TK , χ = 4(u2 − v 2 )uv − XM (3u2 − v 2 )v + 2YG uv − ZA v.

(3.115) (3.116)

where

Oscillations, Spinors and the Hopf bifurcation

135

As such, in the 4D space of two complex variable pairs, {φ + iχ, u + iv}, according to the theorem of Sophus Lie, any such holomorphic function produces a pair of conjugate minimal surfaces in the four-dimensional space {φ, χ, u, v}. It follows that there exist a sequence of maps, {x, y, z, t} ⇒ {XM , YG , ZA , TK } ⇒ {φ, χ, u, v},

(3.117)

such that the family of hypersurfaces can be decomposed into a pair of conjugate minimal surface components. The criteria for a minimal surface is equivalent to the idea that XM = 4 · M ⇒ 0. The averaged similarity invariant plays the role of the hypersurface. Examples of conjugate pairs of minimal surfaces The idea is that a complex tangent vector, V = [U, V, W ], in complex 3-space, C3 , can be integrated to produce a complex position vector, R = [X, Y, Z], whose real, or imaginary, parts will map out a minimal surface in real 3-space, R3 , if the quadratic form in C3 vanishes: hV| ◦ |Vi = 0. (3.118) Such direction fields |Vi were defined by Cartan as "Pure Spinors". Spinors will be designated by a change of notation to be complex vectors, |Spi, such that hSp| ◦ [η] ◦ |Spi = 0.

(3.119)

The fundamental quadratic form has been modified to include an arbitrary signature matrix, [η] . The complex tangent vector which has spinor properties (zero, or null, quadratic form) can be generated from the Weierstrass representation [165] in terms of the holomorphic function H ( ) = φ + iχ, R X( ) = (1 − 2 )H( )d , (3.120) R 2 (3.121) Y ( ) = (1 + )H( )d , R Z( ) = (2 )H( )d . (3.122) Rewriting H( ) in the form

H( ) = (b − ia)/2

2

, with

= −i exp(η + iξ),

(3.123)

and substituting into the Weierstrass formulas yields the position vector to a family of minimal surfaces of the form X = a sinh(η) cos(ξ) − b cosh(η)sin(ξ), Y = a sinh(η)sin(ξ) + b cosh(η)cos(ξ), Z = a ξ + b η.

(3.124) (3.125) (3.126)

136

The Ubiquitous Topological van der Waals gas

For a = 0 the surface is a catenoid; for b = 0 the surface is a helicoid ([163] p.70). For a and b non-zero, the minimal surface so generated consists of two conjugate minimal surfaces intertwined (the example has a = b = .5)

Figure 3.12 Minimal Chiral Helicoids Note that the conjugate pairs have different chirality. Example of a fractal minimal surface As a second example, consider the Holomorphic function and its functional iterates, H1 ( ) = (

2

− D), H2 ( ) = ((

2

− D)2 − D), ... .

(3.127)

According to the minimal surface theorem, this vector field represents a one (complex) parameter family of minimal surfaces in four dimensions. It follows that the Mandelbrot set, which is given by the values of D for which the function H1 ( ) fails to iterate the origin ( = 0) to infinity is the fractal envelope of a family of minimal surfaces in four dimensions parameterized by D = a + ib. The compliment to the Mandelbrot set is a minimal surface with a fractal boundary where all functional sequences iterate to infinity. Hence the "fractal" minimal surface is complete. The "Gibbs entropy" minimal surface As another example, consider those functions of a complex variable such that H( ) = (∂F ( )/∂ )3 . All functions F ( ) that have the form, F ( ) = {α

ln( ) + C

+ (B − D

2

)},

(3.128)

Oscillations, Spinors and the Hopf bifurcation

137

will generate the same function, H( ) = (∂F ( )/∂ )3 = 2A/

2

.

(3.129)

The format of F ( ) is strikingly reminiscent of those formulas that appear in the literature to describe the Gibb’s entropy. The coefficients α, B, C and D are presumed to be complex constants. Rewriting H( ) in the form H( ) = (b − ia)/2

2

, with

= −i exp(η + iξ),

(3.130)

and substituting into the Weierstrass formulas yields the position vector to a family of minimal surfaces. When α is real then extremal minimal surface is a catenoid again; when α is imaginary, the minimal surface is a helicoid. The interesting features are : 1. All wave functions are related to a minimal surface by this technique. 2. The primitive function, F ( ), is related to the Helmholtz free energy, and it is the entropy term, α ln( ), that generates the family of minimal surfaces. 3. The resulting minimal surface is independent of the linear term C "Mandelbrot germ", (B − D 2 ).

and the

4. The Petrov type D classifications (which yield the only known black hole solutions to the Einstein gravity theory [47] are related to minimal surfaces. Envelopes of the Thermodynamic Phase function The theory of implicit hypersurfaces focuses attention upon the possibility that the Universal Phase function has an envelope. The existence of an envelope depends upon the possibility of finding a simultaneous solution to the two implicit surface equations of the family, Θ(x, y, z, t; ξ) = ξ 4 − XM ξ 3 + YG ξ 2 − ZA ξ + TK ⇒ 0, ∂Θ/∂ξ = Θξ = 4ξ 3 − 3XM ξ 2 + 2YG ξ − ZA ⇒ 0.

(3.131) (3.132)

For the envelope to be smooth, it must be true that ∂ 2 Θ/∂ξ 2 = Θξξ 6= 0, and that the exterior 2-form, dΘˆdΘξ 6= 0 subject to the constraint that the family parameter is a constant, dξ = 0. The envelope as a smooth hypersurface does not exist unless both conditions are satisfied (see Chapter 9.10). The envelope function is to be used as an implicit surface function, and therefore admits an arbitrary factor. Other related algebraic functions are called the resultant function, or the discriminant function. These functions have been programmed into symbolic math packages, such as Maple. The discriminant of the Phase Function polynomial, which, as a zero set, is equal to a universal implicit

138

The Ubiquitous Topological van der Waals gas

hypersurface, DISCΘ ⇒ 0, in the four-dimensional space of similarity variables {XM , YG , ZA , TK }. The discriminant is equal to to the envelope, to within a constant factor. The 4D discriminant function can be written in terms of the similarity "coordinates" (suppressing the subscripts) as, DISCΘ = +Z 2 {−4X 3 Z + 18XZY + Y 2 X 2 − 4Y 3 − 27Z 2 } − 4Y 3 X 2 T +T {+(256)T 2 + (144Y X 2 − 192XZ − 128Y 2 − 27X 4 )T (3.133) −Z 2 (6X 2 + 144Y ) + 16Y 4 + XZY (18X 2 − 80Y )}. The discriminant function times (-1)D is equal to the resultant function. The envelope is a constant factor times the resultant. All three functions have zero sets (implicit hypersurfaces) that have eliminated the family order parameter, ξ. It is important to note that the first bracket in the formula above is equal to the discriminant of the cubic polynomial, and is proportional to the infamous Cardano function,

Ψ = ξ 3 − XM ξ 2 + YG ξ − ZA , (108) · Cardano = (−DISCΨ) = {4X 3 Z − 18XZY − Y 2 X 2 + 4Y 3 + 27Z 2 }.

(3.134) (3.135)

When T = 0, the determinant of the Jacobian matrix is of rank 3, not rank 4. The Cardano function determines the root structure of the characteristic polynomial. In particular when Cardano > 0, then there are two complex roots and one real root. Such a case corresponds to domains of pure gas phase, above the critical isotherm: DISCΘ = −Z 2 · 108 · Cardano > 0.

(3.136)

Domains in the pure liquid or pure vapor or mixed liquid-vapor phases occur when Cardano < 0 . The reduced polynomial (3.75) can also be used to construct an envelope function. For example the discriminant of the Reduced Phase phase function, Φ = Θreduced is equal to DISCΦ = −27a4 + 4(−g 2 + 36k)ga2 + 16k(4k − g2 )2 .

(3.137)

The hypersurface defined by the discriminant, DISCΦ,of the reduced phase function Θreduced = Φ yields the universal swallowtail hypersurface. A plot of the universal envelope Φ = 0, in terms of the coordinates (g, a, k), is given in the Figure 3.13. In classical thermodynamics, the surface is recognized as that of a deformed Gibbs’ function for a van der Waals gas.

Oscillations, Spinors and the Hopf bifurcation

139

Figure 3.13 Universal Swallowtail defect It is apparent that the van der Waals gas is a deformation of the universal swallowtail implicit hypersurface formed as the envelope-discriminant of the reduced phase function, Θreduced . It is remarkable that the envelope-discriminant of the universal phase function, Θ, and the envelope-discriminant of the reduced phase function, Φ, are the same (in the space of intrinsic coordinates {XM , YG , ZA , TK }). When the variables for {a, g, k} in terms of the original similarity variables {X, Y, Z, T } are made, the discriminant function for DISCΦ is identical to the formula for DISCΘ,

: DISCΘ{X, Y, Z, T } ⇐ DISCΦ{a, g, k} = DISCΦ{a{X, Y, Z, T }, g{X, Y, Z, T }, k{X, Y, Z, T }}.

(3.138) (3.139)

Remarkably, choosing the constraint condition in terms of the hypothetical condition that the Mean similarity invariant (related to Mean Curvature) vanishes, XM ⇒ 0, leads to an equivalent formulation for the universal swallowtail surface homeomorphic (deformable) to the Gibbs surface of a van der Waals gas (subscripts suppressed): Minimal Surface : Universal Swallowtail Envelope XM ⇒ 0 DISCΘ(XM =0) = −27Z 4 + 144T Z 2 Y − 4Y 3 Z 2 + 16Y 4 T (3.140) −128Y 2 T 2 + 256T 3 , (3.141) ≈ DISCΘreduced = DISCΦ ⇒ 0. It must be remembered that this Minimal surface is a hypersurface in the space of Pfaff topological dimension 4. Examples are given in that which follows. In other words,

140

The Ubiquitous Topological van der Waals gas

the Gibbs function for a van der Waals gas is a universal idea associated with minimal hypersurfaces, XM = 0, of thermodynamic systems of Pfaff topological dimension 4. The similarity coordinate TK plays the role of the Gibbs free energy written in terms of the abstract pressure (∼ ZA ) and the abstract temperature (∼ YG ). The Spinodal line, as a limit of phase stability, and the critical point are ideas that come from the classic study of a van der Waals gas, but herein it is apparent that these concepts are universal topological concepts that remain invariant with respect to deformations. Another choice would be to constrain the 4D envelope such that it resides in a domain where the 1-form of Action is of Pfaff topological dimension 3. The physical system is closed, but it is not necessarily in equilibrium. An equilibrium or isolated physical system consists of a single topological component, or phase (the Cartan topology is a connected topology). Domains where the Pfaff topological dimension represent mixed phases imply more than 1 topological component, and are to be associated with regions where the Pfaff topological dimension is ≥ 3. The case of Pfaff dimension 3 would correspond to regions where the 3-form of Topological Torsion is not zero (the Cartan topology becomes a disconnected topology - see Chapter 6). Such non-equilibrium domains correspond to the situation where the determinant of the 4 × 4 Jacobian matrix vanishes. That is, set TK = 0, to obtain the (3D constrained) formula for the envelope, which is a multiple of DISC(TK =0) : DISC(TK =0) = Z 2 {−4X 3 Z + 18XZY + Y 2 X 2 − 4Y 3 − 27Z 2 } ⇒ Z 2 {Cardano } ⇒ 0.

(3.142) (3.143)

It is remarkable that the bracketed formula (in terms of X, Y, Z coordinates) is precisely 27 times the Cardano cubic formula that separates the topological features of the generalized cubic equation. When the Cardano function [79] is greater than zero, the cubic portion of the phase function has one real root and two complex roots. When the Cardano function is less than zero, the cubic portion of the polynomial has three real roots. The Cardano function will be examined again in Chapter 5. It is important to recognize that this development of a universal non-equilibrium van der Waals gas has not utilized the concepts of metric, connection, statistics, relativity, gauge symmetries, or quantum mechanics. The Edge of Regression and Self Intersections The envelope is smooth as long as ∂ 2 Θ/∂Ψ2 = Θξξ 6= 0, and that the exterior 2-form, dΘˆdΘξ 6= 0 subject to the constraint that the family parameter is a constant, dξ = 0. If dΘˆdΘξ 6= 0, but Θξξ = 0, then the envelope has a self intersection singularity. If dΘˆdΘξ = 0, but Θξξ 6= 0, there is no self intersection, and no envelope. If the envelope exists, further singularities are determined by the higher order partial derivatives of the Universal Phase function with respect to ξ: ∂ 2 Θ/∂ξ 2 = Θξξ = 12ξ 2 − 6XM ξ + 2YG , ∂ 3 Θ/∂ξ 3 = Θξξξ = 24ξ − 6XM .

(3.144) (3.145)

Oscillations, Spinors and the Hopf bifurcation

141

When ∂ 3 Θ/∂ξ 3 = Θξξξ 6= 0, and dΘˆdΘξ ˆdΘξξ 6= 0, the envelope terminates in a edge of regression. The edge of regression is determined by the simultaneous solution of Θ = 0, Θξ = 0 and Θξξ = 0. Solving for ξ in Θξξ = 0 yields YG = ξ(3XM − ξ). Reduced Phase Functions Reconsider the reduced phase function, Φ, in terms of coordinate coefficients {g, a, k}, and its partial derivatives with respect to the family parameter, s: Φ Φs Φss DISCΦ

= = = =

s4 + gs2 − as + k = 0, ∂Φ/∂s = 4s3 + 2gs − a, ∂ 2 Φ/∂s2 = 12s2 + 2g, −27a4 + 4(−g2 + 36k)ga2 +16k(4k − g2 )2 .

(3.146) (3.147) (3.148) (3.149)

The reduced formula is more tractable for, if the family parameter is fixed, then the equation represents a implicit surface in the space of coordinates, {g, a, k}. A representation for this implicit surface DISCΦ = 0 was given in the previous figure. It is an obvious deformation equivalent to the Gibbs function for a van der Waals gas. The edge of regression is given by the zero set of Φss = 0 or g = −6s2 . Using this value in Φs = 0 permits a solution for a in terms of s. Using these values for a and g in Φ = 0 gives the three components of a position vector R =[−6s2 , −8s3 , −3s4 ] in {g, a, k} space for the edge of regression. The result for the edge of regression in the g − a plane is plotted below:

Figure 3.14 Gibbs edge of regression defect The same function is plotted as the edge of regression for the Universal Swallowtail in Figure 3.13.

142

The Ubiquitous Topological van der Waals gas

Universal Phase Function Minimal Surfaces For the minimal surface representation of the Gibbs surface for a van der Waals gas, the edge of regression defines the Spinodal line of ultimate phase stability. The edge of regression is evident in the Swallowtail figure (Figure 3.2 and Figure 3.13) describing the Gibbs function for a van der Waals gas. If Θξξ = 0, then for XM = 0 the envelope has a self intersection. It follows from Θξξ = 0, that ξ 2 = −YG /6, which when substituted into (3.150) Θξ = 4ξ 3 + 2YG ξ − ZA ⇒ 0, yields the constrained (XM = 0) Gibbs function, Universal (XM =0) Gibbs function : ZA2 + YG3 (8/27) = 0.

(3.151)

The function defines the Spinodal line, of the minimal surface (Gibbs surface) representation for a universal non-equilibrium van der Waals gas, in terms of "similarity" coordinates. It is important to note that the roots for such a constraint can be complex imaginary. This is a signal that the eigendirection fields (associated with the Jacobian matrix from which the phase function is constructed) are complex spinors. Within the swallowtail region the "Gibbs" surface has 3 real roots; outside the swallowtail region there is a unique real root. The zero set of the Cardano function also furnishes an edge of regression that is, in effect, equivalent to the critical isotherm. Below the critical isotherm, there exist three real density roots. Above the critical isotherm, the root structure has one real root and two complex roots. The details of the universal non-equilibrium van der Waals gas in terms of envelopes and edges of regression with complex molal densities or order parameters will be given more attention in the next chapter. These systems are not equilibrium systems for the Pfaff dimension is not 2. Of obvious importance is the idea that a zero value for both ZG and TK are required to reduce the Pfaff dimension to 2, the necessary condition for an equilibrium system. Ginsburg Landau Currents With a change of notation (ξ ⇒ Ψ), the Universal Phase function can be solved for the determinant of the Jacobian matrix, which is equal to the similarity invariant TK , TK = −{Ψ4 − XM Ψ3 + YG Ψ2 − ZA Ψ}.

(3.152)

The similarity invariant, TK , represents the determinant of the Jacobian matrix. All determinants are in effect N-forms on the domain of independent variables. All N-forms can be related to the exterior differential of some (N-1)-form or current, J. Hence dJ = TK Ω4 = (divJ + ∂ρ/∂t)Ω4 = −(Ψ4 − XM Ψ3 + YG Ψ2 − ZA Ψ)Ω4 .

(3.153)

Oscillations, Spinors and the Hopf bifurcation

143

For currents of the form, J = grad Ψ, ρ = Ψ,

(3.154) (3.155)

the Universal Phase function generates the universal Ginsburg Landau equations ∇2 Ψ + ∂Ψ/∂t = −(Ψ4 − XM Ψ3 + YG Ψ2 − ZA Ψ).

(3.156)

3.5.3 Singularities as defects of Pfaff dimension 3 The family of hypersurfaces can be topologically constrained such that the topological dimension is reduced, and/or constraints can be imposed upon functions of the similarity variables forcing them to vanish. Such regions in the four-dimensional topological domain indicate topological defects or thermodynamic changes of phase. It is remarkable that for a given 1-form of Action there are an infinite number rescaling functions, λ, such that the Jacobian matrix [Jscaled ] = [∂(A/λ)j /∂xk ] is singular jk (has a zero determinant). For if the coefficients of any 1-form of Action are rescaled by a divisor generated by the Holder norm, Holder Norm: λ = {a(A1 )p + b(A2 )p + c(A3 )p + e(A4 )p }m/p , then the rescaled Jacobian matrix, £ scaled ¤ = [∂(A/λ)j /∂xk ], Jjk

(3.157)

(3.158)

will have a zero determinant, for any index p, any set of isotropy or signature constants, a, b, c, e, if the homogeneity index is equal to unity, m = 1. This homogeneous constraint implies that the similarity invariants become projective invariants, not just equi-affine invariants. Such species of topological defects can have the image of a three-dimensional implicit characteristic hypersurface in space-time, Singular hypersurface in 4D: det[∂(A/λ)j /∂xk ] ⇒ 0.

(3.159)

The singular fourth order Cayley-Hamilton polynomial of [Jjk ] then will have a cubic polynomial factor with one zero eigenvalue. For example, consider the simple case where the determinant of the Jacobian vanishes, TK ⇒ 0. Then the Phase function becomes (for Pfaff Dimension 3), Universal Equation of State Θ({XM , YG , ZA , TK = 0}; ξ) = ξ(ξ 3 − XM ξ 2 + YG ξ − ZA ) ⇒ 0.

(3.160) (3.161) (3.162)

The space has been topologically reduced to three dimensions (one eigenvalue is zero), and the zero set of the resulting singular Universal Phase function becomes a

144

The Ubiquitous Topological van der Waals gas

universal cubic equation that is homeomorphic to the cubic equation of state for a van der Waals gas. When the rescaling factor λ is chosen such that p = 2, a = b = c = 1, m = 1, then the Jacobian matrix, [Jjk ] , is equivalent to the "Shape" matrix for an implicit hypersurface in the theory of differential geometry. Recall that the homogeneous similarity invariants can be put into correspondence with the linear Mean curvature, XM ⇒ CM , the quadratic Gauss curvature, YG ⇒ CG , and the cubic Adjoint curvature, ZA ⇒ CA , of the hypersurface. The characteristic cubic polynomial can be put into correspondence with a nonlinear extension of an ideal gas not necessarily in an equilibrium state. 3.5.4 The Adjoint Current and Topological Spin £ ¤ From the singular Jacobian matrix, Jscaled = [∂(A/λ)j /∂xk ], it is always possible to jk construct the Adjoint matrix as the matrix of cofactors transposed, h i £ ¤ Adjoint Matrix : b Jkj = adjoint Jscaled . (3.163) jk When this matrix is multiplied times the rescaled covector components, the result is the production of an adjoint current, ¯ E h i ¯ bk = b Jkj ◦ |Aj /λi . (3.164) Adjoint current : ¯J

It is remarkable that the construction is such that the Adjoint current 3-form, if not zero, has zero divergence globally: bk )Ω4 , Jb = i(J dJb = 0.

(3.165)

b Adjoint current : Jb = dG.

(3.167)

Topological Spin 3-form : AˆG.

(3.168)

(3.166)

From the realization that the Adjoint matrix may admit a non-zero globally conserved b it follows abstractly that there exists a 2-form density 3-form density, or current, J, b of "excitations", G, such that b is not uniquely defined in terms of the adjoint current, for The excitation 2-form, G, b could have closed components (gauge additions G bc , such that dG bc = 0), which do G b not contribute to the current, J. From the topological theory of electromagnetism [222] [216] there exists a fundamental 3-form, AˆG, defined as the "topological Spin" 3-form,

The exterior differential of this 3-form produces a 4-form, with a coefficient energy density function that is composed of two parts, b d(AˆG) = F ˆG − AˆJ.

(3.169)

Oscillations, Spinors and the Hopf bifurcation

145

The first term is twice the difference between the "magnetic" and the "electric" energy density, and is a factor of 2 times the Lagrangian usually chosen for the electromagnetic field in classic field theory, Lagrangian Field energy density : F ˆG = B ◦ H − D ◦ E.

(3.170)

The second term is defined as the "interaction energy density" b − ρφ. Interaction energy density : AˆJb = A◦J

(3.171)

{XM , YG , ZA , TK } ⇒ {4M(linear) , 6G(quadratic) , 4A(cubic) , 0}.

(3.172)

For the special (Gauss) choice of integrating denominator, λ with (p = 2, a = b = c = 1, m = 1) it can be demonstrated that the Jacobian similarity invariants are equal to the classic Mean, Gauss, and Adjoint curvatures:

It can be demonstrated (with the use of Maple) that the interaction density is exactly equal to the Adjoint curvature energy density: [219], Interaction energy AˆJb = 4A(cubic) Ω4

(Adjoint Cubic Curvature).

(3.173)

The conclusion reached is that a non-zero interaction energy density implies the thermodynamic system is not in an equilibrium state (but it could be in a "steady state" far from equilibrium). b However, it is always possible to construct the 3-form, S: b Topological Spin 3-form : Sb = AˆG.

(3.174)

The exterior differential of this 3-form leads to a cohomological structural equation similar to the first law of thermodynamics, but useful for non-equilibrium systems. Cohomology is based upon the idea that the difference of πtwo p-forms can be a perfect differential. This result, now recognized as a statement applicable to nonequilibrium thermodynamic processes, was defined as the "Intrinsic Transport Theorem" in 1969 [184] : Intrinsic Transport Theorem : b − AˆJ, b (Spin) dSb = F ˆG First Law of Thermodynamics : (Energy) dU = Q − W.

(3.175) (3.176)

If one considers a collapsing system, then the geometric curvatures increase with smaller scales. If Gauss quadratic curvature, 6G(gauss_quadratic) , is to be related to gravitational collapse of matter, then at some level of smaller scales a term cubic in curvatures, 4A(adjoint_cubic) , would dominate. It is conjectured that the cubic curvature produced by the interaction energy effect described above could prevent the collapse to a black hole. Cosmologists and relativists apparently have ignored such cubic curvature effects associated with non-equilibrium thermodynamic systems.

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The Ubiquitous Topological van der Waals gas

3.5.5 Non-Equilibrium Examples In order to demonstrate content to the thermodynamic topological theory, two algebraically simple examples are presented below. The algebra can become tedious for the rescaled Action 1-forms. Maple programs can be found in Volume 6, "Maple programs for Non-Equilibrium systems" (see HopfPhase.mws and Holder4d.mws). The first program corresponds to a Jacobian characteristic equation that has a cubic polynomial factor, and hence can be identified with a van der Waals gas. The second example exhibits the features associated with a Hopf bifurcation, where the characteristic equation has a quadratic factor with two pure imaginary roots, and two null roots. Example 1: van der Waals properties from rotation and contraction In this example, the Action 1-form is presumed to be of the form A0 = a(ydx − xdy) + b(tdz + zdt).

(3.177)

The 1-form of Potentials depends on the coefficients a and b. The similarity invariants of the Jacobian matrix, J [(A0 )], formed from A0 , are:

Based on XM YG ZA TK Eigenvalues Torsion Current Parity

: = = = = : = =

the 1-form A0 0, a2 − b2 , 0, −a2 b2 , √ = ±b, ± −1a, 2[0, 0, −z, t]ab, 0.

(3.178) (3.179) (3.180) (3.181) (3.182) (3.183) (3.184)

√ The eigenvalues of the Jacobian matrix are global complex constants: ±b, The real eigenvalues, ±b, refer to the "expansion" and the imaginary eigen± −1a. √ values, ± −1a, refer to the rotation. When the 1-form of Action, A0 , is rescaled by the specialized Holder norm, λGauss , (3.185) A0 ⇒ A = A0 /λGauss , p λGauss = (ax)2 + (ay)2 + (bz)2 + (bt)2 , (3.186) 2 2 2 2 2 (3.187) r = (ax) + (ay) + (bz) + (bt) , ¤ £ then the Jacobian matrix, [J] = ∂A/∂xk , becomes the equivalent of the shape matrix, and the similarity invariants of the shape matrix are related to the average

Oscillations, Spinors and the Hopf bifurcation

147

curvatures of the implicit Phase hypersurface, in a space of one less dimension. In the 3D subspace induced by the Gauss map (ξ 4 = 0) the shape matrix gives: 3D Shape Matrix Linear Mean curvature

: Curvatures : CM = XM /3, = (ξ 1 + ξ 2 + ξ 3 )/3, Quadratic Gauss curvature : CG = YG /3, = (ξ 1 ξ 2 + ξ 2 ξ 3 + ξ 3 ξ 1 )/3, Cubic Adjoint curvature : CA = ZA , = ξ 1ξ 2ξ 3, Quartic Curvature : CK = 0.

(3.188) (3.189) (3.190) (3.191) (3.192) (3.193) (3.194) (3.195)

The computations for the given 1-form of Action yield the results, Based on : Linear Mean curvature : Quadratic Gauss curvature :

the 1-form A0 /λGauss CM = −2b3 tz/(r2 )3/2 CG = −a2 b2 {(x2 + y 2 ) −(z 2 + t2 )}/(r2 )2 Cubic Adjoint curvature : CA = −2a2 b3 tz/(r2 )5/2 Quartic Curvature : CK = 0.

(3.196) (3.197) (3.198) (3.199)

The Determinant (4th order curvature) vanishes by construction of the renormalization in terms of the Gauss map. This null result does not mean the Pfaff dimension of A is less than 4 globally, but the constraint defines a singular set upon which there is a closed Current. This current is the Adjoint current of the previous section. However, the rescaled 1-form A is still of Pfaff dimension 4 and has a non-zero topological torsion 3-form and a non-zero topological torsion 4-form: T op_T orsion = 2ab · [0, 0, −z, t]/(r2 ) = AˆdA, Pfaff Dimension 4 : dAˆdA = −4b3 a(t2 − z 2 )/(r2 )2 Ω4 .

(3.200) (3.201)

The Gauss map permits the construction of the "Adjoint conserved current", which combined with the components of the Action 1-form yield an interaction energy density exactly equal to the cubic curvature CA , Adjoint Current Interaction energy density:

: Js = (−a2 b2 )[x, y, z, t] /(r2 )2 , A ◦ Js − ρφ = CA .

(3.202) (3.203)

The rescaled Jacobian matrix has 1 zero eigenvalue and 3 non-zero eigenvalues. Hence, the cubic polynomial will yield an interpretation as a van der Waals gas. The Adjoint current represents a contraction in space-time, while the flow associated with the 1-form has a rotational component about the z-axis.

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The Ubiquitous Topological van der Waals gas

Example 2: A Hopf 1-form In this example,the Hopf 1-form is presumed to be of the form AHopf = a(ydx − xdy) + b(tdz − zdt).

(3.204)

The 1-form of Potentials depends on the chirality coefficients a and b. There are two cases corresponding to left and right handed "polarizations": a = b or a = −b. The results of the topological theory are : Based on XM YG ZA TK Eigenvalues Torsion Current Parity

: = = = = : = =

the 1-form AHopf 0, a2 + b2 , 0, a2 b2 , √ √ ± −1b, ± −1a, −2[x, y, z, t]ab, 8ab.

(3.205) (3.206) (3.207) (3.208) (3.209) (3.210) (3.211) (3.212)

The 4 eigenvalues come in two imaginary pairs. The elements of each pair are equal and opposite in sign. What is remarkable for this Action 1-form is that both the linear similarity invariant XM and the cubic similarity invariant ZA of the implicit phase hypersurface in 4D vanish, for any real values of a or b. The quadratic similarity invariant is non-zero, positive real and is equal to a2 = b2 . The quartic similarity invariant TK is non-zero, positive real and is equal to a2 b2 . The 1-form also supports a Topological Torsion current, with a non-zero divergence. The rescaled Jacobian matrix has 1 zero eigenvalue and 3 non-zero eigenvalues. Hence, the cubic polynomial will yield an interpretation as a van der Waals gas. The Adjoint current represents a contraction in space-time, while the flow associated with the 1-form has a rotational component about the z-axis. However, if the 1-form AHopf is scaled by the Gauss map, the resulting Hopf implicit surface is a single 4D imaginary minimal two-dimensional hypersurface in 4D and has two non-zero imaginary curvatures, but a positive Gauss curvature! This is a most unusual result, for the usual 2D minimal surface has equal and opposite real curvatures, with a negative Gauss curvature. Based on Linear Mean curvature Quadratic Gauss curvature Cubic Adjoint curvature Quartic Curvature Eigenvalues

: the 1-form AHopf /λGauss : CM = 0, : CG = +a2 b2 {r2 }/(λ2 )2 , : CA = 0, : CK = 0, √ √ : [0, 0, + −1, − −1](abr/λ2 ).

(3.213) (3.214) (3.215) (3.216) (3.217) (3.218)

Oscillations, Spinors and the Hopf bifurcation

149

Strangely enough the charge-current density induced by the Adjoint current is not zero, but it is proportional to the Topological Torsion vector that generates the 3-form AˆF. The topological Parity 4-form, dAˆdA, is not zero, and depends on the sign of the coefficients a and b. In other words the "handedness" of the different 1-forms determines the orientation of the normal field with respect to the implicit surface. It is known that a process described by a vector proportional to the topological torsion vector in a domain where the topological parity is non-zero, 4ba/(x2 + y 2 + z 2 + t2 ) 6= 0, is thermodynamically irreversible. However, the rescaled 1-form A is still of Pfaff dimension 4 and has a non-zero topological torsion 3-form and a non-zero topological torsion 4-form: T op_T orsion = −2ab · [x, y, z, t]/(r2 ) = AˆdA, Pfaff Dimension 4 : dAˆdA = (4ba/r2 ) Ω4 .

(3.219) (3.220)

The Gauss map permits the construction of the "Adjoint conserved current", which combined with the components of the Action 1-form yield an interaction energy density exactly equal to the cubic curvature CA , Adjoint Current Interaction energy density:

: Js = (−a2 b2 )[x, y, z, t] /(r2 )2 , A ◦ Js − ρφ = CA ⇒ 0.

(3.221) (3.222)

The interplay between the Pfaff topological dimension and the Projective dimension is an area of active research. Hopefully a chapter on this topic will appear in Volume 5.

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The Ubiquitous Topological van der Waals gas

Chapter 4 ELECTRODYNAMIC, HYDRODYNAMIC, AND MECHANICAL THERMODYNAMIC SYSTEMS In this chapter, the topological foundations of thermodynamics developed in the preceding chapter will be applied to several different physical systems. The objective is to demonstrate how the common thread of topology resides in the different disciplines. The implication is that familiar features of one discipline can be utilized in the study of other disciplines. The universal features of thermodynamics based upon the theory of continuous topological evolution are independent of the symbols used to describe the various disciplines. To repeat the fundamental ideas: Axiom 1. Thermodynamic physical systems can be encoded in terms of a 1-form of covariant Action∗ Potentials, Ak (x, y, z, t...), on a ≥ fourdimensional abstract variety of ordered independent variables, {x, y, z, t...}. The variety supports a differential volume element Ω4 = dxˆdyˆdzˆdt... Axiom 2. Thermodynamic processes are assumed to be encoded, to within a factor, ρ(x, y, z, t...), in terms of contravariant Vector and/or complex isotropic Spinor direction fields, V4 (x, y, z, t...). Axiom 3. Continuous topological evolution of the thermodynamic system can be encoded in terms of Cartan’s magic formula (see p. 122 in [148]). The Lie differential, when applied to an exterior differential 1form of Action, A = Ak dxk , is equivalent abstractly to the first law of thermodynamics.

Cartan’s Magic Formula L(ρV4 ) A First Law Inexact Heat 1-form Q Inexact Work 1-form W Internal Energy U ∗

= : = = =

Recall that A is a 1-form of Action per unit source.

i(ρV4 )dA + d(i(ρV4 )A), W + dU = Q, W + dU = L(ρV4 ) A, i(ρV4 )dA, i(ρV4 )A.

(4.1) (4.2) (4.3) (4.4) (4.5)

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Axiom 4. Equivalence classes of systems and continuous processes can be defined in terms of the Pfaff topological dimension of the 1-forms of Action, A, Work, W , and Heat, Q. 4.1

Physical (Contact) Systems of Pfaff Topological Dimension 3

Consider those abstract physical systems that are represented by 1-forms, A, of Pfaff topological dimension 3. The concept implies that the topological features can be described in terms of 3 functions (of perhaps many geometrical coordinates and parameters) and their differentials. For example, if one presumes the fundamental independent base variables are the set {P, q, τ }, with an exterior differential oriented volume element consisting of a product† of exact 1-forms Ω3 = +dP ˆdqˆdτ , then a Darboux representation for a physical system could have the appearance, (4.6)

A = P dq + dτ .

The objective is to use the features of Cartan’s magic formula to compute the possible evolutionary features of such a system. The evolutionary dynamics is essentially the first law of thermodynamics: (4.7)

LρV A = i(ρV)dA + di(ρV)A) = W + dU = Q. The elements of the Pfaff sequence for this Action become, A dA AˆdA dAˆdA

= = = =

P dq + dτ , dP ˆdq, dP ˆdqˆdτ , 0.

(4.8) (4.9) (4.10) (4.11)

4.1.1 The Vector Processes Relative to the position vector R = [P, q, τ ] of ordered topological coordinates {P, q, τ }, consider the 3 abstract, linearly independent, orthogonal (supposedly) vector direction fields: ¯ + ¯ 0 ¯ (4.12) E = ¯¯ 0 , ¯ 1 ¯ + ¯ 0 ¯ (4.13) V = ¯¯ 1 , ¯ 0 ¯ + ¯ 1 ¯ V⊥ = ¯¯ 0 . (4.14) ¯ 0 †

More abstract systems could be constructed from differential forms which are not exact.

Physical (Contact) Systems of Pfaff Topological Dimension 3

153

These direction fields can be used to define a class of Vector processes, but they do not exhaust the class of eigendirection fields for the 2-form, dA. The Spinor eigendirection fields are missing from this basis frame. The important fact is that thermodynamic processes defined in terms of a real basis frame (and its connection) is incomplete, as such processes ignore spinor direction fields. The extremal vector E is the unique eigenvector with eigenvalue zero relative to the antisymmetric matrix generated by the 2-form, dA. The associated vector V⊥ is orthogonal to the q, τ plane. Recall that associated vectors are adiabatic. Next deform the vector direction fields by an arbitrary function, ρ. Then construct the contractions (the internal energy), UE = i(ρE)A = ρ, UV = i(ρV)A = ρP, UV⊥ = i(ρV⊥ )A = 0.

(4.15) (4.16) (4.17)

The linearly independent Work 1-forms for evolution in the direction of the 3 basis vectors are written as, WE = i(ρE)dA = 0, WV = i(ρV)dA = −ρdP, WV⊥ = i(ρV⊥ )dA = +ρdq.

(4.18) (4.19) (4.20)

From Cartan’s Magic Formula, L(V) A = i(ρV)dA + d(i(ρV)A) ≡ Q, it becomes apparent that, QE = dρ QV = P dρ, QV⊥ = +ρdq,

dQE = 0, dQV = dP ˆdρ, dQV⊥ = dρˆdq.

(4.21) (4.22) (4.23)

All processes in the extremal direction satisfy the conditions that QE ˆdQE = 0. Hence, all extremal processes are reversible. It is also true that evolutionary processes in the direction of the other basis vectors, separately, are reversible, as the 3-form QˆdQ vanishes for V, V⊥ , or E. Hence all such piecewise continuous, transitive, processes are thermodynamically reversible. 4.1.2

The Spinor processes

The eigendirection fields of the antisymmetric matrix representation of dA, ⎡

⎤ 0 1 0 [dA] = ⎣ −1 0 0 ⎦ , 0 0 0

(4.24)

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Electrodynamic, Hydrodynamic, and Mechanical Thermodynamic systems

are given by the equations:

¯ + ¯ 0 ¯ Eigenvalue = 0 EigenVector1 ¯¯ 0 ¯ 1 ¯ ¯ 1 + ¯ √ EigenSpinor1 ¯¯ -1 Eigenvalue = + ¯ 0 ¯ ¯ 1 + ¯ √ Eigenvalue = EigenSpinor2 ¯¯ − -1 ¯ 0

,

(4.25)

√ -1,

(4.26)

√ -1.

(4.27)

Note that the (supposedly) Vector processes of the preceding subsection are combinations of the Spinor processes, √ (4.28) V = − -1(Sp1 − Sp2)/2, (4.29) V⊥ = (Sp1 + Sp2)/2, and neither Vector process is irreversible in the sense that QˆdQ = 0. Note further that the "rotation" induced by the antisymmetric matrix [dA] acting on V yields V⊥ and the 4th power of the matrix yields the identity rotation, [dA] ◦ |Vi = |V⊥ i , [dA]2 ◦ |Vi = − |Vi , [dA]4 ◦ |Vi = + |Vi .

(4.30) (4.31) (4.32)

Consider the processes defined by ρ times the eigendirection fields. Compute the change in internal energy, dU, the Work, W and the Heat, Q, for each eigendirection field: UρV1 = i(ρV1 )A = ρ d(UρV1 ) = dρ, (4.33) √ √ UρSp1 = i(ρSp1 )A = −1ρP d(UρV1 ) = −1(ρdP + P dρ), (4.34) √ √ UρSp2 = i(ρSp2 )A = − −1ρP d(UρV1 ) = − −1(ρdP + P dρ). (4.35) WρV1 = i(ρV1 )dA = 0,

(4.36)

WρSp1

(4.37)

WρSp2

√ = i(ρSp1 )dA = ρ(dq − -1dP ), √ = i(ρSp2 )dA = +ρ(dq + -1dP ).

(4.38)

QρV1 = Li(ρV1 ) A = dρ,

(4.39)

QρSp1

(4.40)

QρSp2

√ √ = Li(ρSp1 ) A = ρ(dq − -1dP ) + -1d(ρP, √ √ = Li(ρSp2 ) A = ρ(dq + -1dP ) − -1d(ρP ).

(4.41)

Physical (Contact) Systems of Pfaff Topological Dimension 3

155

4.1.3 Irreversible Spinor processes Next compute the 3-forms of QˆdQ for each direction field, including the spinors: QρV1 ˆdQρV1 = 0, QρSp1 ˆdQρSp1 QρSp2 ˆdQρSp2

√ = − -1ρdρˆdP ˆdq, √ = + -1ρdρˆdP ˆdq.

(4.42) (4.43) (4.44)

It is apparent that evolution in the direction of the Spinor fields can be irreversible in a thermodynamic sense, if dqˆdP ˆdρ is not zero. However, evolution in the direction of smooth combinations of the base vectors may not satisfy the reversibility conditions, QˆdQ = 0, when the combination involves a fixed point. For example, it is possible to consider expansions or rotations in the P, q plane: √ Vexpansion = V⊥ + -1VV , (4.45) √ QˆdQ = − -1ρdρˆdP ˆdq. (4.46) √ Vrotation = V⊥ − -1VV , √ QˆdQ = + -1ρdρˆdP ˆdq.

(4.47) (4.48)

The non-zero value of QˆdQ for the continuous expansions and rotations are related to the non-zero Godbillon-Vey class [171]. A key feature of both the rotation and expansion processes is that they have a fixed point; they are not transitive. If the physical system admits an equation of state of the form, θ = θ(P, q, ρ) = 0, then the rotation and expansion processes are not irreversible. Remark 34 The facts that piecewise (sequential) C1 transitive evolution along a set of direction fields in odd (3) dimensions can be thermodynamically reversible, QˆdQ = 0, while C2 evolution as composed of linear intransitive combinations of these same direction fields is thermodynamically irreversible, QˆdQ = 0, is a remarkable result which appears to have a relationship to Nash’s theorem on C1 embedding. Physically, the results are related to tangential discontinuities such as hydrodynamic wakes. It should be remarked that for Action 1-forms of odd Pfaff topological dimension, addition of a closed form whose format contains new independent variables does not change the Pfaff topological dimension of the composite. On the other hand, if the original 1-from is renormalized by some non-zero factor, then the Pfaff topological dimension does change. In the next section it will be demonstrated that this piecewise equivalence class of processes in 4D does not produce a thermodynamically irreversible process. In this sense it may be said that thermodynamic irreversibility is an artifact of dimension 4. It is remarkable that a rotation and an expansion can be combined (eliminating the fixed point) to produce a thermodynamically reversible process.

156

4.2

Electrodynamic, Hydrodynamic, and Mechanical Thermodynamic systems

Physical (Symplectic) Systems of Pfaff Topological Dimension 4

Consider those abstract physical systems that are represented by 1-forms, A, of Pfaff dimension 4. For example, if one presumes the fundamental independent base variables are the space-time set {x, y, z, t}, then a representation for a physical system would consist of four functions whose arguments are the base variables, and the Action 1-form would have the appearance: A=Σ3k=1 Ak (x, y, z, t)dxk − φ(x, y, z, t)dt.

(4.49)

This representation has applicability to the study of electromagnetic systems, where the functions Ak (x, y, z, t) and φ(x, y, z, t) play the role of the vector and scalar potentials of classic electromagnetic theory [222]. However, the Darboux theorem says that there exists a map from the set {x, y, z, t} into four independent functions, {P, Mc, q, s} such that the elements of the Pfaff sequence become, A dA AˆdA dAˆdA

= = = =

P dq − (Mc)d(s), dP ˆdq − d(Mc)ˆd(s), (P d(Mc) − (Mc)dP )dqˆd(s), 2dP ˆd(Mc)ˆdqˆd(s),

with an oriented volume element Ω4 = +dP ˆd(Mc)ˆdqˆd(s). The antisymmetric matrix corresponding to dA is‡ : ⎡ ⎤ 0 0 1 0 ⎢ 0 0 0 −1 ⎥ ⎥ [dA] = ⎢ ⎣ −1 0 0 0 ⎦ . 0 +1 0 0

(4.50) (4.51) (4.52) (4.53)

(4.54)

Relative to the ordered "position" vector R = [P, (Mc), q, s], consider the 4 linearly independent orthogonal vector direction fields: V V⊥ T T⊥

= = = =

[0, 0, P, −(Mc)]T , [0, 0, (Mc), P ]T , [P, (Mc), 0, 0]T , [−(Mc), P, 0, 0]T .

(4.55) (4.56) (4.57) (4.58)

These four "vectors" must be composites of the Spinor eigendirection fields of the antisymmetric matrix, [dA]. The vectors V and V⊥ reside in the subspace of {0, 0, x, s} and therefore will be called "tangent" vectors. The vectors T and T⊥ reside in the subspace {P, (Mc), 0, 0} and will be called "normal" or "vertical" vectors. These vectors can be used a basis frame for any vector in the four-dimensional space. ‡

Compare to the Dirac γ 3 matrix.

Physical (Symplectic) Systems of Pfaff Topological Dimension 4

157

However, [dA] ◦ |Vi [dA]2 ◦ |Vi [dA] ◦ |V⊥ i [dA]2 ◦ |V⊥ i [dA] ◦ |Ti [dA] ◦ |T⊥ i

= = = = = =

|Ti , − |Vi , |T⊥ i , − |V⊥ i , − |Vi , − |V⊥ i .

(4.59) (4.60) (4.61) (4.62) (4.63) (4.64)

The [dA] rotation in four-dimensional space takes horizontal direction fields into vertical direction fields. Each of the direction fields acts like a composite of two spinors. Note that the operator [dA]4 = [I], which is typical of Spinor rotations being of period 4, where vector rotations are of period 2.

Conjecture 35 Somehow, this period 4 phenomena has to be related to the concept of non-orientability.

Construct the contractions, i(V)A i(V⊥ )A i(T)A i(T⊥ )A

= = = =

(P 2 + (Mc)2 ), 0, 0, 0,

(4.65) (4.66) (4.67) (4.68)

and note that there are three associated (hence locally adiabatic) vectors, V⊥ , T and T⊥ . Next construct the linearly independent Work 1-forms for evolution in the direction of these 4 basis vectors, WV WV⊥ WT WT⊥

= = = =

i(V)dA = −d{P 2 + (Mc)2 }/2, i(V⊥ )dA = P d(Mc) − (Mc)dP, i(T)dA = P dq − (Mc)d(s) = A, i(T⊥ )dA = −(Mc)dq − P d(s).

(4.69) (4.70) (4.71) (4.72)

In each case, the Work 1-form is not zero, and if the Pfaff dimension is 4, then the process must be composed from Spinor eigendirection fields. From Cartan’s Magic Formula, L(V) A = i(V)dA + d(i(V)A) ≡ Q, it becomes

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Electrodynamic, Hydrodynamic, and Mechanical Thermodynamic systems

apparent that: QV dQV QV⊥ dQV⊥ QT dQT QT⊥ dQT⊥

= = = = = = = =

+ d(P 2 + (Mc)2 )/2, 0, P d(Mc) − (Mc)dP, 2dP ˆd(Mc), P dq − (Mc)d(s) = A, dP ˆdq − d(Mc)ˆd(s) = dA, −(Mc)dq − P d(s), −d(Mc)ˆdq − dP ˆd(s).

(4.73) (4.74) (4.75) (4.76) (4.77) (4.78) (4.79) (4.80)

It follows that the 3-forms QˆdQ become: QV ˆdQV QV⊥ ˆdQV⊥ QT ˆdQT QT⊥ ˆdQT⊥

= = = =

0, 0, ((Mc)dP − P d(Mc))ˆdqˆd(s) = AˆdA, −((Mc)dP − P d(Mc))ˆdqˆd(s) = −AˆdA.

(4.81) (4.82) (4.83) (4.84)

The vector T is known as the Topological Torsion vector, and motion in the direction of the Torsion vector (or its orthogonal compliment T⊥ ) yield a non-zero value for the 3-form Q. It follows that evolution in the direction of the Torsion vector is thermodynamically irreversible. For any vector with components constructed from V and V⊥ with arbitrary functional coefficients, it follows that the Heat 1-form Q satisfies the Frobenius integrability theorem, and such processes are reversible. Only evolutionary processes with components constructed from T⊥ and T will represent thermodynamically irreversible processes. Note that a process which is the sum of T⊥ and T is reversible. The difference between the 3D case and the 4D case is that piecewise evolution in the direction of the basis vectors is always reversible in the 3D case, but piecewise evolution in the direction of the 4D basis vectors is not always thermodynamically irreversible. So the work of the last two subsections establishes the following theorem: Theorem 4 Thermodynamic irreversibility is an artifact of topological Pfaff dimension > 4. It is of some interest to recognize that the 1-form, (Mc)dP − P d(Mc), has a rotational polar coordinate (elliptic) representation (Mc)dP − P d(Mc) = (P 2 + (Mc)2 )δ(Θ), under the mapping

p ((Mc)2 + P 2 ) cos Θ, p P = ((Mc)2 + P 2 ) sin Θ.

(Mc) =

(4.85)

(4.86) (4.87)

Electromagnetism as a topological theory

159

The 1-form δ(Θ) = ((Mc)dP − P d(Mc))/(P 2 + (Mc)2 ) ,

d(δ(Θ) = 0,

(4.88)

is not an exact 1-form, but it is closed mod the origin of (Mc) and P. Such closed but not exact forms are called harmonic forms and have values, when integrated over closed cycles, which form rational ratios. It is also true that the differential 1-form ((Mc)dP − P d(Mc))/(P 2 − (Mc)2 ) is closed, but now the "rotation" is hyperbolic. It should be remarked that for Action 1-forms of even Pfaff topological dimension 2n+2 it is possible to reduce the Pfaff dimension by finding a functional factor of the 1-form, A, that becomes an integrating factor for the (2n+1)-form, AˆdAˆ..dA. The integrating factor (of which there are an infinite number) causes the Topological Torsion (2n+1)-form to have zero divergence. 4.3

Electromagnetism as a topological theory

4.3.1 Introduction In this chapter, the topological theory of Electromagnetism will be summarized in the language of exterior differential forms. Much of the symbolism will be recognized as being similar to the presentation made in Chapter 1, which summarized the thermodynamic features of continuous topological evolution. Many different physical systems enjoy this universal correspondence to topological thermodynamics. This correspondence will be utilized to determine and study those electrodynamic systems which are not in thermodynamic equilibrium. The starting point of non-equilibrium topological thermodynamics is the assumption that a thermodynamic system can be encoded in terms of a differentiable 1-form of Action per unit source (In electromagnetism, the unit source is charge). In Topological Electromagnetism, not only is a 1-form of Action potentials, A, utilized to produce the 2-form, F, of field intensities (B and E), but also a 2-form density, G, is required to define the charge current sources in terms of the field excitations (D and H). The Electromagnetic system is a refinement of the thermodynamic system such as to include the concept of charge and currents. These ideas lead to the formulation of the two postulates (two exterior differential systems) introduced in section 2.2, below. The exact 2-form of thermodynamic field "intensities" F (E, B) = dA is defined in terms of inexact 1-form of potentials, A, which has physical units of ~/e. The exact 3-form of charge current density, J = dG, is defined in terms of the inexact 2-form density of thermodynamic field quantities, or "excitations", G(D, H), with physical units of ~. The 2-form, F, historically is associated with forces, and the 2-form density, G, historically is associated with sources. The first postulate, the existence of a differentiable 1-form of Action, A, leads to a the thermodynamic concept of a topological torsion 3-form, AˆF , whose nonzero value is an indicator of a non-equilibrium system. The second postulate, the

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existence of a differentiable 2-form density, leads to another 3-form of topological spin, AˆG., whose non-zero value will also be an indicator of non-equilibrium. Each of these 3-forms have exterior differentials (divergences) producing 4-forms whose closed integrals have been defined as the Poincare-Bateman 4-forms. If the divergences are not zero, the non-zero values act as sources of topological defects. 4.3.2 The classical Maxwell-Faraday and the Maxwell-Ampere equations Using the notation and the language of Sommerfeld and Stratton [246] [250], the classic definition of an electromagnetic system is a domain of space-time {x, y, z, t} which supports both the Maxwell-Faraday equations, curl E + ∂B/∂t = 0,

div B = 0,

(4.89)

div D = ρ.

(4.90)

and the Maxwell-Ampere equations, curl H − ∂D/∂t = J,

This formulation of two sets of Partial Differential Equations makes no statement about metric, or connections with geometric constraints of "gauge". The conservation of charge current In every case, the charge current density for the Maxwell system satisfies the conservation law, div J + ∂ρ/∂t = 0.

(4.91)

The charge-current densities are subsumed to be zero [J, ρ] = 0 for the vacuum state. For the Lorentz vacuum state, the field excitations, D and H, are assumed to be linearly connected to the field intensities, E and B, by means of the Lorentz (homogeneous and isotropic) constitutive relations: D = εE , B = µH.

(4.92)

The two vacuum constraints imply that the solutions to the homogeneous Maxwell equations also satisfy the vector wave equation, typically of the form grad div B − curl curl B − εµ∂ 2 B/∂t2 = 0.

(4.93)

The constant wave phase velocity, vp , is taken to be vp2 = 1/εµ ≡ c2 .

(4.94)

Similar results can be obtained for the solid state where the constitutive constraints can be more complex [206], and for the plasma state where the charge-current densities are not zero. It should be emphasized that the Lorentz constitutive equations

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form a severe topological (and not necessary) constraint on the general Maxwell electromagnetic system. The existence of potentials It is further subsumed that the classic Maxwell electromagnetic system is constrained by the statement that the field intensities are deducible from a system of twice differentiable potentials, [A, φ]: B = curl A,

E = −grad φ − ∂A/∂t.

(4.95)

This constraint topologically implies that domains that support non-zero values for the covariant field intensities, E and B, can not be compact domains without a boundary. It is this constraint that distinguishes classical electromagnetism from Yang Mills theories, which have propensity for topological manifolds that are compact without boundary. Two other classical 3-vector fields are of historical interest, the Poynting vector, E × H, representing the flux of electromagnetic radiative energy, and the field momentum flux, D × B (in the sense that D × B = c2 E × H ). 4.3.3 The Fundamental Exterior Differential Systems The formulation of Maxwell theory given above is relative to a choice of ordered independent variables {x, y, z, t}, and utilized the notation of classical vector analysis as developed in Euclidean 3-space. The topological features of the formalism are not immediately evident. However, electromagnetism has a formulation in terms of Cartan’s exterior differential forms (see Volume 1 [273] for more details and references about differential forms). Exterior differential forms do not depend upon a choice of coordinates, do not depend upon the a choice of metric, and are independent of the symmetry constraints imposed by gauge groups and connections. In such a formulation the equations of an electromagnetic system become recognized as consequences of topological constraints on a domain of independent variables. The use of differential forms should not be viewed as just another formalism of fancy. The technique goes beyond the methods of tensor calculus, and admits the study of topological evolution, while tensor based theories do not. Recall that if an exterior differential system is valid on a final variety of (ordered) independent variables, {x, y, z, t}, then it is also functionally well defined on any initial variety of independent variables, {ξ 1 , ξ 2 , ξ 3 , ξ 4 ...}, that can be mapped onto {x, y, z, t}. The map need only be differentiable, such that the Jacobian matrix elements of the mapping are well defined functions. The Jacobian matrix does not have to have an inverse, so that the exterior differential system is not restricted to the equivalence class of diffeomorphisms. (Tensor analysis and coordinate transformations require that the Jacobian map from initial to final state has an inverse - the method of exterior differential forms does not.) However, the field intensities on the initial variety are functionally well defined by the pullback mechanism, which involves algebraic composition with components of the Jacobian matrix transpose and adjoint, and the

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process of functional substitution. Covariant antisymmetric tensor fields pullback retrodictively with respect to the transpose of the Jacobian matrix (of functions) and functional substitution, and contravariant tensor densities pullback retrodictively with respect to the adjoint of the Jacobian matrix, and functional substitution [192]. The transpose and the adjoint of the Jacobian exist, even if the Jacobian inverse does not. This independence from a choice of independent variables (or coordinates) for Maxwell’s equations was reported early on by Van Dantzig [271]. It is a surprise to many, that the Maxwell differential system of PDE’s is well defined in a covariant manner for both Galilean transformations as well as Lorentz transformations, or any other diffeomorphism. The singular solution sets to the Maxwell differential system of PDE’s do not enjoy this universal property. Maxwell’s PDE’s are topological statements that can be deduced from an exterior differential system. The two postulates are: The Postulate of Potentials F − dA = 0,

(4.96)

which defines the Maxwell Faraday exterior differential system, F − dA = 0, (leading to the concept of conserved flux), and The Postulate of Field Excitations: J − dG = 0,

(4.97)

which defines the Maxwell Ampere exterior differential system, J − dG = 0, (leading to the concept of conserved charge). The first postulate involves a 2-form of thermodynamic intensities whose coefficients transform as a covariant tensor with respect to diffeomorphisms. The second postulate involves a 2-form density of thermodynamic quantities [245] with coefficients that transform as a contravariant tensor density with respect to diffeomorphisms. Each postulate is to be recognized as an exterior differential system [34] constraining the topology of the functions defined in terms of the independent variables. For example, from Stokes theorem, the (two dimensional) domain of finite support for F can not, in general, be compact without boundary, unless the Euler characteristic vanishes. There are two exceptional cases for two-dimensional domains, the torus and the Klein-Bottle, but these situations require the additional topological constraint that F ˆF ⇒ 0. The fields in these exceptional cases must reside on these exceptional compact surfaces, which form topological coherent structures in the electromagnetic field. No constraints of geometrical connection or metric are required explicitly. Such geometric constraints can be used to refine the Maxwell topology for different specific physical systems. For example, constitutive equations of constraint between the two 2-forms F and G can be used to distinguish birefringent media from optically active media. The Maxwell-Faraday PDE’s are not restricted to spaces of topological dimension N = 4. For an exterior differential system, F − dA = 0, on a space

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of any dimension N > 3, the closure conditions, ddA = dF = 0, always yield the same identical Maxwell-Faraday PDE’s for the first 4 variables. Additional PDE’s are also generated for N > 4, but the system of PDE’s form a nested set, with the Maxwell-Faraday equations as topological kernel, of invariant format for any dimension N. A remarkable result is that Faraday induction is a topological idea, and does not depend upon metric or connection. The concept of Faraday induction applies to any system that satisfies the Postulate of Potentials, including the fundamental axiom of topological thermodynamics which encodes a physical system in terms of a 1-form of Action. As demonstrated below, the Postulate of Potentials establishes the field of thermodynamic intensities, E and B, (think forces), and the Postulate of Field Excitations establishes the field of thermodynamic quantities, D and H, (think sources). The topological perspective subsumes that the two species are independent ideas. The experimental justification of such ideas can be demonstrated with a simple parallel plate capacitor experiment. First connect the plates to a battery of constant potential and let it remain connected. Insert a slab of plastic dielectric halfway between the plates. Release the plastic slab. Does the slab remain motionless, or is the motion such that the slab is expelled or attracted? For a second experiment, attach the plates of the capacitor to a battery and then disconnect the battery after charging the capacitor. Now insert the plastic slab halfway, and release it. Does the slab remain motionless, or is the motion such that the slab is expelled or attracted? In the first case, the E field remains constant (the potential does not change), and motion of the dielectric slab causes the D field to change (the battery adjusts the charge distribution). In the second experiment, the charge distribution is constant, so that the D field remains constant, but the E field changes. Consider the simple constitutive constraint, D = εE. In the first experiment, insertion would cause the average ε to increase, hence even though E remains constant, the D field would increase. However, the total energy density D ◦ E would decrease if the slab was expelled, and that is what happens. In the second experiment, motion of the slab would cause the E field to change. As the D field remains constant, the minimum energy density occurs when the slab is fully inserted. Current electromagnetic dogma presents the idea that from a given charge current density distribution, [J, ρ], it is possible to deduce the E and B fields. However, the postulate of conserved Charge-Current densities indicates that it is D and H that are the related quantities, not E and B. The Postulate of Potentials indicates that the field intensities E and B are deduced from the potentials [A, φ]. It takes some constitutive constraint to convert D and H into E and B, or [J, ρ] into [A, φ]. However, it has been discovered that there exists an adjoint mapping that will produce a conserved charge current algebraically (see Chapter 3.5). Both types of constraints appear in the literature in great detail and variety. Such assumptions obscure the topological basis and differences between exterior differential forms and exterior differential form densities.

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There are two types of differential forms considered in this monograph. The first type transforms as a scalar with respect to diffeomorphisms. The second type transforms as a scalar density, which is proportional to the determinant (or the magnitude of the determinant) of the diffeomorphism. The coefficients of the first type pullback with respect to the transpose of a differential Jacobian mapping, whether it is a diffeomorphism or not. The coefficients of the second type, the differential form densities, pullback with respect to the adjoint of a differential Jacobian mapping, whether it is a diffeomorphism or not. The Postulate of Potentials indicates that the domain of support for the 2form F is not compact without boundary. The postulate also demonstrates that magnetic monopoles are not compatible with the assumption of C2 differentiability. Such a statement does not apply to the density (N-2)-form, G, which can have closed and non-closed components. The closed but not exact components of G lead to the quantization of charge as a topological result. If G is a density with a pullback related to the magnitude of the determinant, it follows that quantized charge is a pseudoscalar [175]. The historical assumptions of charge as a scalar are not compatible with the topological format. Experiments with piezo magnetic crystals indicate that volume deformations can cause electrical phenomena. If charge was not a pseudo scalar, the concept of magnetic permeability would have to vanish in crystals with centers of symmetry, a result not compatible with experiment [174]. 4.3.4 The Maxwell Ampere system: J − dG = 0 The D H field excitations: Differential (N-2)-form densities For example consider the exterior differential of the (N-1)-form density, D, in three dimensions, given by the expression, dD = d(Dx dyˆdz − Dy dzˆdx + Dz dxˆdy) = div3 (D)dxˆdyˆdz ⇒ ρ(x, y, z)dxˆdyˆdz,

(4.98)

where ρ has been defined as the resultant of the action of the exterior differential, div3 (D). The usual interpretation of Gauss’ law is that the field lines of the vector (density) D terminate (or have a limit or accumulation point) on the charges, Q. Gauss’ law generates both the intuitive idea that sources are related to limit points, and demonstrates the novel concept that the exterior differential is a limit point operator. The exterior differential creates limits points when the operation is applied to a differential form. However, as demonstrated above, the concept that the exterior differential is a limit point operator relative to the Cartan topology is a general idea, and is not restricted to Gauss’ law. Extending this idea to four dimensions for the (N-2)-form density, G, of Maxwell excitations (D, H), G = −Dx dyˆdz + Dy dzˆdx − Dz dxˆdy + H x dxˆdt + H y dyˆdt + H z dzˆdt, (4.99)

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the exterior differential dG of G yields a 3-form, J, defined as the electromagnetic current 3-form, J = J x dyˆdzˆdt − J y dxˆdzˆdt + J z dxˆdyˆdt − ρdxˆdyˆdz,

(4.100)

where in 3-vector language, curl H − ∂D/∂t = J

div D = ρ.

(4.101)

The charge current density act as the "limit points" of the Maxwell field excitations. Note that dJ ⇒ 0 for C2 functions by Poincare’s lemma (see section 4.1). However, consider the N-1 current, C (not necessarily equal to J as defined above), in four dimensions, C = ρ{V x dyˆdzˆdt − V y dxˆdzˆdt + V z dxˆdyˆdt − 1dxˆdyˆdz},

(4.102)

and its exterior differential as given by the expression, dC = {div3 (ρV) + ∂ρ/∂t}dxˆdyˆdzˆdt = Rdxˆdyˆdzˆdt = R Ω4vol .

(4.103) (4.104)

When the 4-form with coefficient R vanishes, the resultant expression is physically interpreted as the "equation of continuity" or as a "conservation law". Over a closed boundary, that which goes in is equal to that which goes out (when dC = 0). Note that the concept of the conservation law is a topological constraint; the "limit points" of the "current 3-form" in four dimensions must vanish if the conservation law is to be true. If the RHS of the above expression (4.103) is not zero, then the current 3-form is said to have an "anomaly", or a source (or sink). The anomaly acts as the source of the otherwise conserved quantity. The limit points, R, of the 3-form, C, are generated by its exterior differential, dC = {div3 (ρV) + ∂ρ/∂t}Ω4 . When the RHS is zero, the current "lines" do not stop or start within the domain. (It is possible for the current lines to be closed on themselves in certain topologies.) The conservation of charge The Maxwell Ampere exterior differential system is a topological constraint for by Stokes theorem the support for G can be compact without boundary only if the domain is without charge-currents. The closure of the exterior differential system, dJ = 0, generates the charge-current conservation law. The integral of J over a closed three-dimensional domain is a relative integral invariant (a deformation invariant) of any process that can be described in terms of a vector direction field, V . The formal statement is given by Cartan’s magic formula [148], which describes continuous topological evolution in terms of the action of the Lie differential, with respect to a vector field, acting on the exterior differential 3-form, J. The Maxwell-Ampere

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exterior differential system leads to the evolutionary Conservation of Charge theorem. Charge is defined as the integral of the 3-form, J, over a three-dimensional integration chain M3. Using Stoke’s theorem, RRR RR Conservation of Charge : J= G = charge, dJ = 0, (4.105) M3 ∂M3 RRR RRR RR LβV4 J = d{i(βV4 )J} = G ⇒ 0. (4.106) M3

M3

∂M3

When the Lie differential of the integral vanishes, charge is an evolutionary invariant; e.g., charge is conserved. There are three cases:

i(βV4 )J = 0, i(βV4 )J = dΦ, i(βV4 )J = dΦ + ζ, dζ = 0.

(4.107) (4.108) (4.109)

If i(βV4 )J = 0, then the integral vanishes (the extremal case) independent from the deformation function β. The process, βV4 , is an extremal field relative to J, and the integral is a deformation invariant which represents a topological property. 4.3.5 The Maxwell-Faraday system: F − dA = 0 The E B Field Intensities: differential 2-forms On a four-dimensional space-time of independent variables, (x, y, z, t) the 1-form of Action (constrained by the postulate of potentials, F − dA = 0) can be written in the form (4.110) A = Σ3k=1 Ak (x, y, z, t)dxk − φ(x, y, z, t)dt = A◦dr−φdt. This 1-form of Action defines (part of) the physical system of electromagnetism. Subject to the constraint of the exterior differential system, the 2-form of field intensities, F, becomes, F = dA = {∂Ak /∂xj − ∂Aj /∂xk }dxj ˆdxk = Fjk dxj ˆdxk (4.111) = Bz dxˆdy + Bx dyˆdz + By dzˆdx + Ex dxˆdt + Ey dyˆdt + Ez dzˆdt, where, in usual engineering notation, E = −∂A/∂t − gradφ,

B =curl A ≡ ∂Ak /∂xj − ∂Aj /∂xk .

(4.112)

The closure of the exterior differential system, dF = 0, vanishes for C2 differentiable p-forms, to yield, dF = ddA = {curl E + ∂B/∂t}x dyˆdzˆdt − .. + .. − div Bdxˆdyˆdz} ⇒ 0. (4.113)

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Equating to zero all four coefficients leads to the Maxwell-Faraday partial derivative equations, {curl E + ∂B/∂t = 0, div B = 0}. (4.114) This topological development of the Maxwell-Faraday equations has made no use of a connection nor of a metric. Be aware that the engineering notation, where the six components of the second rank covariant tensor, Fjk , are grouped into two 3-component vectors, is deceptive, for the diffeomorphic transformational properties of the field intensities (E and B) are not that of Cartesian rank 1 three-dimensional vectors, but are that of a second rank covariant tensor field. The component functions (E and B) of the 2-form, F, transform as covariant tensor of rank 2. The topological constraint that F is exact, implies that the domain of support for the field intensities cannot be compact without boundary, unless the Euler characteristic vanishes. These facts distinguish classical electromagnetism from Yang-Mills field theories, where in addition it is assumed that G is dual or anti-dual to F . Moreover, the fact that F is subsumed to be exact and C1 differentiable excludes the concept of magnetic monopoles from classical electromagnetic theory on topological grounds. Note that the Poincare lemma applied to the topological constraint (ddA = dF = 0) always leads to the Maxwell pair of (Faraday Induction) PDE equations (4.114). This result is actually true for a variety of any dimension ≥ 4 and for any set of covariant symbols. Even if the dimension of the ordered domain exceeds 4, the concept of ddA = dF = 0, always yields the same set of partial differential equations relative to the first four base variables. The first Maxwell set of equations forms a nested set of PDE’s on varieties of pre-geometric dimension > 4. The addition of new independent base variables does not change the format of the first four Maxwell PDE equations, but just adds to the set new PDE equations involving field components defined over the new ordered set of base variables. The concept of Faraday Induction is universal, and should not restricted to the science of electromagnetism. It is valid for any physical system which can be described by a 1-form of Action with a Pfaff dimension 2 or larger, such as a fluid with vorticity. The very existence of the E and B fields implies that the 2-form F = dA does not vanish. Hence, the 2-form defines a symplectic manifold of at least Pfaff dimension 2. As the 2-form is exact, the symplectic 2-manifold cannot be compact without boundary. This result follows from Stokes theorem, with 2 exceptions, the Klein bottle and the torus. Except for these two exceptions, the exact symplectic domain of the electromagnetic field intensities, E and B, must either be open, or if compact it has a boundary. The conservation of Flux The Maxwell Faraday exterior differential system is a topological constraint, for by Stokes theorem the support for F can not be compact without boundary. The

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closure of the exterior differential system, dF = 0, generates the flux conservation law. The integral of F over a closed two-dimensional domain is a relative integral invariant (a deformation invariant) of any process that can be described in terms of a vector direction field, V . The formal statement is given by Cartan’s magic formula [148], which describes continuous topological evolution in terms of the action of the Lie differential, with respect to a vector field, acting on the exterior differential 2form, F. The Maxwell-Faraday exterior differential system leads to the evolutionary Conservation of Flux theorem. Flux is defined as the integral of the 2-form, F , over a two-dimensional integration chain M2. Using Stokes’ theorem, RR R Conservation of Flux : F = A = Flux, dF = 0, M2 ∂M2 RR RR R LβV4 ( F ) = d(i(βV4 )F ) = A ⇒ 0. M2

M2

(4.115) (4.116)

∂M2

When the Lie differential of the integral vanishes, flux is an evolutionary invariant; e.g., flux is conserved. There are three cases: Extremal i(βV4 )F = 0, Bernoulli i(βV4 )F = dΘ, Stokes i(βV4 )F = dΘ + γ, dγ = 0.

(4.117) (4.118) (4.119)

If i(βV4 )F = 0, then the last integral in equation (4.116) vanishes (the extremal case) independent from the deformation function β. The process βV4 is an extremal field relative to F, and the integral is a deformation invariant which represents a topological property. Thermodynamically, the extremal case implies that the Work 1-form, W = i(βV4 )F , is of Pfaff topological dimension 0, otherwise W is of Pfaff topological dimension 1. Note that the Pfaff topological dimension of the Action 1-form is not explicitly defined, but it must be 2 or greater to produce the 2-form, F. It is remarkable that the Maxwell-Faraday differential system can also be used to encode hydrodynamic and thermodynamics systems, both of the equilibrium and non-equilibrium variety. Limit Points This now almost classic generation of the Maxwell field equations has another less familiar interpretation. The E and B field intensities are the topological limit "points" of the 1-form of potentials, {A, φ}, relative to the Cartan topology! The limit points of the 2-form of field intensities, F , are the null set. For C2 vector fields, the Cartan topology admits flux quanta, charge quanta, and spin quanta, but excludes magnetic monopoles. When the differential system of interest is built upon the forms A, F

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and G, it is possible to show that superconductivity is to be associated with the constraints on the limit point sets of A, AˆF, and AˆG [205]. That is, superconductivity has its origins in topological, not geometrical, concepts. This remarkable idea that the exterior differential is a limit point operator is based upon Kuratowski’s closure operator (see p.72 [141]) is equivalent to the union of the identity and the exterior differential. 4.3.6 The Lorentz force Consider the 1-form of electromagnetic action, A, given above (4.110). In electromagnetic format, for all processes, it follows that curl E + ∂B/∂t = 0, div B = 0.

(4.120)

Consider an abstract process, V4 ⇒ ρ{V k , 1} = ρ{V, 1}, on the space, {xk , t}. Evaluate the evolution of the electromagnetic action in terms of Cartan’s magic formula, L(V4 )A = i(V)dA + d(i(V)A) = W + dU = Q, W = i(V)dA = −{ρ(E + V × B)k dxk − ρ(V ◦ E)dt}, U = ρ(V ◦ A−φ).

(4.121) (4.122) (4.123)

The covariant spatial components of the Work 1-form, W , are to be recognized as the Lorentz force per unit charge to within the parametrization factor, ρ, f(Lorentz) = −ρ(E + V × B).

(4.124)

Note that the time-like component becomes the "local dissipative" power, P = ρ(V ◦ E). The classic formula for the Lorentz force has been derived on topological grounds as a consequence of the perspective of continuous topological evolution. It has not been injected into the theory as a constraint of special relativity. 4.3.7 Non-Equilibrium Features of Electromagnetism The exterior differential forms that make up the electromagnetic system on a geometric domain of four dimensions consist of the primitive 1-form, A, and the primitive (N-2)-form density, G, their exterior differentials, and their algebraic intersections defined by all possible exterior products. The complete Maxwell system of exterior differential forms (the Pfaff sequence for the Maxwell system on 4 geometric dimensions) is given by the set: {A; F = dA, G; J = dG, AˆF, AˆG, AˆJ; F ˆF, GˆG, F ˆG}.

(4.125)

These differential forms and their unions may be used to form a topological base on the domain of independent variables. The Cartan topology constructed on this system of forms has the useful feature that the exterior differential may be interpreted as a limit point, or closure, operator in the sense of Kuratowski (see p. 72 in [141]).

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The exterior differential systems that define the Maxwell-Ampere and the MaxwellFaraday equations above are essentially topological constraints of closure. The complete Maxwell system of differential forms (which assumes the existence of A and G and C2 differentiability) also generates two other exterior differential systems, (F ˆG − AˆJ) − d(AˆG) = 0, F ˆF − d(AˆF ) = 0,

(4.126) (4.127)

which prolong the primary (exact) exterior differential systems, F − dA = 0, J − dG = 0.

(4.128) (4.129)

The terms AˆF and AˆG are zero for equilibrium systems. The existence of these 3-forms are indicators that the electromagnetic system is NOT in equilibrium. Each of the forms, A, G, AˆG, AˆF , can have closed but not exact components. The two 4-forms (F ˆG − AˆJ) and (F ˆF ) are exact and have closed integrals which are evolutionary (relative) invariants of continuous deformations. The closed integrals therefore describe topological properties. The first 3-form density, AˆG, with physical units of ~, is called the "topological spin" (or chirality) [193] and the second 3-form, AˆF, with physical units of (~/e)2 , is called the "topological torsion" (or helicity) [202]. These two exterior 3-forms, AˆG and AˆF are not usually found in discussions of classical electromagnetism. The 3-forms are abstractly defined (on a space of 4 geometric dimensions with a volume element, Ω4 = dxˆdyˆdzˆdt) in terms of exterior multiplication, but can be given realization in terms of 4-component engineering variables, S4 and T4 : Topological Spin density AˆG = i(S4 )Ω4 S4 = [S, σ] = [A × H + Dφ, A ◦ D],

(4.130) (4.131) (4.132)

Topological Torsion vector AˆF = i(T4 )Ω4 T4 = [T, h] = −[E × A + Bφ, A ◦ B].

(4.133) (4.134) (4.135)

These constructions should be compared with the (closed and exact) 3-form of chargecurrent (4-vector) density, J = dG , which has a 4-component engineering representation as, J4 = [J, ρ]. The concepts of the topological Spin density (current) and the topological Torsion vector have had almost no utilization in applications of classical electromagnetic theory. Each construction depends explicitly on the existence of the 1-form of Action-potentials.

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Recall that the closed components of the 1-form of Action do not effect the components of the 2-form of intensities, F = dA = d(Ac + A0 ) = 0 + d(A0 ). However, these "gauge" additions do influence the topological dimension of the 1-form of Action. For example, let A0 be of Pfaff Topological dimension 2, representing an equilibrium system where A0 ˆdA0 = 0. Then by addition of a closed component to the original action, A = Ac + A0 could have a topological dimension of 3, as AˆdA = (Ac + A0 )ˆdA0 = Ac ˆdA0 6= 0.

(4.136)

So the addition of a closed component to the 1-form of Action could change the system from an equilibrium system to a non-equilibrium system. The 4-form dAˆdA is not influenced by the (gauge) addition to the original 1-form of Action, as dAˆdA = dA0 ˆdA0 .

(4.137)

4.3.8 The Poincare Topological 4-forms The exterior differentials of the 3-forms of topological Spin and topological Torsion produce two exact 4-forms, F ˆG − AˆJ and F ˆF , whose closed integrals are topological objects which generalize the conformal invariants [290] of a Lorentz system, as discovered by Poincare and Bateman. Note that these topological properties of invariance with respect to continuous deformations are valid even in the non-equilibrium domain of dissipative charge-currents and radiation. In the format of independent variables {x, y, z, t}, with a volume element Ω4 , the exterior differential, acting on the 3-forms as a topological limit point generator, can be related to the classic 4-divergence of the 4-component Topological Spin and Topological Torsion vectors, S4 and T4 : Poincare 1 = d(AˆG) = F ˆG − AˆJ = {div3 (A × H + Dφ) + ∂(A ◦ D)/∂t} Ω4 = {(B ◦ H − D ◦ E) − (A ◦ J − ρφ)} Ω4 . Poincare 2 = d(AˆF ) = F ˆF = {div3 (E × A + Bφ) + ∂(A ◦ B)/∂t} Ω4 = {2E ◦ B} Ω4 .

(4.138)

(4.139)

The Poincare 4-forms are, in effect, the evolutionary source terms for the 3-forms of topological spin, AˆG, and topological torsion, AˆF . When the Poincare 4forms are zero, the closed integrals of the electromagnetic 3-forms of AˆG and AˆF become additional topologically coherent configurations invariant with respect to all evolutionary processes of continuous deformation. The first term in the first Poincare 4-form has a coefficient function which represents twice the difference between the magnetic energy density and the electric energy density of the electromagnetic field in a Lagrangian sense, Topological Field Lagrangian: F ˆG = (B ◦ H − D ◦ E) Ω4 .

(4.140)

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The second term in the first Poincare 4-form has a coefficient function which is defined as the interaction energy density: Topological Interaction: AˆJ = (A ◦ J − ρφ) Ω4 .

(4.141)

In Lagrangian variational methods, the 4-form F ˆF, which defines the second Poincare 4-form, has been related to the concept of Topological Parity: Topological Parity: F ˆF = +{2E ◦ B} Ω4 .

(4.142)

4.3.9 Topological Torsion and Spin quanta When either Poincare deformation invariant vanishes, the corresponding closed threedimensional integrals of AˆG and AˆF become deRham period integrals. The closed, but not exact, components of each 3-form can be put into correspondence with "quantized" topological defects. The topological Spin quantum is defined as the closed integral of those closed but not exact components of the 3-form AˆG (which represent the kernel of the first Poincare 4-form), RRR Spin quantum = AˆG with units n (~). (4.143) z3d

RRR

The period integrals AˆG are deformation invariants (hence define a topologz3d ical property) with rational ratios. The notation z3d designates a closed integration chain defined in regions where d(AˆG) = 0. Similarly, when the second Poincare 4-form vanishes, the closed integral of the 3-form of Torsion-Helicity becomes a deformation invariant with quantized values: Torsion quantum =

RRR

AˆF with units m (~/e)2 .

(4.144)

z3d

RRR The period integrals AˆF are deformation invariants (hence define a topologiz3d cal property) with rational ratios. In this case, the notation, z3d, designates a closed integration chain defined in regions where d(AˆF ) = 0. It is important to realize that the topological conservation laws (deformation invariants with respect to homeomorphisms) are valid in a plasma as well as in the vacuum, subject to the conditions of zero values for the Poincare 4-forms. On the other hand, topological evolution and transitions between "quantized" states of Spinchirality or Torsion-helicity require that the respective Poincare 4-forms are not zero. The 3-forms, AˆG and AˆF, are not necessarily closed, nor exact. Their exterior differentials (divergences) are not necessarily zero. The values of the 4forms created by exterior differentiation of these 3-forms define the integrands of the topological Poincare 4-forms. As these 4-forms are exact by construction, their closed integrals are always relative integral deformation invariants and thereby define

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173

topological properties. The 3-forms are not necessarily, in themselves, deformation invariants. However, when the Poincare 4-forms vanish (zero divergence) the closed integral of the corresponding 3-form generates a topological quantity (Topological Spin or Topological Torsion respectively) which is also a deformation invariant. In such situations, the 3-forms are closed, but not necessarily exact. Hence their closed integrals generate deRham period integrals [73] [182], and have rational ratios. Such is the stuff of topological quantization, which is independent from scales. The theory of 3-forms and their period integrals was investigated with respect to electromagnetism and other field theories, first with respect to the 3-form defined below as topological spin, AˆG, and then later with respect to the 3-form of topological torsion, defined as AˆF. The first application of AˆF was in the field of turbulence [190], where it was conjectured that the transition from streamline flow (uniquely integrable in the sense of Frobenius, such that AˆF = 0) to a turbulent flow (not uniquely integrable in the sense of Frobenius, AˆF 6= 0) must involve a topological change. Although the interest was focused on hydrodynamics, the electromagnetic format was always used to establish a credence level in the computations that were done by hand. In the modern world of symbolic calculators on your desktop, this algebraic tedium has been alleviated. See http://www22.pair.com/csdc/pdf/maxwell.pdf and Volume 6 in this series [278]. On domains where the Pfaff topological dimension is 3 (and not 4) there exists a three-dimensional period integralRRR of the topological torsion, which is related to the Hopf invariant (see p. 228 in [30]), AˆF. It will be demonstrated below that 3_cycle if the domain is of Pfaff dimension 3, then evolutionary processes in the direction of the electromagnetic charge current 4D vector density, J, leave the integral of the Topological Torsion current over a three-dimensional boundary as an evolutionary invariant. Even more remarkable is the fact that such a statement is valid in domains where the Pfaff dimension is 4, not 3, if the current flow is on the surface defined by (E • B) = 0. To summarize, given a 1-form of Action in Electromagnetic format, A = k Ak dx − φdt, 1. If (E • B) 6= 0 then the thermodynamic system is in a turbulent non-equilibrium state of Pfaff topological dimension 4. 2. If (E • B) = 0, and the Topological Torsion 3-form is not zero, such that all 4 components of 4D vector, T4 = −[E × A + Bφ, A ◦ B], do NOT vanish, then the thermodynamic system is in a non-equilibrium state of Pfaff topological dimension 3, and can support "steady states" far from equilibrium. 3. If all components of the Topological Torsion 3-form vanish, then the thermodynamic system is in an isolated state of Pfaff topological dimension 2.

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4. If the components of E and B vanish, then the thermodynamic system is in an equilibrium state of Pfaff topological dimension 1. 4.4

Hydrodynamics as a topological theory

An abbreviated summary of the topological features of hydrodynamic systems is presented in the next section. A more detailed set of examples and applications is to be found in Volume 3 of this series, "Wakes, Coherent Structures and Turbulence" [275]. 4.4.1 Euler flows and Hamiltonian fluids Consider the Action 1-form per unit source (in thermodynamics, the unit source is mole number, or sometimes mass), A = v ◦ dr−{v.v/2}dt.

(4.145)

Compute the exterior differential dA and define the following functions, ω = curl v

and a = −{∂v/∂t + grad(v · v/2)},

(4.146)

such that, F = dA = {∂Ak /∂xj − ∂Aj /∂xk }dxj ˆdxk = Fjk dxj ˆdxk (4.147) = ωz dxˆdy + ω x dyˆdz + ωy dzˆdx + ax dxˆdt + ay dyˆdt + az dzˆdt. These vector fields always satisfy the Poincare-Faraday induction equations, dF = ddA = 0, or, curl a + ∂ω/∂t = 0, div ω = 0. (4.148) Consider a process created by the vector field, V4 = [Vx , Vy , Vz , 1] and use Cartan’s magic formula, L(ρV4 ) A = i(ρV4 )dA + d(i(ρV4 )A) = W + dU = Q,

(4.149)

to compute the Work 1-form, W ,. The expressions for Work, W, and internal energy, U, become: W = i(ρV4 )dA = ρ{∂v/∂t + grad(V · v/2) − V × w} ◦ dr −ρV ◦ {∂v/∂t + grad(v · v/2)}dt, U = i(V4 )A = ρ(V · v − v.v/2).

(4.150) (4.151)

At first, topologically constrain the Work 1-form to be of the Bernoulli class in terms of the exterior differential system: W − dP = 0, with i(V4 )dP = 0.

(4.152) (4.153)

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175

The last equation states that the Bernoulli function is invariant along a trajectory, but can vary (transversely) from trajectory to trajectory. Assume that V = v and set the coefficients of spatial components of the Bernoulli exterior differential system to zero. The result is the partial differential equations that represent the LagrangeEuler fluid: {∂v/∂t + grad(v · v/2) − v × w} − grad(P )/ρ = 0. (4.154) This formula should be compared to the derivation of the Lorentz force in electromagnetic systems. Note that the Bernoulli pressure, P , is an evolutionary invariant along a trajectory. The flow is Hamiltonian (but not extremal Hamiltonian) and reversible, as Q is exact, of Pfaff topological dimension, 1, and QˆdQ = 0. The time-like component of the exterior differential system W − dP = 0 leads to the equation, ∂P/∂t = −ρv ◦ {∂v/∂t + grad(v · v/2) = ρ(v ◦ a).

(4.155)

It is apparent that if the velocity and the acceleration are orthogonal, then the time rate of change of the Bernoulli pressure is zero. It also follows that the "Master" equation is valid, with the only difference being that curl v is defined as ω, the vorticity of the hydrodynamic flow. The master equation becomes, curl(v × ω) = ∂ω/∂t, (4.156) and this equation is to be recognized as the equivalent of Helmholtz’ equation for the conservation of vorticity. In the hydrodynamic sense, conservation of vorticity implies uniform continuity. In other words, the Eulerian flow is not only Hamiltonian, it is also uniformly continuous, and satisfies both the master equation and the conservation of vorticity constraints. In addition, it may be demonstrated that such systems are at most of Pfaff dimension 3, and admit a relative integral invariant which generalizes the hydrodynamic concept of invariant helicity. In the electromagnetic topology, the Hamiltonian constraint is equivalent to the statement that the Lorentz force vanishes, a condition that has been used to define the "ideal" plasma or "force-free" plasma state. 4.4.2 The Navier-Stokes fluid From the theory of topological fluctuations, it must be true that the 1-form of Work must have a format of the type W = i(ρV4 )dA = −dP +

j j (dx

− vj dt),

(4.157)

such that for a pressure, P, of the Bernoulli class (4.153), the work done is transverse to the process trajectory, (i(ρV4 )W = 0.

(4.158)

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Electrodynamic, Hydrodynamic, and Mechanical Thermodynamic systems

The coefficients, j , of the velocity topological fluctuations act in the manner of Lagrange multipliers. If j /ρ is defined (arbitrarily) as υ curl curl v then the spatial components of the Work 1-form, W , are constrained to yield the partial differential equations for a constant density Navier-Stokes fluid: {∂v/∂t + grad(v · v/2) − v × w} = −grad(P )/ρ + υ curl curl v.

(4.159)

Density variations can be included by adding a term λdiv(V) to the potential {v · v/2} to yield: ∂v/∂t + grad{v.v/2} − v × curl v = −gradP/ρ +λ{grad(div v)} +υ{curl curl v}.

(4.160) (4.161) (4.162)

Classically, v can be identified with the geometric kinematic shear viscosity, and λ = µB − υ. The coefficient µB can be identified with the topological (space-time) bulk viscosity. It is thereby demonstrated that the Navier-Stokes equations correspond to a refinement of the Cartan topology [186]. The Navier-Stokes constraint implies that the Work 1-form need not be closed. There are solutions to the Navier-Stokes equations that are thermodynamically irreversible. The 1-form of Action will generate a 3-form of Topological Torsion, AˆdA = i(T4 )dxˆdyˆdzˆdt, of the form, T4 = [a × v + {v.v/2} curl v, (v ◦ curl v)], = [a × v + {v.v/2} ω,(v ◦ ω)].

(4.163) (4.164)

Use the Navier-Stokes equations (4.161) to solve for a, a = −[grad{v.v/2} + ∂v/∂t] = −v × curl v + gradP/ρ −λ{grad(div v)} + υ{curl curl v},

(4.165) (4.166)

and then substitute in the expression for T4 , to yield T = [hv − (v ◦ v/2)curl v − v × (gradP lρ) +λ{v × grad(div v)} − υ{v × (curl curl v)}, h], h = v · curl v,

(4.167) (4.168)

which persists even for Euler flows, where υ = 0, if the flow is baroclinic. The measurement of the components of the Torsion vector, T4 , have been completely ignored by experimentalists in hydrodynamics.

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177

By a similar substitution, the topological parity pseudo-scalar (equivalent to the second Poincare 4-form) becomes expressible in terms of engineering quantities as, K = {2(a ◦ ω)}Ω4 = {−2{gradP/ρ ◦ curl v −λ{ grad(div v) ◦ curl v} −υ{curl v ◦ (curl curl v)}}Ω4 .

(4.169)

The coefficient K is a measure of the space-time bulk dissipation coefficient (not λ), and it is the square of this number which must not be zero if the process is irreversible. Recall that turbulent dissipative irreversible flow is defined when the Pfaff dimension of the Action 1-form is equal to 4, which implies that K 6= 0. From this expression it is apparent that if the vorticity is of Pfaff dimension 2, then the last term vanishes, and the there is no irreversible dissipation due to shear viscosity. Other useful situations and design criteria for dissipation, or the lack thereof, can be gleaned from the formula. If the vector field is harmonic, then an irreversible process requires that, K = {2(a ◦ ω)}Ω4 = {[2gradP/ρ − µB grad(div v)] ◦ curl v}Ω4 6= 0.

(4.170)

(Recall that harmonic vector fields are generators of minimal surfaces.) For fluids where (µB ) ⇒ 0, and if the pressure gradient is orthogonal to the vorticity and the flow field is harmonic, then there is no irreversible dissipation, and the flow is not turbulent. Note that for many fluids the bulk viscosity is much greater than the shear viscosity. When K = 0 no topological torsion defects are created; the acceleration and the vorticity of the Navier-Stokes fluid are colinear. The integral of K over {x,y,z,t} gives the Euler Index of the flow. See the discussions in Volume 3 [275]. These results should be compared to those generated by Lamb and Eckart [66] for the fluid dissipation function, defined by the requirement that the dissipative flow has a (geometric) entropy production rate greater than or equal to zero. More will be said about these classical equations in the section below on dynamical systems. 4.5

Mechanics as a topological theory

4.5.1 Cartan’s development of Hamiltonian Mechanics In 1922 [43] Cartan developed the idea that if the closed integrals of an Action 1form, A, are subjected to processes, ρV, such that the closed integrals are deformation invariants (and therefore behave as topological properties relative to the evolution), then the process has a Hamiltonian representation for a system of ordinary differential equations describing the process. The idea was that if Z Z Z L(ρV) A= i(ρV)dA + d(i(ρV)A) ⇒ 0, any ρ > 0, (4.171) z1

z1

z1

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Electrodynamic, Hydrodynamic, and Mechanical Thermodynamic systems

then there was a function, H, which generated a set of first order differential equations that defined the admissible processes, V, as a dynamical system. Such processes preserved topological properties defined by the closed integrals. The classic CartanHamiltonian Action is written on a space of 2n+1 variables as, A = pk dq k + H(pk , qk , t)dt.

(4.172)

The second integral in the expression above vanishes for C1 functions, as the closed integral of an exact differential is always zero. However, the first integral can be equal to zero in several ways. The first solution is given by extremal processes, ρV ⇒ρVE such that the virtual Work 1-form, W , vanishes. The Cartan constraint for extremal Hamiltonian processes is given by the equation, Extremal Hamiltonian processes: W = i(ρVE )dA = 0.

(4.173)

The Cartan proof in dimension 2n+1 is quite straightforward, and follows by evaluating the constraint for a process direction field with the assumed functional components, VE = [fk , v k , 1]: (4.174) i(ρVE )dA = i(ρVE ){dpk ˆdqk + dHˆdt = k k k i(ρVE ){dpk ˆdq + (∂H/∂pk dpk + ∂H/∂q dq )ˆdt}, = ρ{fk (dqk − ∂H/∂pk dt) − vk (dpk + ∂H/∂q k dt)} + (∂H/∂pk dpk + ∂H/∂qk dq k ), = ρ{fk ∆q k − vk ∆pk } + (∂H/∂pk dpk + ∂H/∂q k dqk ) ⇒ 0. (4.175) The dynamical system so generated by Cartan’s constraint is said to be Hamiltonian, for the components of ρVE are determined by the partial derivatives of the Hamiltonian function, H(pk , qk , t) when the system of (fluctuation) 1-forms ∆qk and ∆pk vanish. If ∆qk = dq k − ∂H/∂pk dt ⇒ 0, and ∆pk = dpk + ∂H/∂q k dt ⇒ 0, then 0 = (∂H/∂pk dpk + ∂H/∂qk dq k ).

(4.176) (4.177) (4.178)

The extremal solution to the Cartan constraint equation requires that the components of VE = [fk , v k , 1] relative to the coordinates {p, q, t} are given by the equations, [−∂H/∂q, ∂H/∂p, 1]. The dynamical system is defined by the equations, dp dq dt = = . −∂H/∂q ∂H/∂p 1

(4.179)

The extremal constraint is severe and implies that the antisymmetric matrix representing dA has at least one eigenvector of zero eigenvalue. If the domain of

Mechanics as a topological theory

179

independent variables is odd, 2n+1, and the 1-form A is of maximal Pfaff dimension, then the null eigenvector is uniquely defined (to within a factor) by the functional components of the 1-form. That is to say, the functional description of the physical system determines the unique extremal direction field. This result is at the foundation of Hamiltonian mechanics on state spaces of 2n+1 dimensions. Note that the constraint of Hamiltonian mechanics to processes which are extremal fields, W = i(V)dA = 0, implies that the processes constructed from individual spinors are not permitted. However, the extremal possibility is not the only process by which the closed integrals of the Action behave as invariant topological properties. It is also possible that the integrand in the first integral is exact. Such constraints are very significant to the study of symplectic spaces of topological dimension 2n+2. The Hamiltonian "energy function" is replaced by the "Bernoulli-Casimir-Hamiltonian" generating function. The processes considered are again constrained, and now the dynamics of the evolutionary process is NOT determined uniquely in terms of the functional representation of the physical system. Exterior terms are required in terms of the Bernoulli function, but once chosen, the process vector field is determined by linearity in symplectic spaces, for the antisymmetric matrix representation of the 2form dA is invertible in the (2n+2)-dimensional space. The Cartan constraint for Bernoulli-Symplectic Hamiltonian processes is given by the equation, Bernoulli-Symplectic processes: W = i(ρVB )dA = −dB.

(4.180)

Mathematicians call the function B a "Hamiltonian function", but herein due respect is given to those many engineers and applied scientists involved with hydrodynamics who have known about this function, the Bernoulli function, for more than 200 years. Students in sophomore hydrodynamics learn that the rule, B = P + ρgh + 1/2ρV 2 is a constant along any any streamline. Along any trajectory, the Bernoulli function is a path invariant, for L(ρVB ) B = i(ρVB )i(ρVB )dA + d(i(ρVB )B) = 0 + 0.

(4.181)

In hydrodynamics, the Bernoulli function is a constant along any "stream" line, but the constant can have different values from one neighboring stream line to another stream line. The invariance of the Bernoulli function is path dependent, and its value is not global (not the same for all streamlines). Both the Extremal and Bernoulli processes are subsets of the Helmholtz processes which satisfy the equations, L(ρV) dA = 0, W = i(ρVS )dA = −dP + γ,

(4.182) (4.183)

where γ is a closed but not exact 1-form. When γ is not zero, then the closed integrals of Action can change during the evolutionary processes, and they are no

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Electrodynamic, Hydrodynamic, and Mechanical Thermodynamic systems

longer deformation invariants. Helmholtz processes preserve the fundamental 2-form dA as a process invariant. Hence the symplectic structure generated by the 2-form dA is an invariant of a Helmholtz process. Mathematicians call such Helmholtz processes symplecto-morphisms§ . It is important to note that Bernoulli-Symplectic processes, that are not extremal, generate Work 1-forms of Pfaff dimension 1, W 6= 0. This fact implies that that such Bernoulli-Symplectic processes must have Spinor components. 4.5.2 A Generalized Hamiltonian Formalism The independent variables will be assumed to be "Action conjugate" set {q k , t, Pk , Mc2 }. The 1-form of Action will be assumed to be of Pfaff topological dimension 2n+2, and will have the classic Darboux format, A = Pk dq k − Φ(q k , t, Pk , Mc2 )dt.

(4.184)

The function Φ(qk , t, Pk , Mc2 ) is defined as the "total energy" function in Hamiltonian coordinates. For a Bernoulli process, the Work 1-form is constrained by the equation: W = i(ρVB )dA = −dB(q k , t, Pk , Mc2 ).

(4.185)

On the symplectic 2n+2 space, the 2-form dA has a Pfaff dimension 2n+2. Given a function, B, a vector direction field on the symplectic space is uniquely defined, as the 2n+2 × 2n+2 non-degenerate antisymmetric matrix of dA has an inverse. As dA = dPk ˆdqk − dΦˆdt,

(4.186)

i(ρ[V k , 1, Fk , R, ])dA, ρ(Fk dq k − V k dPk ) + ρ(dΦ − Rdt), −dB(q k , t, Pk , Mc2 ), −∂(B + Φ)/∂q k , ρV k = +∂(B + Φ)/Pk , ∂(B + Φ)/∂t, ρ = −∂(B + Φ)/∂(Mc2 ).

(4.187) (4.188) (4.189) (4.190) (4.191)

it follows that, i(ρVB )dA = = = hence, ρFk = ρR =

Other constraints may be placed upon the 1-form of Action, in order to reduce the Pfaff topological dimension to 2n+1. The strongest constraint would require that the "total energy" function, be a global constant, dΦ = 0. For such constrained systems, the Bernoulli function, B, takes over the role of the "Hamiltonian" generator: dPk = −∂B/∂q k dt, §

dq k = +∂B/∂pk dt,

d(Mc2 ) = −(∂B/∂t)dt.

A symplectic process, or symplecto-morphism, is not the same as a Symplectic space (manifold). A symplecto-morphism requires that dA is a process invariant. There are many processes on a symplectic manifold that are not symplecto-morphisms.

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181

In both the contact 2n+1 system (where the constant total energy constraint has reduced the 2n+2 symplectic system to a 2n+1 contact system), and for the symplectic 2n+2 system itself, the Bernoulli evolution of the 2-form dA obeys the Helmholtz conservation of vorticity theorem, L(ρV) dA = d(−dB + dU) = dQ = 0.

(4.192)

Hence it follows that QˆdQ = 0, such that: Theorem 5. All Helmholtz-Symplectic, Bernoulli, Hamiltonian, or extremal processes are thermodynamically reversible. It is no wonder, that the historical theory of mechanics, almost always restricted to the study of Helmholtz processes, is incapable of describing the concept of thermodynamical irreversibility. The baby has been thrown out with the wash. It is important to note that Helmholtz processes are not necessarily adiabatic. Remark It is remarkable that from a thermodynamic point of view, the concept of a constant "total energy" dΦ = 0, for the Universe is not tenable if the Universe is open, dAˆdA 6= 0, and moreover need not be valid if the Universe is closed. The 3D geometries of closed thermodynamic systems, for which dAˆdA = 0, can be distinct. In that which follows, other distinct constraints will be placed upon the Symplectic system to reduce the topological dimension of the Action 1-from from 4D (dAˆdA 6= 0) to 3D, (dAˆdA = 0). Such constraints appear to be related to the eight distinct geometries of Thurston. 4.5.3 A Generalized Lagrangian Formalism The Lagrangian set of independent variables will be assumed to be of the format {q k , t, Pk , v k }. The 1-form of Action will be assumed to be of Pfaff topological dimension 2n+2, and will have the classic Cartan-Hilbert format, A = L(qk , v k , t)dt + Pk (dq k − vk dt) = Pk dq k − Φ(q k , t, Pk , v k )dt.

(4.193) (4.194)

Φ(q k , v k , Pk, t) = Pk v k − L(qk , v k , t), such that ∂Φ/∂v k = (Pk − ∂L/∂v k ) = (~kk ) 6= 0.

(4.195) (4.196)

The function Φ(q k , t, Pk , v k ) is defined as the "total energy" function in Lagrangian coordinates,

The momenta have non-canonical components, ~kk . The two generalized formalisms (Hamiltonian and Lagrangian) are almost the same, but the independent variables are different. For example, the 4D natural volume elements are distinct: ΩL = dΦˆdP ˆdqˆdt = (∂Φ/∂v)dvˆdP ˆdqˆdt, ΩH = dΦˆdP ˆdqˆdt = (∂Φ/∂Mc2 )d(Mc2 )ˆdP ˆdqˆdt.

(4.197) (4.198)

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Electrodynamic, Hydrodynamic, and Mechanical Thermodynamic systems

The Lagrangian method emphasizes the kinematic variables, the Hamiltonian method emphasizes the energy-momentum variables. The variation of total energy with velocity is an artifact of the existence of non-canonical momenta, {~kk }, in the Lagrangian formalism By direct computation, the Pfaff dimension of the Cartan-Hilbert Action 1form is 2n+2 and the symplectic manifold volume element is given by the expression: dAˆ(dA)n = (n + 1)!{(Pk − ∂L/∂vk )dv k }dP1 ˆ...dPn ˆdq 1 ˆ...dq n ˆˆdt.

(4.199)

The assumption that dAn+1 is a volume element implies that the bracketed factor represents a mass variation dependent upon velocity, and is a perfect differential (to within a non-zero factor). This result is encoded as: {∂Φ/∂(M)d(M)} = {(Pk − ∂L/∂vk )dv k } = {~kk dv k } = T dS.

(4.200)

Remark The result that "mass" is velocity dependent is not explicitly determined from a geometric constraint determined by the Lorentz equivalence class of metrics defined for special relativity theory. The variation of mass with velocity is established when the Cartan-Hilbert Action generates a symplectic domain and the entropy of the physical system is not a constant. At first glance it would appear that the Cartan-Hamiltonian 1-form of Action (4.172) is equivalent to the primitive Lagrange function integrand of the Calculus of variations, L(q k , v k , t)dt, constrained by the anholonomic differentials, (dq k − v k dt), with Lagrange multipliers, Pk , in the format of the Cartan-Hilbert Action (4.193). However, the Cartan-Hilbert Action is of Pfaff topological dimension 2n + 2, while the Cartan-Hamiltonian Action is of Pfaff topological dimension 2n + 1. The classical format presumes that the Hamiltonian function is a function only of {q, t, P }. However, H(q, t, P ) 6= Φ(q, t, P, Mc2 ), (4.201)

and the equivalence is not justified unless the generalized total energy function Φ is used as the Hamiltonian (4.184). From an alternate point of view, the Cartan-Hilbert system must have an additional constraint in order to reduce the Pfaff dimension of the Action from 2n+2 to 2n+1. That constraint is precisely equal to the equation, {∂Φ/∂(M)d(M)} = {(Pk − ∂L/∂vk )dv k } = {~kk dv k } = T dS ⇒ 0.

(4.202)

It follows that the condition that the classic Cartan-Hamiltonian Action be equivalent to the Cartan-Hilbert Action is the constraint that the entropy change is zero; dS = 0. Hence the following theorem has been established: Remark 36 Classical Hamiltonian processes on contact manifolds of dimension 2n+ 1 are either isentropic or are at zero temperature.

Mechanics as a topological theory

4.5.4

183

Physical Systems on four dimensions again

The objective is to set up the theory in a form that can be applied to some simple examples. The four-dimensional variety will be expressed as above except the running index k will be suppressed. The 2 "conjugate" pairs of independent variables will be written in the generalized Hamiltonian format as {q, t, P, Mc2 }, where c is a constant. The 1-form of Action representing the physical system in each case will be written in terms of the generalized Hamiltonian format, A = P dq − Φdt.

(4.203)

The energy function Φ will be defined in terms of the independent variables, {q, t, P, Mc2 }, and defined as the "total energy" function in Hamiltonian coordinates: Total energy : Φ = Φ(q, t, P, Mc2 ), with dΦ = Φq dq + Φt dt + ΦP dP + ΦMc2 d(Mc2 ), with Φq = ∂Φ/∂q, Φt = ∂Φ/∂t, ΦP = ∂Φ/∂P, ΦMc2 = ∂Φ/∂(Mc2 ).

(4.204) (4.205) (4.206) (4.207)

The Pfaff sequence has the terms: Topological Action A Topological Vorticity dA Topological Torsion AˆdA Topological Parity dAˆdA

= = = =

P dq − Φdt, dP ˆdq − dΦˆdt, {P dΦ − ΦdP }ˆdqˆdt, −2dΦˆdP ˆdqˆdt.

(4.208) (4.209) (4.210) (4.211)

The 3-form of topological torsion yields the components of the Topological Torsion vector, relative to the ordered volume element Ω = dqˆdtˆdΦˆdP, T = [0, 0, −Φ, −P ],

(4.212)

from which it is obvious that i(ρT)A = 0. However, the 4-Volume, Ω4 , element in terms of the independent variables is given by the expression, Ω4 = dqˆdtˆd(Mc2 )ˆdP. (4.213) The Pfaff sequence becomes: Top. Action Top. Vorticity

A = P dq − Φdt, dA = dP ˆdq −{Φq dq + ΦP dP + ΦMc2 dMc2 }ˆdt, Top. Torsion AˆdA = {P ΦP − Φ}dP ˆdqˆdt +{P ΦMc2 }d(Mc2 )ˆdqˆdt, Top. Parity dAˆdA = −2{ΦMc2 }d(Mc2 )ˆdP ˆdqˆdt.

(4.214) (4.215) (4.216) (4.217)

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The 3-form of topological torsion yields the components of the Topological Torsion vector, relative to the ordered natural volume element Ω4 = dqˆdtˆd(Mc2 )ˆdP, in the format, T4 = [0, 0, {P ΦP − Φ}, −{P ΦMc2 }]. (4.218)

The expression for the 1-form of Work for an adiabatic process in the direction of the Topological Torsion vector becomes: W = = such that L(ρT4 ) A = Γ = U = and QˆdQ =

i(ρT4 )dA {−ρΦMc2 }(P dq − Φdt) − ρΓA, ρΓA = Q, {ΦMc2 } = d(i(T4 )Ω4 = (Div4 T4 )Ω4 , i(ρT4 )A = 0, +ρ2 Γ2 AˆdA 6= 0.

(4.219) (4.220) (4.221) (4.222) (4.223) (4.224)

Hence adiabatic evolution in the direction of the Topological Torsion vector is irreversible if Γ 6= 0, and a non-zero result implies that the Pfaff dimension of the Action must be 4. Evolution in the direction of T4 describes a "conformal" or "homogeneous [140]" process that is governed by the equation, L(T4 ) A = −ΓA.

(4.225)

Consider a "rescaled" action given by the function β L(T4 ) βA = {L(T4 ) β}A + βL(T4 ) A, = {L(T4 ) β}A − {βΓ}A.

(4.226) (4.227)

Then a solution to the equation, {L(T4 ) β} − {βΓ} = 0, or i(T4 )dβ − {βΓ} = 0,

(4.228) (4.229)

implies that the rescaled 1-form, βA, is invariant, and reversible, such that L(T4 ) βA = Q = 0.

(4.230)

The idea that a rescaling or a renormalization of the action changes a conformal process into a globally adiabatic process is a general result [189]. A conformal expansion or contraction can compensate for the divergence and irreversibility induced by evolution in the direction of the topological Torsion vector. In effect, an integrating factor can be found such that the divergence of the direction field induced by the topological torsion 3-form vanishes. As an example, if the amplitude of the damped harmonic oscillator was measured in terms of rulers that contracted at the appropriate rate, then the observer would measure no decay. This problem was treated by Robertson in a relativistic setting (see page 192 in [229]).

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4.5.5 The Harmonic Oscillator (W is of Pfaff Topological Dimension 0) The 4D Action 1-form will be assumed to be of the format given in equation (4.203). Assume that the "mass energy" function is defined by a non-zero function, Φ : The Harmonic Oscillator Φ(q, t, P ) = P 2 /2m0 + kq2 /2, with A = P dq − Φdt.

(4.231) (4.232)

The physical system is governed by the 1-form of Action, A, given in equation (4.203), where m0 and c are constants. If there is to be an extremal Hamiltonian field, the Pfaff dimension of A must be reduced from 4 to 3. This constraint can be accomplished by the sufficient (strong) assumption that Φ is a constant, d(Φ) = 0, d(Φ) = P/m0 dP + kqdq + 0 ⇒ 0.

(4.233)

The equation, Φ(q, t, P, Mc2 ) = constant, establishes a cylindrical subspace of constraint in the 4D domain. It is also true that a weaker constraint is given when Γ ⇒ 0. Then the Pfaff dimension of the Action 1-form is reduced to 3, also. This weaker requirement is satisfied globally when Φ > 0 is independent from Mc2 , and does not require the strong constraint that Φ(q, t, P, Mc2 ) = constant. The topological torsion 3-form in 4D has a presentation as, 4D-Topological Torsion AˆdA = P dΦˆdqˆdt − ΦdP ˆdqˆdt.

(4.234)

When the Action is constrained to be of Pfaff dimension 3, by the strong condition, (dΦ ⇒ 0), the topological torsion 3-form has only 1 component, and generates a volume element in 3D space, AˆdA, without zeros. The 3D contact structure is said to be "tight" without limit cycles. Pfaff Dimension 3 Closed System - Strong constraint 3D-Topological Torsion AˆdA = −(Φ)dP ˆdqˆdt, with the strong constraint dΦ = 0.

(4.235) (4.236)

However, if the mass energy function is such that ∂Φ/∂(Mc2 ) = 0, then the Topological Torsion 3-form has a different format for this weaker constraint. Pfaff Dimension 3 Closed System -weak constraint AˆdA = {P ∂Φ/∂P − Φ}dP ˆdqˆdt, when ∂Φ/∂(Mc2 ) = 0.

(4.237) (4.238)

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Now the volume element AˆdA is not global and can have singular points, indicating that the 3D contact manifold can have limit cycles generated by the zeros of the function, {P ∂Φ/∂P − Φ}. From the classic theory it is known that the Extremal Hamiltonian vector field on the 2n+1=3 contact manifold will have components: vextremal ' ve = dq/dt = ∂Φ/∂P = P/m0 , Fextremal ' Fe = dP/dt = −∂Φ/∂q = −kq.

(4.239) (4.240)

The Work 1-form vanishes for the extremal evolution of the Harmonic Oscillator, and the internal energy becomes, W = i([ve , 1, Fe , −])dA ⇒ 0, U = i([ve , 1, Fe , −])A = P 2 /2m0 − kq 2 /2.

(4.241) (4.242)

The Heat 1-form, Q, becomes, Q = dU = (P dP/m0 − kqdq) = d(P 2 /2m0 − kq2 /2), dQ = 0.

(4.243) (4.244)

This result is valid for both the strong and the weak constraint. The factor in brackets is to be recognized as the Lagrangian for the Harmonic Oscillator. When the bracket factor is a constant, then the extremal evolutionary process is globally adiabatic, as with such a constraint, Q ⇒ 0. Solve equation (4.239) for P = m0 ve and substitute in equation (4.240) to obtain the second order differential equation for the Harmonic Oscillator, m0 d2 q/dt2 + kq = 0.

(4.245)

4.5.6 The Damped Harmonic Oscillator - a Sol Geometry This next example is extraordinary, for the geometry so generated by the nonintegrable 3-form of Pfaff dimension 3 is an example of the "Sol" class of Thurston’s eight 3D geometries. The 4D Action 1-form will be assumed to be of the format given in equation (4.203). Suppose the Hamiltonian is time dependent and of the form: Damped Harmonic Oscillator Φ = (P 2 /2m0 ) exp−(b/m0 )t +(kq 2 /2) exp+(b/m0 )t , dΦ = P/m0 exp−(b/m0 )t dP + kq exp+(b/m0 )t dq −(b/m0 ){(P 2 /2m0 ) exp−(b/m0 )t −(kq 2 /2) exp+(b/m0 )t }.

(4.246) (4.247) (4.248)

As the mass-energy function is not explicitly dependent upon Mc2 the Pfaff topological dimension of the Action 1-form (4.198) is 3. A contact structure is

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established, and there exists an extremal Hamiltonian vector field, generated from Φ: Fe = dP/dt = −∂Φ/∂q = −kq exp+(b/m0 )t . ve = dq/dt = +∂Φ/∂P = +(P/m0 ) exp−(b/m0 )t .

(4.249) (4.250)

Use the second equation (4.249) to solve for P = mv exp+(b/m)t , and then substitute into the first equation, dP/dt = d(m0 ve exp+(b/m0 )t )/dt, = {m0 d2 q/dt2 + bdq/dt} exp+(b/m0 )t = −kq exp+(b/m0 )t , which, upon re-arrangement and ignoring the exponential factor, becomes the classic kinematic second order equation for the damped harmonic oscillator, (m0 d2 q/dt2 + bdq/dt + kq) exp+(b/m0 )t = 0.

(4.251)

It is important to note that the momentum, P, is not equal to the classic representation m0 V, but is explicitly dependent upon time, P = (m0 ve exp+(b/m0 )t ).

(4.252)

The Work 1-form vanishes for the extremal evolution of the damped Harmonic Oscillator, and the internal energy becomes: V4 = ([ve , 1, Fe , −]), W = i([ve , 1, Fe , −])dA ⇒ 0, U = i([ve , 1, Fe , −])A, = P ve − Φ = P 2 /2m0 exp−(b/m0 )t −kq2 /2 exp+(b/m0 )t ,

(4.253) (4.254) (4.255) (4.256)

such that the evolutionary system of equations becomes: (4.257)

L(λV4 ) A = Q = dU, −(b/m0 )t

+(b/m0 )t

= P/m0 exp dP − kq exp dq −(b/m0 )Φdt, = −P kq/m0 dt − kqP/m0 dt − (b/m0 )Φdt = (−2Fe ve − b/m0 )dt = Γe (q, t, P )dt. L(λV4 ) dA = dQ = dΓe (q, t, p)ˆdt 6= 0. QˆdQ = 0. Reversible.

(4.258) (4.259) (4.260) (4.261) (4.262)

The non-intuitive result is that the damped harmonic oscillator is thermodynamically reversible! The 1-form of heat, Q, is not zero and is of Pfaff dimension 1 in the undamped case, but is of Pfaff dimension 2 in the damped case. Damped and Undamped Harmonic Oscillators

(4.263)

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Electrodynamic, Hydrodynamic, and Mechanical Thermodynamic systems

In both cases Q 6= 0, Undamped (Pfaff dimension Q = 1) dQ = 0, Damped (Pfaff dimension Q = 2) dQ = 6 0.

(4.264) (4.265) (4.266)

The damped Harmonic Oscillator is an example of physical system for which the Helmholtz conservation law is not valid, L(λV4 ) dA 6= 0. The process cannot be a symplectomorphism. 4.5.7

Rayleigh dissipation

The 4D Action 1-form will be assumed to be of the format given in eq. (4.203). Suppose the Hamiltonian is time dependent and of the form, Φ = (P 2 /2m0 ) exp−γq/m0 +(kq 2 /2) exp+γq/m0 .

(4.267)

Then a Hamiltonian solution is of the form: Fe = dP/dt = −∂Φ/∂q = [+γP 2 /2m20 ] exp−γq/m0 +[−kq − (kγ/2m0 )q2 ] exp+γq/m0 . ve = dq/dt = +∂Φ/∂P = +(P/m0 ) exp−γq/m0 .

(4.268) (4.269)

Solving the second equation for P and inserting into the first equation yields the kinematic second order equation for the quadratically damped non-linear oscillator; namely, (4.270) m0 d2 q/dt2 + (γ/2)(dq/dt)2 + kq(1 + (γ/2m0 )q) = 0. The Work 1-form vanishes for the extremal evolution of the Rayleigh Oscillator, and the internal energy becomes: V4 = ([ve , 1, Fe , −]), W = i([ve , 1, Fe , −])dA ⇒ 0, U = i([ve , 1, Fe , −])A, = P ve − Φ = P 2 /2m0 exp−(γ/m0 )q −kq2 /2 exp+(γ/m0 )q ,

(4.271) (4.272) (4.273) (4.274)

such that the evolutionary system of equations becomes: L(λV4 ) A = Q = dU = (−2Fe ve )dt = Γe (q, t, P )dt, L(λV4 ) dA = dQ = dΓe (q, t, p)ˆdt, QˆdQ = 0, Reversible.

(4.275) (4.276) (4.277)

The process is not a symplectomorphism, as the 2-form dA is not an evolutionary invariant.

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4.5.8 Wave propagation The 4D Action 1-form will be assumed to be of the format given in equation (4.203). Suppose the Hamiltonian is time dependent and of the form, Φ = (P 2 /2m0 ) exp−αq+βt +(kq 2 /2) exp+αq−βt .

(4.278)

Then a Hamiltonian solution is of the form: dP/dt = Fe = −∂Φ/∂q, = [+αP 2 /2m20 ] exp−αq+βt +[−kq − (kα/2m0 )q2 ] exp+αq−βt , dq/dt = ve = +∂Φ/∂P = +(P/m0 ) exp−αq+βt .

(4.279) (4.280) (4.281)

Solving the second equation for P and inserting into the first equation yields the kinematic second order equation for the wave propagated non-linear oscillator, {m0 d2 q/dt2 +βm0 (dq/dt)+(γ/2)(dq/dt)2 +kq(1+(γ/2m0 )q)} exp+αq−βt = 0. (4.282) Homework: Compute Q and dQ for the Wave 4D Action 1-form. 4.5.9 Helmholtz processes with variable connectivity Helmholtz type B processes (see Chapter 1.9.4) imply that the one-dimensional period integrals (first Betti numbers) are NOT necessarily evolutionary invariants of the Helmholtz class of processes. Such Type B processes then must represent topological evolution and are NOT homeomorphisms of the Cartan topology. In the sense that the first Betti number is related to the "hole" count in some surface, and as the process is assumed to be continuous, the interior "holes" can disappear by collapse and pasting, but new holes can be created only by distortion and entrapment at a piece of a boundary. However, as QˆdQ = 0, Helmholtz processes of both type A and type B are thermodynamically reversible, even though the type B processes involve topological evolution. In other words, Remark 37 Topological change is necessary, but not sufficient for thermodynamic irreversibility. For C2 functions, the vector fields of the Helmholtz class Part A are constrained by the equations of closure, d[i(W)dA] = dW = 0.

(4.283)

Although, the Helmholtz process indicates that the Work 1-form is closed, it need not be exact. Closed integrals of the Work, where the integration path is a cycle, need

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H not be zero, W 6= 0. According to deRham, the closure condition can be satisfied by exact or harmonic contributions. That is, suppose, W ≡ dP + γ = dP + Γ [(ψ∗ dψ − ψdψ ∗ )/(ψ∗ ψ)].

(4.284)

The symbol, ψ, represents some complex function. The first term dP is the analogue of "pressure gradient" in a hydrodynamic system and gives no contribution to the line integral around a cycle, or a boundary. The second term is a harmonic component that yields integer values for integrations around cycles that enclose the zeros (the holes) of the denominator. The creation or the destruction of the harmonic components represents a change of topology. As the evolution is presumed to be continuous, the production of "holes" can occur only on segments of cycles or boundaries and not in the "bulk" interior. Interior production would be a cutting process which is discontinuous. As dA is an evolutionary invariant for Helmholtz processes, the "holes" must be produced in equal and opposite pairs. 4.5.10 The Master Equation Consider the 1-form of electromagnetic action, A, given above (4.110). In electromagnetic format, for all processes, it follows that, curl E + ∂B/∂t = 0, div B = 0.

(4.285)

Consider an abstract process V4 ⇒ ρ{V k , 1} = ρ{V, 1} on the space {xk , t}. Then for type A Helmholtz processes, W = i(V)dA = {−ρ(E + V × B)k dxk + ρ(V ◦ E)dt} 6= 0.

(4.286)

The covariant spatial components of the Work 1-form are to recognized as the Lorentz force per unit charge to within the parametrization factor, ρ, f(Lorentz) = −ρ(E + V × B).

(4.287)

Note that the time-like component becomes the "local dissipative" power, P = ρ(V ◦ E). The Lorentz force has been derived on topological grounds, and has not been injected into the theory on phenomenological grounds. The Helmholtz closure conditions d[i(W)dω] = 0 require that, curl f(Lorentz) = 0,

(4.288)

∂f(Lorentz) /∂t + grad P = 0.

(4.289)

and For simplicity, let ρ = 1, such that the closure conditions of Type A Helmholtz processes becomes the set of constraints which define the Master Equation of a "perfect plasma": curl f(Lorentz) = −{curl(E) + curl(V × B)} = 0, Master Equation ⇒ ∂B/∂t − curl(V × B) = 0.

(4.290) (4.291)

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The same results and conclusions apply to Hydrodynamics as well as all physical systems that can be encoded in terms of 1-form of Action with coefficient functions defined over a four-dimensional variety. The statement demonstrates the concept of topological universality. 4.5.11

The Isovector Class of Irreversible Processes

On an even-dimensional symplectic space it has been noted that there does not exist a unique extremal field, but there does exist a unique Topological Torsion field, whose components to within a factor are completely determined in terms of the functions that define the physical system encoded as a 1-form of Action. It has been established that evolution in the direction of the Topological Torsion vector can be thermodynamically irreversible. One of the key features of the Topological Torsion vector is that it is "conformal", or "homogeneous", in the sense that L(T4 ) A = ΓA. It is also true that the Torsion direction field is associated and therefore locally adiabatic as i(T4 )A = 0. Now consider those "Isovector" fields such that L(V4 ) A = ΓA , but where the processes they represent are not (necessarily) associative, but where the differential of the internal energy, U, is zero . Let V4 = {vk , 1, Fk , R} be the symbols for the component functions of a direction field relative to the coordinate variables (q k , t, Pk , Φ). Define: Φ(q k , t, Pk , v k ) dΦ −dL A dA AˆdA

= = = = = =

{Pk v k − L(qk , v k , t)}, v k dPk + Pk dv k − dL, −∂L/∂vk dv k − ∂L/∂q k dq k − ∂L/∂tdt, Pk dqk − Φdt, dPk ˆdqk − dΦˆdt, Pk dΦˆdqk ˆdt − ΦdPk ˆdqk ˆdt.

(4.292) (4.293) (4.294) (4.295) (4.296) (4.297)

Next apply the Lie differential operator to the 1-form of Action A and constrain the system to consider those vector fields that are "Isovectors". Then after performing the operations and gathering the algebra, note that: L(V4 ) A Fk dq − v dPk − Rdt + dΦ + dU U dU +dL k

k

k

Fk dq − Rdt + Pk dv

k

= = = = =

ΓA, Γ{Pk dq k − Φdt}, {Pk vk − Φ}, v k dPk + Pk dv k − dΦ, ∂L/∂qk dq k +∂L/∂v k dv k +∂L/∂tdt, = Γ{Pk dq k − Φdt}.

(4.298) (4.299) (4.300) (4.301)

(4.302) (4.303)

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The equations that define these "conformal-homogeneous" direction fields are given by solutions to the Pfaffian expression, Pk dv k + (Fk − ΓPk )dq k − (R − ΓΦ)dt = 0.

(4.304)

Note that this expression can be interpreted as a second order equation if it is assumed that dq k = ±v k dt. First, assume the plus sign. Then, Pk dv k − (R − Fk v k + ΓL)dt = 0.

(4.305)

Next assume the minus sign. Then, Pk dv k − (R + Fk v k − 2ΓPk vk + ΓL)dt = 0.

(4.306)

Remark 38 It appears that evolution (defined by the "Isovector" direction fields) of the dynamical system in the forward direction is not the same as motion in the backwards direction, a result which demonstrates the irreversibility of motion along the torsion vector. The two Pfaffian equations are the same only when, (2ΓPk vk − Fk vk ) = Fk v k ,

(4.307)

(Fk − ΓPk )vk = 0.

(4.308)

which requires that

Remark 39 Motion in the forward and backward directions are not the same unless the dissipative Forces are proportional to the momenta, or the deviation from this form of dissipation is orthogonal to the velocities. The result does not invoke the criteria of local or global adiabaticity. In engineering terms, either the force is viscously dissipative, (Fk − ΓPk ) = 0, or the velocities vk are orthogonal to the difference (Fk − ΓPk ), thereby defining a special type of "no-slip" condition associated with "rolling". The evolution proceeds from the initial state irreversibly to the "steady state" case of rolling without slipping in the presence of friction. Open question: Under what circumstances does (Fk −Γpk ) ≈ µ(Normal))? This idea should be put into correspondence with a bowling ball which initially is given rotational energy - as overspin or underspin - as well as translation energy before making contact with the bowling alley surface. This problem is discussed in detail in Volume 2 [274], and in Chapter 2.9 of this volume. There are four evolutionary possibilities, that come in two pairs. In one circumstance the linear momentum reverses sign; in the second circumstance the angular momentum reverses sign. The choice depends upon the relative amounts of rotational and translational

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193

energy given in the initial state. In the "steady state" case of no slipping, the motion is such that v = λω. Then the rotational energy = βmλ2 ω 2 /2 and the translation energy is mλ2 ω2 /2 = mv 2 /2. At first glance it would appear that the "frictional" forces do not equilibrate to where the two degrees of freedom have "equal" energy. In the underspin case the motion must correspond to the dq + vdt = 0 constraint, and the overspin case corresponds to the dq − vdt = 0 assumption. The extremal path must ultimately be where (Fk − Γpk )vk = 0, and it must be approached from two different initial conditions. (If the coordinates were closed the constraint leads to a limit cycle. The limit cycle constraint must be the analogue of (Fk − Γpk )v k = 0.) A special subclass of processes belong to the characteristic class. In addition to satisfying the isovector condition with Γ = 0, characteristic vectors satisfy the "associated" equation, U = i(λViso )A = λ(V k Pk − ε) = 0.

(4.309)

4.5.12 Mass in symplectic systems For a Lagrange system, the 3-form of Topological Torsion becomes, AˆdA = Pk dΦˆdq k ˆdt − ΦdPk ˆdq k ˆdt, = (v j dPj + Pj dv j ) − ∂L/∂v j dv j )Pk ˆdq k ˆdt −{Pj vj − L(q k , v k , t)}dPk ˆdq k ˆdt, = {Pk (+Pj dv j − dL) + L(qk , v k , t)dPk }ˆdqk ˆdt.

(4.310) (4.311) (4.312)

The reduction of the symplectic manifold to a Lagrangian submanifold requires that the 3-form of Topological Torsion should vanish. A second possibility exists for the Frobenius integrability constraint to be valid, but now the condition is not valid globally over the 4D domain. However, the 3-form of topological torsion will vanish when, [LdPk + Pk (Pj − ∂L/∂v j )dv j ] = [α(t, q, v, P )dqk + β(t, q, v, P )dt] ⇒ 0.

(4.313)

This constraint can be satisfied on domains even when (Pj − ∂L/∂v j ) 6= 0. Hence such systems admit an integrable submanifold in a symplectic 4D domain. It is these integrable submanifolds of a symplectic system that are of interest to this subsection. Consider the case where the RHS of the above equation vanishes. Then the non-zero function m(t; q, v, p) defined in a symplectic domain as, m(t; q, v, p) = dPk /dv k = Pk (Pj − ∂L/∂vj )dv j ,

(4.314)

plays the role of a "mass" (at least for closed systems). The mass can be zero, if the momenta are canonical, or non-zero, which is the non-canonical (symplectic) case. For constant mass, the resulting singular hypersurface in the 4D domain yields a constrained subspace of (t; q, v, p) upon which the 1-form of Action satisfies the

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Frobenius complete integrability theorem. Examples will be given below where the symplectic orbits reside on this constant mass surface. The constant mass hypersurface in symplectic systems is the analogue of the constant energy hypersurface in extremal systems.

Chapter 5 THE THERMODYNAMICS OF DYNAMICAL SYSTEMS OF NON-LINEAR ODE’S 5.1

Introduction

In previous Chapters, especially Chapter 3.3, it was demonstrated how the Jacobian matrix of the covariant components Ak of any 1-form of Action, A (which were used to encode the thermodynamic properties of a particular physical system), could be used to construct a hypersurface in the space of variables, {x, y, z, t; ξ}. The hypersurface was defined in terms of the zero set of a Cayley-Hamilton function, a 4th order polynomial in the matrix eigenvalue (possibly complex) parameter, ξ. It was demonstrated that this function could be described in terms of similarity invariant coefficients, {XM , YG , ZA , TK }, and could be put into correspondence with a universal Phase function of a (deformed) van der Waals gas. The universality is associated with the assumption that the similarity coefficients, {XM , YG , ZA , TK }, can be considered as mapping functions from a base space {x, y, z, t} to a set of universal "coordinates", {XM , YG , ZA , TK }. The method is similar to the constructions used in the universal formulations of Frenet - Serret space curves in terms of the arclength, s, curvature, κ, and torsion coefficients τ . In this section, a similar analysis will be applied to dynamical systems, where the input information is often a system of N-1 component functions, V k , of a direction field, V = [V k , 1], in a space of N dimensions. The N-dimensional direction field, V , can be used to define (to within a factor) an (N-1)-form, C = −i([V, 1])Ω4 , on the domain of interest. The conjecture to be exploited is that thermodynamic properties and experimental experience of a van der Waals gas are topological properties of all C2 continuous systems. If a mapping could be determined such that the covariant coefficients Ak can be determined from the contravariant functions, [V k , 1], then the methods of the previous chapters could be applied directly. If the correspondence could be made, then there exists properties of the dynamic system that correspond to the thermodynamic (topological) singularities, such as the existence of a Critical Point, the Spinodal lines of ultimate instability (an edge of regression), and the Binodal decomposition into two thermodynamic phases representing a bifurcation change of phase. In addition, the critical isotherm could be determined indicating domains where complex spinor direction fields, not only real vector fields, should be

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considered. The imposition of a metric field on the domain is a possible constraint. However, the theme of this monograph is to NOT to utilize a metric, or a connection, and consider properties that are independent from size and shape. What alternate choices are there? The objective is to find the proper format for the functional form of the 1-form of Action A that represents a physical system. Once the Action 1-form, A, is formulated, then the theory of Pfaff Topological dimension can be used to extract topological information and the possibility of irreversible topological evolution. From the Jacobian matrix of the functional coefficients, similarity invariants can be determined that lead to universal thermodynamic phase properties of the physical system. The question then arises: if given a dynamical system, what is the corresponding 1-form of Action? Unfortunately, a procedure for extracting the 1-form of Action is not unique. In that which follows, a projective correspondence between the 1-form of Action, A, and the dynamical system (N-1)-form, C, will be assumed, such that AˆC = 0. The method will then be applied to a number of familiar examples. Note that the direction field C may have a global zero divergence, and in such situations it would be better described as a 4D "current". Also it is true that all (N-1)-forms admit an infinity of integrating factors, ρ, such that the product, ρC, has zero divergence. The N coefficients of the 1-form of Action, A, and the N coefficients of the 3-form, ρC, of the dynamical system can be related by assuming some projective measure function, ρ, on the four-dimensional variety. In this sense, the two formulations are projectively "dual" to one another. It follows that there should exist recognizable topological features of a van der Waals gas in dynamical systems, and that these £ features ¤ can be determined from the similarity invariants of the Jacobian k matrix, J(V ) , for the dynamical system, by comparison to£ the results obtained by ¤ k the Jacobian matrix [J(Ak )] . Note that Jacobian matrix, J(V ) = [∂(V k )/∂xm ], is a collineation, while the Jacobian matrix, [J(Ak )] = [∂(Ak )/∂xm ] , is a correlation. In that which follows the two dual points of view are to be exploited. In summary, given a dynamical system, a projective dual 1-form, A, can be constructed to yield thermodynamic non-equilibrium features of the dynamical system. 5.2

The Dynamical System as an Exterior Differential System

The classical format of a dynamical system is given by a set of velocity vectors that are presumed (constrained to 3 spatial dimensions in this section) to satisfy the derivative equations, dx/dt = dy/dt = dz/dt =

Vx (x, y, z, t), Vy (x, y, z, t), Vz (x, y, z, t).

(5.1) (5.2) (5.3)

The Dynamical System as an Exterior Differential System

197

The unstated assumption is that a topological limit process exists such that the derivative, or differential quotient, dxk /dt, makes sense. In the language of exterior differential (not derivative) forms, the limit process is not specified explicitly, and the dynamical system becomes restated as a set of exterior differential 1-forms, not necessarily equal to zero. If the exterior differential system is to reduce to the dynamical system, then the 1-forms must vanish to produce what is defined herein as the state of "Kinematic Perfection". ωx = dx − Vx (x, y, z, t)dt ⇒ 0, ω y = dy − Vy (x, y, z, t)dt ⇒ 0, ωz = dz − Vz (x, y, z, t)dt ⇒ 0.

(5.4) (5.5) (5.6)

The exterior differential system (all three 1-forms vanish) becomes equivalent to a dynamical system on a 1-D "solution submanifold" of the 4D domain {x, y, z, t}. The variable, t, is considered as an arbitrary parameter. For autonomous vector fields the velocity components are independent from time, ∂Vk /∂t = 0. The solution to the dynamical system is in effect a parametrization of the parameter, t, to the space curve, Cparametric , in 4D space, where for Kinematic Perfection, [Vk , 1] is a tangent vector to the curve Cparametric . Off the solution submanifold, the non-zero values for the 1-forms, ωk , can be interpreted as topological fluctuations from "Kinematic Perfection". If "Kinematic Perfection" is not exact, then the three 1-forms ω k are not zero, and have a finite triple exterior product that defines a (N-1=3)-form in the 4 D space. From the theory of exterior differential forms it is the intersection of the three hypersurfaces ω k that creates an implicit curve Cimplicit in 4D space: Cimplicit = ωx ˆωy ˆωz , = dxˆdyˆdz − Vx dyˆdzˆdt + Vy dxˆdzˆdt − Vz dxˆdyˆdt = −i([V, 1])Ω4 .

(5.7) (5.8) (5.9)

It is possible to consider multiple parameter systems of dynamical systems, which can also be considered to be a topologically constrained exterior differential system [34] of Pfaffian 1-forms. That is, on an N = 4D domain with a volume element, Ω4 , it is possible to specify a set of N − 1 = 3 Pfaffians, which are presumed to vanish on a 1-D submanifold. These constraints lead to a multiple parameter Dynamical System: ω1 (x, y, z, t, dx, dy, dz, dt) ⇒ 0, ω2 (x, y, z, t, dx, dy, dz, dt) ⇒ 0, ω3 (x, y, z, t, dx, dy, dz, dt) ⇒ 0.

(5.10) (5.11) (5.12)

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The intersection of these three hypersurfaces ω k again forms an implicit curve in 4D. The discussion brings to mind the dualism between points (rays) and hypersurfaces (hyperplanes) in projective geometry. If a ray (a point in the projective 3 space of four dimensions) is specified by the 4-components of the 4D vector [V, 1] multiplied by any non-zero factor, κ, (such that [V, 1] ≈ κ[V, 1]), then the equation of a hyperplane is given by the expression [A, −φ] such that hγ[A, −φ]| ◦ |κ[V, 1]i = 0.

(5.13)

The principle of projective duality [168] implies that (independent from the factors γ and κ) φ = A ◦ V.

(5.14)

The symbolism has been chosen deliberately to make a correspondence between the dual 1-form, A, and the dual 3-form, C, such that the projective duality condition AˆC = 0 yields a 1-form (to within a factor) that is to be recognized as the format of an electromagnetic system (see section 3.3). A particularly easy choice is to assume that (to within a factor, say m)

Ak ⇔ Vk , and φ = V ◦ V, A = Ak dxk − Ak V k dt.

(5.15) (5.16)

such that the functions Ak (x, y, z, t) ⇔ V k (x, y, z, t), representing the dynamical system. The E and B fields can be computed for this Action 1-form for arbitrary direction fields, but the construction can rapidly lead to enormous algebraic difficulty that hides some of the central issues. 5.2.1

Frenet-Serret theory

When the Velocity field admits parametrization, such that dω k = 0, then the analysis reduces to the familiar format of the Frenet - Serret theory, where the differential dt is converted into arclength, ds: arclength ds =

p Vm V m dt.

(5.17)

(The arclength coordinate will turn out to be useful in terms of chemical reactions where it is to be compared with De Donder’s "extent" of a chemical reaction.) The √ m arclength parameter requires that the product Vm V be a function of time alone, such that ds is a perfect differential. The constraint is valid for all parametric formulations of kinematics which map t into x, y, z, but may not be valid for implicit

The Dynamical System as an Exterior Differential System

representations. It follows that this "Frenet-Serret" format yields: p p A = Vk dxk − Vm V m ds = Vm V m {tk dxk − ds}, p tk = Vk / Vm V m a unit velocity vector, A = γA0 = γ{tk dxk − ds}. 5.2.2

199

(5.18) (5.19) (5.20)

Implicit theory

However, it is not necessary to assume a parametrization map exists, for under the rules of projective geometry, the factor γ can be any non-zero function, and the 1form γA is considered to be (projectively) equivalent to A. In particular re-write the dual 1-form as A = Vk dxk − Vm V m dt = Vk dxk − λdt, = Ak dxk − φdt, in EM notation, λ = (Vm V m ) = β 2 c2 for some constant c2 .

(5.21) (5.22) (5.23)

Now construct the Pfaff sequence for this 1-form, borrowing from the notation of an electromagnetic system, where all the terms are worked out. The vector potential becomes the functions that make up the velocity field, and the scalar potential becomes λ. Observe that if the components of E and B vanish, then dA = 0, and the maximum Pfaff dimension of A is 1. Theorem 6 The dual 1-form "E" fields and "B" fields of a dynamical system must vanish if the dynamical system represents an equilibrium thermodynamic system. If dA is not zero, then construct the 3-form, AˆdA. If the 3-form is zero, then the Pfaff topological dimension is 2. A necessary and sufficient condition for the 3-form to vanish can be written in EM vector format:

Pfaff dimension 2 requires AˆdA = 0, In EM notation, E × A + Bφ = 0.

(5.24) (5.25)

If the Topological Torsion 3-form does not vanish, the Pfaff dimension is 3 or more, and the dynamical system has a dual 1-form that is equivalent to a non-equilibrium thermodynamic system. The Pfaff dimension remains 3 unless the 4-form dAˆdA does not vanish. If dAˆdA is not zero, then the Pfaff dimension is 4. The criteria can be expressed in EM vector format as:

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

Pfaff dimension 3 requires dAˆdA In EM notation, K Pfaff dimension 4 requires dAˆdA In EM notation, K

= = = =

K = 0, 2E ◦ B = 0, K 6= 0, 2E ◦ B 6= 0.

(5.26) (5.27) (5.28) (5.29)

The implicit hypersurface defined by the equation, 2(E ◦ B) = 0, represents a hypersurface which divides the two possible orientations in the 4D space-time. In addition, the implicit function may contain chiral parameters, such as a term involving either right handed or left handed rotation. Then the hypersurface has more than one component depending upon the chirality. Examples will be given below where the chiral effects of rotation can be related to Hopf bifurcations. 5.2.3 The dynamical system defines an adiabatic process It should be noted that any "process" with the direction field of the dynamical system, V4 = [V x , V y , V z , 1] acting on the projective dual 1-form, A, is adiabatic, for by construction, adiabatic (associated) i(V4 )A = 0.

(5.30)

Such vector fields were defined as "associated" vector fields. This fact does not mean that the kinematic process, V4 , is necessarily a thermodynamically reversible process. Moreover it does not mean that the process is isentropic, for it is not necessary that Q be of Pfaff dimension 2 (which is required if Q = T dS). By duality, given a 1-form of Action, it is possible (to within a factor) to construct the projective dynamical system. A very special case arises if the kinematic vector direction field V4 represents an extremal process, for then it is required that: If extremal, i(V4 )dA = 0, and therefore L(V4 ) A = 0.

(5.31) (5.32)

Such an adiabatic extremal process generates no thermodynamic heat flow, as Q = 0. Such vector fields have been defined as "characteristic" vector fields. Such a constrained result should focus attention on the limitations of those ubiquitous extremal processes used in the calculus of variations. 5.2.4 Conservative distributions It should be pointed out that there are an infinite number of distributions, ρ, such that the divergence of dC = d(ρi(V4 )Ω4 ) ⇒ 0, thereby reducing the 4D symplectic manifold to a 3D contact submanifold. The question is: Can this integrating factor be related to the thermodynamic molar density n/V ? Are there certain molar distributions that generate integrating factors which will reduce the dynamical system

The Dynamical System as an Exterior Differential System

201

from a turbulent Pfaff topological dimension 4 to a steady state of Pfaff topological dimension 3? Are there molar distributions that represent the thermodynamics of a closed system, decaying to the equilibrium state? At the time of writing, these questions do not have a completely satisfactory answer. In section 2.10.8 it was demonstrated that given a physical system as a 1-form, A, it was possible to construct the adjoint matrix of the rescaled Jacobian matrix b such that the J[Ak /λ] and thereby produce a closed "adjoint" current 3-form, J, b product (Ak ˆJ)/λ was precisely equal to the cubic similarity invariant, ZA of J[Ak /λ]. The fourth order similarity invariant of J[Ak /λ] became zero by the appropriate choice for λ. This special situation implied that the thermodynamic system (scaled by 1/λ) was of Pfaff dimension 3. If it is possible to reverse the procedure, then given a dynamical system, from the 3-form C with appropriated integrating factor it should be possible to deduce the 1-form Ak /λ. Another procedure would be to devise a Hamiltonian function such that the coordinate velocities of the dynamical system are generated by the classic Hamiltonian procedures. A 1-form of Action can then be defined as: A = Pk dxk + Hdt = L(xk , V k )dt + Pk (dxk − V k dt), = Pk dxk + (L(xk , V k ) − Pk V k )dt.

(5.33) (5.34)

5.2.5 Characteristic Polynomial Yet another procedure that yields interesting results follows from the construction of the Jacobian matrix, ⎡ ⎤ ∂V 1 /∂x ∂V 1 /∂y ∂V 1 /∂z ∂V 1 /∂t ⎢ ∂V 2 /∂x ∂V 2 /∂y ∂V 2 /∂z ∂V 2 /∂t ⎥ ⎥ (5.35) J[V k , −λ] = ⎢ ⎣ ∂V 3 /∂x ∂V 3 /∂y ∂V 3 /∂z ∂V 3 /∂t ⎦ , −∂λ/∂x −∂λ/∂y −∂λ/∂z −∂λ/∂t

and studying the similarity invariants of its characteristic polynomial, Θ(x, y, z, t; ξ) = ξ 4 − XM ξ 3 + YG ξ 2 − ZA ξ + TK ⇒ 0. If the kinematic dynamical system is autonomous, the format of the transposed affine map, ⎡ ∂V 1 /∂x ∂V 1 /∂y ⎢ ∂V 2 /∂x ∂V 2 /∂y J0 [V k , −λ] = ⎢ ⎣ ∂V 3 /∂x ∂V 3 /∂y −∂λ/∂x −∂λ/∂y

(5.36)

the Jacobian matrix reduces to ⎤ 0 0 ⎥ ⎥, 0 ⎦ 0

(5.37)

Θ0 (x, y, z, t; ξ) = (ξ 3 − XM ξ 2 + YG ξ 1 − ZA )ξ ⇒ 0.

(5.38)

∂V 1 /∂z ∂V 2 /∂z ∂V 3 /∂z −∂λ/∂z

and the characteristic polynomial has the simpler format (as TK = 0),

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

The cubic polynomial permits all autonomous dynamical systems to be put into correspondence with the van der Waals gas. In particular, a search for the critical point, binodal and spinodal lines, and the critical isotherm can be evaluated in terms of the points in the space, {x, y, z}. In general, the universal features of the similarity invariants of the characteristic polynomial and their zero sets can be used to define domains of topological singularities. These evolutionary solutions can have topological singularities where they become structurally unstable, and the topology changes (a change of thermodynamic phase). It is remarkable that systems with TK = 0 can still support thermodynamic irreversible processes, for which the dual 1-form has Pfaff dimension 4. It might be concluded that as TK = 0 the topological dimension of the physical system is reduced from 4 to 3. Examples given below demonstrate that this is not the case. The moral of this observation is that conclusions based on symmetries (such as statistical averages) sometimes hide physical features. Remark 40 All of the features associated with the Cartan topological development of non-equilibrium thermodynamics applies to the analysis of a Dynamical System. The Dynamical System has the equivalent of a critical point, a Spinodal line of ultimate instability of phase transitions, a Binodal line of coexistence, an entropy integrating factor, and a critical isotherm above which there can exist complex Hopf and spinor bifurcations leading to minimal surfaces, as submanifolds. 5.3

The van der Waals Gas as a Dynamical System

As a first example, and in order to give credence to the ideas that a dynamical system has thermodynamic properties, a van der Waals gas will be formulated explicitly in terms of a Dynamical System [202]. The dynamical system chosen for demonstration purposes is integrable. First, the properties of the van der Waals gas will be written in terms of specific (molar) values of the extensive variables, S ⇒ s = S/n, U ⇒ u = U/n, V ⇒ v = V /n.

(5.39) (5.40) (5.41)

The classic phase function will be given the notation of a Lagrangian function, L(s, v, u), written in terms of the rescaled extensive variables: L(s, v, u) = 8 exp(3s/8)/(3v − 1) − 3/v − u ⇒ 0, or u = 8 exp(3s/8)/(3v − 1) − 3/v, such that Te = ∂L/∂s = 3exp(3s/8)/(3v − 1)}, Pe = ∂L/∂v = {24 exp(3s/8)/(3v − 1)2 − 3/v2 } = 8Te/(3v − 1) − 3/v2 , −1 = ∂L/∂u.

(5.42) (5.43) (5.44) (5.45) (5.46) (5.47)

The van der Waals Gas as a Dynamical System

203

Consider the non-linear flow on the variety with coordinates (s, v, u) leading to the dynamical system, ds/dt = ∂L/∂s = Te = (3s/8)/(3v − 1)}, dv/dt = ∂L/∂v = −Pe = +3exp(3s/8)/(3v − 1)2 − 3/v 2 ,

du/dt = ∂L/∂u = −1.

(5.48) (5.49) (5.50)

The pressure equation becomes −Pe = +3exp(3s/8)/(3v − 1)2 − 3/v2 ,

(5.51)

e ρ3 − 3e ρ2 + ({8Te + Pe}/3)e ρ − Pe = 0.

(5.52)

P ⇒ Pe = P/Pc , T ⇒ Te = T /Tc .

(5.53)

e ρ3 − 3e ρ2 + ({8Te + Pe}/3)e ρ − Pe = 0, 3 2 e ρ + YG e ρ − ZA = 0, e ρ − XM 3 2 ρ − 3Mρ + 3Gρ − A = 0.

(5.55) (5.56) (5.57)

M = 1 = XM /3, G = ({8Te + Pe}/3) = YG ,

(5.58)

and with the substitutions (3v − 1)}exp(3s/8)/3 = Te, and v = 1/e ρ, becomes the classic cubic expression for the van der Waals gas: The intensive variables are scaled by values of the form,

(5.54)

The Jacobian of an arbitrary 3D vector field has rank 3, unless the determinant of the Jacobian vanishes. The implication is that the Pfaff topological dimension of such a system is 3, so in principle the "gas" representing the dynamical system is not globally in equilibrium. It is useful to compare the van der Waals gas equation of state with the two formats of the Cayley-Hamilton polynomial of the Jacobian matrix of an arbitrary 3D vector field:

By comparing the formulas, where M, G, K are the 3D averages of the similarity invariants XM , YG , ZA of the 3 × 3 Jacobian matrix,

A

= Pe = ZA ,

(5.59) (5.60)

a correspondence can be made between the arbitrary dynamical system and a van der Waals gas. Recall that the average similarity invariants take on the values

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

M = 1, G = 1, A = 1 at the critical point. It becomes apparent that the van der Waals gas presumes that the Mean curvature, M , remains fixed at its critical point value. The isotherms (fixed values of Te on the implicit surface defined by this cubic polynomial) have a projection onto the Pe, e ρ plane and are displayed in the following figures (repeated from Chapter 3):

Figure 5.1 Classical van der Waals gas isotherms

Figure 5.2 Classical van der Waals gas critical isotherm

The van der Waals Gas as a Dynamical System

205

The critical isotherm has a fundamental shape that should be memorized for it is a topological deformation invariant (and is the central line of hysteresis phenomena). The implicit surface L(s, v, u) = 0, defined in the space of variables, {s, v, u}, can be used to define a gradient vector field, C(s, v, u) = [∂L/∂s, ∂L/∂v, ∂L/∂u] = [Te, −Pe, β],

(5.61)

n = C(s, v, u)/(β 2 + Te2 + Pe2 )1/2 , ns = Te/(β 2 + Te2 + Pe2 )1/2 ,

(5.62)

leading to a unit "surface normal" vector field, n, by means of the Gauss map:

nv = −P/(β + Te2 + Pe2 )1/2 , nu = −1/(β 2 + Te2 + Pe2 )1/2 , λ = (β 2 + Te2 + Pe2 )1/2 . 2

(5.63) (5.64) (5.65) (5.66)

For the van der Waals gas, the term β 2 = 1. The 3D Jacobian Dyadic of unit normal field n always has a determinant equal to zero, so the domain of topological dimension 3 has been reduced to the case of isolated systems of Pfaff dimension 2 by use of the Gauss map normalization. The Cayley-Hamilton phase function for the normalized system becomes ξ 3 − 3Mξ 2 + 3Gξ − A = 0.

(5.67)

However, now A = 0 (Pfaff dimension 2), and M and G are related to the Mean and Gauss curvatures of the implicit surface with the a surface normal direction field, n. The dynamical system for this vector field is represented by the equations, ω x = ds − Te(s, v, u)dτ ⇒ 0, ω y = dv −(− Pe(s, v, u))dτ ⇒ 0, ω

z

= du − (−1)dτ ⇒ 0.

(5.68) (5.69) (5.70)

Note that by construction the differential du is symbolically equivalent to dτ . It is possible to proceed in 2 ways. The first method uses the technology of Frenet-Serret space curves, using the unit surface normal field as a unit tangent field n ⇒ t. The second method studies the shape matrix formed from the Jacobian of the unit normal field (normalized by the Gauss map). The Jacobian matrix of the unit normal field becomes, ⎤ λTev − Teλv /2 0 λTes − Teλs /2 [J(t)] = ⎣ −(λPes − Peλs /2) −(λPev − Peλv /2) 0 ⎦ /λ3 , −λs /2 −λv /2 0 ⎡

(5.71)

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

where Tev = ∂ Te/∂v = ∂ 2 L/∂s∂v Tes = ∂ Te/∂s = ∂ 2 L/∂s2 . Pes = ∂ Pe/∂s = −∂ 2 L/∂s∂v Pev = ∂ Pe/∂v = −∂ 2 L/∂v 2

(5.72)

The similarity (curvature) invariants of the Jacobian matrix become defined in terms of a set of partial differential equations: M(x, y, z) = trace[J]/2 = div n = {(1 + Pe2 )Tes − (1 + Te2 )Pev − TePe(Tev − Pev )}/λ3 ,

G(x, y, z) = trace[adjointJ] = −{Tes Pev − Tev Pes }/λ4 , A (x, y, z) = det[J] = 0.

(5.73) (5.74) (5.75) (5.76)

The product of the position vector, R = [s, v, u] and gradient of the (Lagrangian) phase function defines (for β 2 = 1) a new phase function, Gibbs(Te, Pe), (the Gibbs function) Gibbs(Te, Pe) = −R ◦ [∂L/∂s, ∂L/∂v, ∂L/∂u] = u − Tes + Pev,

(5.77)

which is a Legendre dual of the Lagrangian L(s, v, u). The zero set of the Gibbs function defines an implicit surface in the space of variables, {Gibbs, Te, Pe}. The Gibbs implicit surface mimics a swallowtail bifurcation. Where the Lagrangian is a function of the extensive variables, the Gibbs function is more like a Hamiltonian function of the intensive variables. The (multi-valued) Gibbs surface can be mapped out in terms of a family of solution curves generated by the dynamical system of the vector, C(s, v, u)/λ. Remember that G(s, v, u) and M(s, v, u) are first computed for the implicit surface determined by the phase function of extensive variables, L(s, v, u) = 0, and then re-expressed in terms of the intensive variables (Gibbs, Te, Pe) = (0, Te, Pe). The differential equations (5.73) that describe the zero set of the Gauss Curvature, G(Te, Pe), and the Mean Curvature, M(Te, Pe), for the renormalized Jacobian matrix can be evaluated by numerically integration. These results are displayed below along with the Gibbs surface constructed from the integration of the tangent lines. The intersection of the Gibbs surface, Gibbs = 0, and the surface, G(s, v, u) = 0, of zero Gauss sectional curvature forms a line in space which is the envelope, or tac locus, of the cuspoids of the Gibbs (swallowtail) surface. In thermodynamic terms, this intersection of the Gibbs Legendre dual surface and the surface of zero Gauss curvature, G(Te, Pe) = 0, defines the Spinodal line of absolute phase instability for the van der Waals gas.

The Similarity and Brand Scalar Invariants for an arbitrary 3D vector field

207

Figure 5.3 Gibbs swallowtail for a dynamical system It is apparent that the cuspoidal edge corresponds to the Cardano condition Cardano = 0 for the cubic polynomial, where the three eigenfunctions of the Jacobian matrix are real, but two are degenerate. Moreover, the intersection of the set ,{Gibbs(Te, Pe) = 0, G(Te, Pe) = 0, and M(Te, Pe) = 0}, defines the thermodynamic critical point. Note that the critical point for the cubic polynomial occurs when {G(s, v, u) = 1, M(s, v, u) = 1, A (s, v, u) = 1}, but {G(Te, Pe) = 0, M(Te, Pe) = 0}. The Gibbs function (Legendre dual) effectively moves the critical point to the zero of the space {Gibbs(Te, Pe)}. This relocation of the critical point to the origin was observed in terms of the reduced phase function, Φ (see (3.75)). These are universal features of all 3D Dynamical Systems. Remark 41 The Mean curvature and Gauss curvature of the Jacobian surface of an integrable dynamical system can be put into correspondence with the Critical Point and the Spinodal line of the Gibbs multivalued surface function for an isolated van der Waals gas. From the Frenet-Serret point of view, the Frenet - Torsion for the dynamical system (in the coordinates, {s, v, u}) equivalent to a van der Waals gas can be written as: Frenet Torsion

t◦ curl t = (Tes + Pev )/λ2 = ⇒ (∂ Te/∂s + ∂ Pe/∂v)/(1 + Te2 + Pe2 ). :

(5.78) (5.79)

Remark 42 The integral paths to a Dynamical System can be used to describe a universal thermodynamic Gibbs surface, which is dual to the Thermodynamic equation of State. 5.4

The Similarity and Brand Scalar Invariants for an arbitrary 3D vector field

For any family (parameter, t) of 3D vector field of functions, Vk (x, y, z; t), it is possible to renormalize (or rescale) the field by dividing each component of Vk (x, y, z; t)

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

by a Holder norm. The Holder norm is defined in terms of arbitrary constants ak , degree p and index m. λHolder3D = (± a1 (V 1 )p ± a2 (V 2 )p ± a3 (V 3 )p )m/p .

(5.80)

The resulting rescaled vector field, Vk ⇒ tk = Vk /λHolder3D , 1 ≤ k ≤ 3,

(5.81) (5.82)

then can be used to produce a 3 × 3 Jacobian matrix (a collineation), [J(t)] = [∂tk /∂xj ].

(5.83)

When the number of independent variables is 3, the Cayley-Hamilton polynomial becomes, Θ([J]) = Θ(x, y, z; ξ) = ξ 3 − XM ξ 2 + YG ξ 1 − ZA = 0. (5.84)

For any three arbitrary constants, ± ak , and for any degree p, it is remarkable that: XM = trace [J(t)] ⇒ 0, for m = 3, YG = trace [J(t)]Adjoint ⇒ 0, for m = 3/2, ZA = determinant [J(t)] ⇒ 0, for m = 1.

(5.85) (5.86) (5.87)

The eigenvalues for the 3 × 3 matrix, [J(t)], are either all real, or consist of one real eigenvalue and two complex eigenvalues. The two complex eigenvalues correspond to spinor, not vector, eigendirection fields. The famous Cardano function can be used to determine the root structure of the Cayley-Hamilton polynomial in 3D. When the number of independent variables is 4, λHolder4D Vk 1 [J(t)]

= ⇒ ≤ =

(± a1 (V 1 )p ± a2 (V 2 )p ± a3 (V 3 )p ± a4 (V 4 )p )m/p , tk = Vk /λHolder4D , k ≤ 4, [∂tk /∂xj ],

(5.88) (5.89) (5.90) (5.91)

the Cayley-Hamilton polynomial becomes, Θ([J]) = Θ(x, y, z, t; ξ) = ξ 4 − XM ξ 3 + YG ξ 2 − ZA ξ + TK = 0.

(5.92)

For any four constants, ± ak , and for any degree p, it is remarkable that∗ : XM YG ZA TK ∗

= = = =

trace [J] ⇒ 0, for m = 4, trace [J] ⇒ 0, for m = 2, trace [J]Adjoint ⇒ 0, for m = 4/3, determinant [J] ⇒ 0, for m = 1.

These ideas can be extended to vector fields in N dimensions

(5.93) (5.94) (5.95) (5.96)

The Similarity and Brand Scalar Invariants for an arbitrary 3D vector field

209

The coefficient functions, {M, G, A , K}, represent the averaged similarity invariants of the "renormalized"Jacobian matrix. In a previous section, the notation was {M = XM /4, G = YG /6, A = ZA /4, K = TK }. The possibly complex function ξ represents the eigenvalues which are functions of the independent variables, say {x, y, z, t}. It should be emphasized that the utilization of the universal Phase function does not describe in all detail the properties of non-equilibrium thermodynamic systems. There are examples of systems that have TK = 0, yet the Pfaff topological dimension is 4. Although the geometry appears to have been reduced to a threedimensional manifold, the topological dimension is still 4. The fundamental difference is that (N-1)-forms always admit integrating factors, while 1-forms do not. The dual 1-form and the dynamical system do not have identical Jacobian properties. 5.4.1 Vector Fields in 3D For the specialized values, degree p = 2, m = 1, and for ak = +1, the resulting rescaled Jacobian matrix becomes the Shape matrix (related to the second fundamental form) of differential geometry describing an implicit surface with a (perhaps non-integrable) normal field, tk . This specialized Holder norm is called the Gauss map, and leads to the constraint that the 4D determinant, K = 0. Hence, the Gauss Map forces the system to be of Pfaff dimension 3, or less. The Cayley-Hamilton polynomial becomes, ξ(ξ 3 − XM ξ 2 + YG ξ − ZA ) = 0, (5.97)

where the coefficients, {M, G, A }, are the 3D averaged similarity coefficients related to the "curvature" eigenvalues of the renormalized Jacobian matrix, [J(t)]. Now consider the 3D system whose characteristic equation is ξ 3 − XM ξ 2 + YG ξ − ZA ) = 0.

(5.98)

From the previous work, the cubic equation can be deformed into the equation of state for a van der Waals gas, where the polynomial parameter ξ plays the role of the molar density, ρ. For a 3D system, there are six primary invariant functions [31] of the 3x3 Jacobian matrix [J], which are to be associated with its complex eigenspectra. The Jacobian is to be viewed as a dyadic of functions rather than as a matrix of values. No linearization process is subsumed. The first three are the "symmetric" similarity invariants, where the second three will be called herein the "Brand" invariants. When the Jacobian is symmetric, the Brand invariants are zero. In 3D, with D = curl t, the six scalar invariants are: Curvature invariants Mean XM (x, y, z) = trace[J] = divt, Gauss YG (x, y, z) = trace[adjointJ], Adjoint ZA (x, y, z) = det[J],

(5.99) (5.100) (5.101)

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Brand Invariants Enstrophy L(x, y, z) = hD| ◦ bDi , Stretch M(x, y, z) = hD| ◦ [J] ◦ bDi , Brand (folds?) N(x, y, z) = hD| ◦ [J] ◦ [J] ◦ bDi .

(5.102) (5.103) (5.104)

The similarity invariants are well known, but the Brand invariants are not. The Brand invariants have applications in hydrodynamics, but have been used only rarely in practice. The Enstrophy is the square of the fluid vorticity, (curl V◦curl V), and is intuitively related to a "twisting" of the flow. The Stretch has been associated with the "stretching of vortex lines" in fluid flow. No practical application of the Brand invariant is known (to me), but it is conjectured that the third Brand invariant is related to "folding" of vortex lines. The Brand invariants help capture properties where the Jacobian matrix has a fixed point (such as the origin in a rotation). Such properties are missed by the similarity invariants. Perhaps in an oversimplified way, similarity transformations are related to translational momentum, and Brand invariants are related to intrinsic angular momentum. Perhaps the most important property of the Brand Invariants (especially the enstrophy) is their relationship to the antisymmetric parts of the Jacobian matrix. It is the antisymmetric parts that contribute to the possibility that the eigendirection fields are complex isotropic spinors. In thermodynamic terms, the system is above the critical isotherm when the enstrophy dominates the fluid motion. In the examples that follow, the topological thermodynamic properties of these invariants will become evident. When evolution (not representable by a similarity transformation) causes these functions to change, topological features change. In dynamical systems, these changes are recognized as bifurcations. In thermodynamics, these changes are recognized as changes of phase. The null sets of these similarity functions, or combinations of these functions, form invariant surfaces of separation, which globally separate domains of different topology. The idea to be developed is that these surfaces of separation (in velocity space) define domains of equivalent thermodynamic phase! What better way to think of a coherent structure, than as deformable domains of pure thermodynamic phase. The similarity invariant surfaces may also be viewed as sets of partial differential equations whose solutions have interesting properties: XM (x, y, z) = 0 defines a minimal surface (soap film of zero mean curvature). YG (x, y, z) = 0 defines a flat surface of zero Gauss sectional curvature, or a singular subset in a curved surface. L(x, y, z) = 0 defines an asymptotic surface of zero enstrophy (where the vorticity of a fluid is zero). M(x, y, z) = 0 defines a surface of null vortex stretching rate.

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5.4.2 Chirality In the examples that follow, one (or more) of the parameters may cause the similarity invariants, and often, the Brand invariants to exhibit chiral invariance. For example, if the sign of the chirality parameter, say left handed or right handed rotation, Ω, does not make a difference, then the object is a chiral invariant. This would imply that the chirality parameter Ω only appears in the expressions as an even function of the parameter. However, in many interesting cases, especially for the hypersurface that defines the zero set of the topological parity 4-form, K = dAˆdA = 0. This function can contain odd powers of the chirality parameter, and therefore will have chiral sensitive components. This phenomena is observed in the exact solutions of Maxwell’s equations [206] and leads to the extraordinary result that the speed of propagation of singular solutions (electromagnetic signals) is not 4-fold degenerate in the presence of both Optical Activity and Faraday effects. Not only is the chiral sensitivity associated with polarization broken, but also the chiral sensitivity associated with propagation direction is observed. Measurements in dual polarized ring lasers verify the theory [194]. 5.4.3 The Gauss map normalization and the Shape matrix for a projection 3D to 2D The invariant functions have a simple representation in 3 D when one of the velocity components is never zero. The constraint permits a projection from 3D to 2D. This simplification was used above for describing the dynamical system representation for a van der Waals gas. In this subsection, the classic geometric formalism of the FrenetSerret method in Euclidean (not thermodynamic) coordinates will be compared to the implicit surface shape matrix method. Consider the 3D velocity vector with components, V = [u, v, w], with the assumption that w 6= 0. Then consider the direction field, v = w[u/w, v/w, 1] = w[p, q, 1]. The unit tangent field of this direction field can be written as p t = [p, q, 1]/ 1 + p2 + q 2 , (5.105) and the Jacobian matrix can be written as (where px = ∂p/∂x, ..., qx = ∂q/∂x, ..λx = ∂λ/∂x..), ⎤ ⎡ λpx − pλx /2 λpy − pλy /2 0 [J(t)] = ⎣ λqx − qλx /2 λqy − qλy /2 0 ⎦ /λ3 . (5.106) −λy /2 0 −λx /2 The 2D similarity (curvature) invariants become defined in terms of a set of Partial differential equations: CM (x, y, z) = = CG (x, y, z) = CA (x, y, z) =

trace[J]/2 = divt {(1 + q 2 )px + (1 + p2 )qy − pq(py + qx )}/λ3 , trace[adjointJ] = {px qy − py qx }/λ4 , det[J] = 0.

(5.107) (5.108) (5.109) (5.110)

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The formula CM (x, y, z) ⇒ 0 defines the classical expression for a Minimal (Monge) surface. The formula CG (x, y, z) ⇒ 0 defines the classic expression for a "flat" surface. The unit tangent vector, t, also has an expression in terms of a space curve in Frenet-Serret theory. The Darboux vector can be written as, Darboux D = curl t = [λy , −λx , (2λ(py − qx ) − (pλy − qλx )2 ]/2λ3 .

(5.111)

The Helicity (Frenet Torsion) may be computed as, t◦curl t = (px − qy )/λ4 .

Frenet Torsion: 5.5

(5.112)

Universal Cayley-Hamilton polynomials and 3D Thermodynamics

The work of the preceding section demonstrates the close correspondence between Cayley-Hamilton invariant theory and thermodynamics. More than twenty five years ago, at the 1977 Aspen conference on “New Frontiers in Thermodynamics" I suggested that there is a connection between the invariants of the shape matrix of differential geometry, critical behavior (topological transitions) in dynamical systems, and the thermodynamics of a Gibbs equilibrium system. In particular, it was determined that the Spinodal line of a van der Waals gas was given by the condition that the sectional Gauss curvature (the determinant of the shape matrix) of the Gibbs function as a surface constraint in {ρ, T, P } space must vanish. The van der Waals Gibbs function is a cubic polynomial which generates the shape of the now classic swallowtail singularity (see Chapter 3). The previous work was related to isolated or equilibrium systems, where the Pfaff topological dimension is 2 or less. This work can now be generalized to include non-equilibrium Pfaff dimension 3 systems with a null point, and Pfaff dimension 4 systems "without" a null point. Non-Equilibrium Phase No null eigenvalues Θ TK One null eigenvalue Θ TK

: = 6= = =

Θ(x, y, z, t..; ξ) ⇒ 0, (ξ 4 − XM ξ 3 + YG ξ 2 − ZA ξ + TK ), 0. (ξ 3 − XM ξ 2 + YG ξ 1 − ZA )ξ, 0.

(5.113) (5.114) (5.115) (5.116)

The equilibrium thermodynamic critical point for the example of the van der Waals gas was determined in the previous subsection as that point where both the Mean curvature and the Gauss curvature of the 2D Gibbs surface vanished. As the intrinsic Spinodal line in thermodynamics is the limit of single phase stability, the thought, in 1977, was to apply such thermodynamic ideas to the stability analysis of dynamical systems (such as the Brusselator chemical system) in degree to gain

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213

insight into their behavior. In modern language, the idea recognized that the first law of equilibrium thermodynamics, ω = dU − T dS + P dV ⇒ 0,

(5.117)

could be viewed as an integral submanifold of an exterior differential system, with the fundamental group as the projective subgroup of the general linear group. The associated vectors of the 1-form, such that i(e)ω = 0, create a tangent space orthogonal to the adjoint field defined by the covariant components of the 1-form, ω. This vector bundle can be used to define a 3 × 3 basis frame, whose Cartan matrix of connection coefficients can be used to determine the differential geometry properties of the thermodynamic submanifold. The interesting thermodynamic properties turned out to be the zero sets of the geometrical similarity invariants of the basis frame. For non-equilibrium thermodynamics, these notions need to be extended to the continuous topological properties of a physical system Turning the idea around, if a dynamical system in three dimensions could be used to define a basis frame, then the associated Cartan matrix of connection 1forms could be used to determine geometrical properties, and the important intrinsic properties would be related to the similarity invariants of the basis frame. As every 3 × 3 matrix satisfies its Cayley-Hamilton polynomial, which in the general case is a cubic polynomial, and as the Jacobian matrix of every three-dimensional dynamical system is a 3 × 3 matrix which can be used as a basis frame, then all such systems must have a universal representation as a van der Waals gas. Indeed, the oscillation frequency of the Brusselator dynamical chemical system was shown by this technique to be related to the square root of the Gauss curvature of the equivalent thermodynamic surface. It is now known that this oscillatory result, in which the square root of the Gauss curvature is an oscillation frequency, is not only a similarity invariant of the dynamical system, but also a projective invariant whereby the dimensionless ratio of the three similarity invariants, XM YG /ZA , of a certain 3 × 3 Jacobian matrix, is equal to unity. In a three-dimensional dynamical system, a Hopf bifurcation occurs when a complex conjugate pair of roots has no real part. Suppose the eigenvalues are [a, iω, −iω]; then XM = a, YG = ω 2 , ZA = aω2 and it follows that XM YG = ZA . The occurrence of complex eigenvalues signals that the Jacobian matrix has antisymmetric components and spinor eigendirection fields. The Hopf condition corresponds to the existence of a surface in the three-dimensional domain defined by the condition, XM = 0, YG ≥ 0, ZA = 0. Oscillatory limit cycles, if they exist, must reside on the surface defined by the Hopf condition, YG ≥ 0, subject to the conditions that XM = 0, ZA = 0. The Hopf bifurcation is not the only set that belongs to the surface constraint, XM YG = ZA , for there is another sheet that belongs to the hyperbolic situation when YG < 0. 5.5.1 Lessons from the van der Waals gas The van der Waals gas was studied in detail in Chapter 3. A point of departure from the classic analysis is realized when the projectivized constraint is chosen such

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that the shape matrix admits complex eigenvalues. In all projective geometries, the fundamental invariants are constructed from six primitive cross ratios, two of which are bounded by negative infinity and zero, two of which are bounded by zero and one (the probability domain) and two of which are bounded by one and infinity. It will be demonstrated below how this signature of the three projective equivalence classes appear in the relationships that relate envelopes to bifurcations in projective space. As mentioned above, others have attempted to use differential geometric methods to analyze thermodynamic systems, but almost always these attempts have tried to construct a suitable metric formalism. For example, Tisza mentions that Blaschke attempted to deduce a differential geometry that would apply to the metric free Gibbs space, but with only limited success. In Blaschke’s geometry, the projective space was confined to the equi-affine group, which forces the shape matrix to be symmetric. Such equi-affine systems admit only real eigenvalues for the shape matrix, where the richness of non-equilibrium thermodynamics, and its possible application to the theory of dynamical systems, requires the existence of domains of both real and complex eigenvalues. In this monograph, a projective geometry without metric is presumed to be the useful basis for non-equilibrium thermodynamics. When the Gibbs primitive phase surface of the van der Waals gas is mapped to its dual by means of a Legendre transformation, the Spinodal line can be interpreted as an edge of regression in the dual surface of “Gibbs free energy". It is this clue that focuses attention on the theory of envelopes, for the edge of regression is a singularity in an enveloping surface (Struik). It is apparent in the dual surface of Gibbs free energy that, in addition to the edge of regression, there exists another topological feature of singularity, a line of self intersection (which is not an intrinsic property that can be determined locally). This non-metrical feature of self intersection was interpreted as the Binodal line in the earlier work mentioned. Usually, the Binodal line is defined through a heuristic Maxwell construction on the PVT surface representing the equation of state. As Tisza states [259] in reference to the Maxwell procedure, ...a “van der Waals gas" (referring to the equation of state) does not constitute a fully defined thermodynamic system. A complete definition would include the specific heat as a function of say temperature and volume.... In the concept of a Van der Waals gas a spurious interpolation (the Maxwell construction) through the instable range (of the equation of state) is substituted for the missing (specific heat) information. In differential geometry, the line of surface self-intersection is a locus of singularities, and as such would offer a projective geometric definition of the Binodal line, without the heuristic Maxwell assumption. Although visually apparent in the equilibrium surface representing the Gibbs free energy, the differential geometry of the extrinsic Binodal line has long eluded algebraic derivation. It is now apparent that the Binodal line represents a Pitchfork bifurcation in the "Higgs" phase function

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215

representing the Gibbs function of a van der Waals gas. This problem was resolved in the previous Chapter 3.4, where the full four-dimensional (not three dimensional) development was utilized to demonstrate that below the critical isotherm, the phase potential was of the 4th degree "Higgs" type, and this permitted analytic solutions for the Binodal line as a representation of a pitchfork bifurcation. 5.5.2 Envelopes In Chapter 3.4 the focus was on four-dimensional (open 4D) thermodynamic systems, but in this section the emphasis is on 3D dynamical systems. (The theory of envelopes in the language of Cartan’s exterior calculus is given in detail in Chapter 9.10.) For any vector field V(x, y, z) the 3x3 Jacobian matrix relative to the variables {x, y, z} always has a characteristic cubic polynomial equation of the form, Characteristic polynomial : for [Jacobian (V(x, y, z))] 3D Θ(x, y, z; ξ) ⇒ (ρ3 − XM ρ2 + YG ρ − ZA ) = 0.

(5.118)

Algebraically, the (possibly complex) variable ρ(x, y, z) corresponds to the eigenvalues of the polynomial expression. How can this universal result be put into correspondence with thermodynamic ideas? Conjecture Thermodynamically, the multivalued eigenvalue function, ρ(x, y, z), is to be interpreted as the molar density of matter. The lesson of the van der Waals gas is that the cubic polynomial can be considered to be a Gibbs function in terms of the intensive variables of pressure and temperature, and the matter density. So consider the function, Gibbs (q k , Ak , ρ) = (1/ρ)L(qk , V k ) = (1/ρ)L(q k , ρAk ), where by definition V k = ρAk .

(5.119) (5.120)

Note that the formulation leads to the Legendre transformation between the Lagrange function L(qk , V k ) and the Hamiltonian function, H(qk , Ak ), ∂Gibbs (qk , Ak , ρ)/∂ρ = = H(qk , Ak ) = =

−1/ρ2 {L − V k ∂L/∂V k } 1/ρ2 {H}, ρ2 ∂Gibbs (q k , Ak , ρ)/∂ρ ρ2 {3ρ2 − 2XM ρ + YG }.

(5.121) (5.122) (5.123)

When ∂Gibbs (q k , Ak , ρ)/∂ρ ⇒ 0, the Lagrange function, L(q k , V k ), is homogeneous of degree 1 in the variables V k . When Gibbs (q k , Ak , ρ) is identified with the cubic polynomial, then the similarity invariants become the universal functions, Ak , representing the potentials per unit source (intensive variables) and the functions V k = ρAk become the "densities" (or extensive variables) for the problem being

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studied. The idea that the Lagrangian is homogeneous of degree 1 in the extensive variables is equivalent to forming a "projectivized Finsler space" [52]. The criteria that Gibbs (q k , Ak , ρ) ⇒ 0 defines a family of implicit hypersurfaces with a family parameter ρ. The criteria that the partial derivative with respect to the family parameter, ∂Gibbs (q k , Ak , ρ)/∂ρ ⇒ 0 adds an additional constraint that an envelope exists. The envelope exists only if the 2-form specified below is not zero: Envelope Condition ∂Gibbs (q k , Ak , ρ)/∂ρ ⇒ 0; but the 2-form, dGibbs ˆd(∂Gibbs /∂ρ) 6= 0.

(5.124) (5.125)

The envelope is smooth if ∂ 2 Gibbs (q k , Ak , ρ)/∂ρ2 6= 0, and will have an edge of regression when ∂ 2 Gibbs (q k , Ak , ρ)/∂ρ2 = 0, but only if the 3-form specified below is not zero: Edge of regression Condition ∂ 2 Gibbs (qk , Ak , ρ)/∂ρ2 = 0; but the 3-form, dGibbs ˆd(∂Gibbs /∂ρ)ˆd(∂ 2 Gibbs /∂ρ2 ) 6= 0.

(5.126) (5.127)

For the premise that the Gibbs surface in 3D is defined by the implicit hypersurface equation (in terms of averaged similarity invariants), Gibbs (q k , Ak , ρ) = ρ3 − XM ρ2 + YG ρ − ZA ⇒ 0,

(5.128)

then an envelope exists when, ∂Gibbs /∂ρ = {3ρ2 − 2XM ρ + YG } ⇒ 0,

(5.129)

and the envelope 2-form condition is satisfied. The condition on the family parameter yields two roots: q ρ± = (XM /3)(1 ±

2 1 − 3YG /XM .

(5.130)

Substitution of each root ρ± into the Gibbs surface equation yields an equation that can be solved for two factors, A± : A+ = {ρ3+ − XM ρ2+ + YG ρ+ − ZA }, A− = {{ρ3− − XM ρ2− + YG ρ− − ZA }}.

(5.131) (5.132)

The product of the two functions, A+ · A− , yields the envelope function in the space of averaged similarity invariants, {XM , YG , ZA }, 2 3 3D Envelope = (ZA )2 − (2/3)ZA YG XM − 1/27XM YG2 + 4/27(XM ZA + YG3 ). (5.133)

The zero set of the envelope function defines an implicit hypersurface in the space of similarity variables. Symbolic math packages algebraically define the discriminant

Universal Cayley-Hamilton polynomials and 3D Thermodynamics

217

and the resultant functions, which are multiples of the envelope function by a factor of +27 for the resultant (in 3D) and a factor of -27 for the discriminant. The 3D Envelope of the cubic polynomial is related, historically, to another function defined in the literature as the Cardano function (p.98 [79]). It is remarkable that 4 times the Cardano function is equal the envelope. Hence, the zero set of all three functions, Cardano, envelope, discriminant, define the same implicit surface. The Cardano function can be evaluated in terms of two "control" parameters: α β Cardano 4 · Cardano Discriminant Resultant

= = = = = =

3 (−2/27XM + 1/3XM YG − ZA ), 2 ) (YG − 1/3XM 2 (α/2) + (β/3)3 = 3D Envelope/4, Envelope −27 · Envelope +27 · Envelope

(5.134) (5.135) (5.136) (5.137) (5.138)

The words "Cardano-Envelope" function, with the notation, Cardano , will be used to describe all such implicit surfaces.

The Universal Cardano function The three dimensionality of the Jacobian with its cubic polynomial immediately focuses attention on the Cardano-Envelope function, Cardano (x, y, z), whose null set generates an implicit surface that separates domains of complex roots from domains of real roots. When Cardano (x, y, z) < 0, the three roots of the cubic polynomial are distinct and real. When Cardano (x, y, z) > 0, there is one real solution, and two conjugate complex solutions. The envelope consists to two sheets joined at an edge of regression. The universality comes about when the Cardano implicit surface is expressed in the 3D space of similarity invariants, {XM , YG , ZA }. To within a factor,

3 2 ZA +27ZA2 −XM YG2 −18·XM YG ZA ⇒ 0. (5.139) Cardano {XM , YG , ZA } = +4YG3 +4XM

A plot of the universal Cardano-Envelope surface, ⇒ 0, Cardano {XM , YG , ZA } = 0, is given in Figure 5.4. Note the edge of regression.

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Figure 5.4 The Universal Cardano Envelope The Cardano-Envelope surface, Cardano (x, y, z) = 0, consists of two sheets, one for (A ), and the other for (A− ). The edge of regression for the two sheets leads to the critical isotherm, which in physical language separates the pure gas phase from the condensate liquid and vapor. The region in between the two sheets corresponds to domains where Cardano < 0. This is the domain that consists of liquid, vapor, or mixtures. Such domains admit three real solutions to the characteristic polynomial. The complex eigenvalues that occur when Cardano > 0 can be associated with spinor eigendirection fields and antisymmetric components of the matrix that generates the thermodynamic phase function. This condition corresponds to the pure gas phase domains of the Van der Waals gas. The degeneracies of the root structure and the transitions from real to imaginary roots occur when Cardano = 0. It is also useful to note that if the complex roots have no real parts, then it must be true that XM YG /ZA = 1, the projective Hopf (necessary) condition. However, this Hopf necessary condition is not sufficient for complex roots without real parts, for suppose that the roots are all real and two real roots are real and opposite in sign, eigenvalues = [u2, −u2, u3]. Then A = −(u2)2 (u3), YG = −(u2)2 /3, XM = (u3/3) satisfies the equation. This implies that there exists a sectional saddle point which is locally a minimal surface (u1 = −u2), and which becomes a minimal surface when M and therefore A simultaneously become zero. When YG > 0 the Hopf oscillatory condition is true. When YG < 0 the (real) Minimal surface condition is true. Of particular interest is the case when YG = −1, for +

Universal Cayley-Hamilton polynomials and 3D Thermodynamics

219

then the necessary conditions exist for a Backlund transformation, and the possible existence of non-dissipative soliton solutions. Also, if ZA 6= 0, then there are no real eigenvalues which are zero. The quest for a Hyperbolic domain (a useful restriction in dynamical systems) is then easily determined by the two conditions, ZA 6= 0 and XM YG = ZA . It is important to note that the Hopf condition, XM YG = ZA , can be satisfied even though there are no zero roots to the characteristic polynomial. This result demonstrates that the bifurcation concepts do not have to be centered about the neighborhood of a fixed point (where V = 0). The criteria of rank less than 3 implies that ZA = 0, and indicates that the zero set of the Cardano function is satisfied by the equation, 2D Cardano constraint : (for ZA = 0) 2 )YG2 ⇒ 0. Cardano {XM , YG , 0} = +(4YG − XM

(5.140) (5.141)

By comparison to the Gibbs surface for a van der Waals gas, the Spinodal constraint is related to the factor YG2 = 0, and the Binodal constraint is related to the factor, 2 {(4YG − XM )} = 0. Returning to the full Cardano formula, equation (5.139), when all three nonzero roots are equal, then ZA = 1, YG = 3, XM = 3. These conditions define the generalized critical point of a generalized Van der Waals gas. For the classical van der Waals gas these values indicate that: Pe = 1, Te = 1,

(5.142) (5.143)

XG = (8Te + Pe)/3 = 3, XM = 3.

(5.144) (5.145)

The Cardano function can be expressed in terms of the properties of the rescaled van der Waals Gas:

(5.146)

XM = 3, β 0 = {(8Te + Pe) − 9}, α0 = {(8Te − 2Pe) − 6}, 2

(5.147) (5.148)

3

27Cardano = (α0 /6) + (β 0 /9) = (Pe + 2/27Pe2 + 1/729Pe3 ) +(8/243Pe2 − 40/27Pe)Te +(64/243Pe − 16/27)Te2 +512/729Te3 .

(5.149)

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For various values of Pe and Te the Cardano function can be evaluated to determine the root structure of the characteristic polynomial. For values of Pe and Te above the critical point there is always one real solution, but there also is the possibility that there are two complex conjugate solutions. Also recall that for a van der Waals gas, when modified by an interaction length cb with c > 1, leads to cohesive forces for which the averaged Mean similarity invariant is not 1, the value at the critical point. For the Van der Waals gas the density is presumed to be a real variable. From the characteristic polynomial, although formally the same, the “molar density" is a set of eigenvalues that are not constrained, a priori, to be real. In other words, the characteristic polynomial represents a complex van der Waals gas. The renormalized eigenvalues of the Jacobian play the role of the “complex molar density relative to the critical point" of a multi-phase system. Note that the characteristic polynomial is universal and deducible for any vector field that can be differentiated once. Examples for several dynamical systems will be presented below. When the system is at “equilibrium", the complex parts of the molar densities are presumed to vanish (by some constraint not given by the Vector field alone). Near by the equilibrium state, the complex portions of the molar densities will be related to global stability phenomena. These features are thermodynamic features that are universal, and apply to all cubic polynomials generated by the vector field that represents the 3D dynamical system. When A ⇒ 0, the topological dimension becomes equal to 2, and the system is "isolated". Non-equilibrium modes correspond to A 6= 0, and can be put into correspondence with the interaction energy between the system and the dynamics [219]. The above analysis indicates that kinematic domains of Dynamical Systems have properties that are topologically equivalent to thermodynamic phases. Borrowing from thermodynamic experience, cooperative and coherent behavior is to be expected in complex kinematic flows, along with kinematic phase transitions, depending on initial conditions and parameters. 5.5.3 Bifurcations and Thermodynamics As noted in previous sections, any 4x4 matrix of maximal rank generates a fourth degree polynomial equation, Θ = 0, in terms of its eigenvalues , ρ. Θ([J]) = Θ(x, y, z, t..; ξ) = ξ 4 − XM ξ 3 + YG ξ 2 − ZA ξ + TK = 0.

(5.150)

When the matrix elements are functions, then the Cayley-Hamilton polynomial equation, Θ = 0, becomes a family of implicit surfaces in 4D with the eigenvalues ξ playing the role of the family parameter. If the matrix is constructed from the Jacobian matrix of a vector which has been rescaled by the Gauss map, the determinant K of the Jacobian matrix vanishes. This constraint implies that (at least) one eigenvalue is zero. The Cayley -Hamilton polynomial takes the form,

Universal Cayley-Hamilton polynomials and 3D Thermodynamics

ξ(ξ 3 − XM ξ 2 + YG ξ − ZA ) ⇒ 0,

221

(5.151)

such that the cubic polynomial is deformably equivalent to the equation of state of a van der Waals gas. There is one obvious solution, ξ = 0. The cubic function may be viewed as a family of hypersurfaces in the space of variables {XM , YG , ZA }. The critical point for the cubic occurs when all roots are degenerately equal to 1. Then the similarity invariant functions have the values {XM = 3, YG = 3, ZA = 3}. This third degree system fits a GLOBAL bifurcation scheme which is said to be: "Hysteretic" in the variable A ∼ ZA ; a Pitchfork in the variable G ∼ YG , and a winged cusp in the variable M ∼ XM [87]. These features can be constructed from the projections of the Cardano Function (see Figures 5.5, 5.6, and 5.7 on the next page).

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Figure 5.5 A Dynamical system and a Critical Isotherm Related to Spinor states and Hysteresis

Figure 5.6 A Dynamical system and a Binodal line

Figure 5.7 A Dynamical system and a Spinodal line Although the critical point is centered on zero in the diagram, the topological features of these bifurcation diagrams are universal. It is apparent that the Hysteritic

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223

line has the same deformable features of the critical isotherm of a van der Waals gas. The similarity invariant ZA , plays the role of "pressure", and the similarity invariant XM plays the role of the "surface tension". Recall that for the van der Waals gas there is no hysteresis in the sense that the similarity invariant XM is fixed in its value to be equal to 3. The critical isotherm separates the gas phase from the vapor or liquid phase. If any mathematical system generates the cubic polynomial, it can be studied in terms of the known behavior of a van der Waals gas. From the study of the Pitchfork bifurcation, it becomes apparent that the similarity invariant, YG , might be associated with "temperature". The 3-prong portion of Pitchfork diagram then can be interpreted as the temperature-Molar density behavior of a van der Waals gas below the critical isotherm, separating liquid, vapor and mixed phases. The Winged Cusp demonstrates the connection between the pressure and the temperature and will map deformably into the swallowtail singularity of the Gibbs function. It is quite apparent that the similarity invariants lead to negative temperatures, pressures and molar density. These non-intuitive results are due to the choice of variables such that the Critical point is a zero. The general case demonstrates that the pressure, temperature and density features of real substances could have regions of negative or complex pressure, temperature and density. Conjecture 43 Are the "negative pressures" required by the dark energy and dark matter theories of current cosmology related to simple extensions of a van der Waals Gas - which experimentally admit transient regions of negative pressure? Or could it be that such effects are due to angular momentum causing the dynamical system to be near to or above the "critical" isotherm. Two Cardano functions, α and β, have been defined in the previous subsection such that: 2 β = (YG − 1/3XM ), 3 α = (−2/27XM + 1/3XM YG − ZA ) Cardano = (α/2)2 + (β/3)3 .

(5.152) (5.153)

The functions, α and β, can be used as the coordinates of the "bifurcation" space. Note that the Cardano function is always greater that zero, when β is greater than zero (a sufficient, but not necessary condition for a single real root, and the existence of spinor direction fields). When β < 0, then it is possible (a necessary but not sufficient condition) that the Cardano solution has three real roots, such that Cardano < 0. This is a rather remarkable and useful result. From the theory of surfaces, YG is related to the Gauss curvature of a hypersurface, and XM is related to the mean curvature. For real surfaces of mean curvature zero (soap films), the Gauss curvature is negative, and therefore β < 0. Even so, it is not sufficient that the

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negative β implies that the Cardano function is negative. A transition from β > 0 to (β/3)3 < (α/2)2 is required to produce condensation from the gas phase. When ZA = 0 the system can be put into correspondence with an isolated thermodynamic system (Pfaff dimension = 2), which is not irreversible in a hysteretic sense. When ZA 6= 0, the system is irreducibly three dimensional, and the thermodynamic system is not in equilibrium, but could be in a "steady" state. When XM = 0, the algebraic constructions given above must be modified, and a new characteristic time (or length) scale arises in terms of the value of (α/β = −ZA /Yg ). The Cardano formula which separates real from complex solutions, and also defines degenerate possibilities, becomes for XM = 0, Cardano = +4YG3 + 27ZA2 ⇒ 0, or ZA2 /YG3 = −4/27.

(5.154) (5.155)

In the case (XM = 0 ) the implicit surface family, or characteristic polynomial, can be written in dimensionless form in terms of only one dimensionless control parameter, δ = YG3 /ZA2 . Then for XM = 0, ρ = (1 − u)ZA /YG , substitute into Gibbs (x, y, z; ρ) = ρ3 + YG ρ − ZA ⇒ 0, to yield, δ = (1 − u)3 /u.

(5.156)

From this equation it is apparent that the measure or scale is determined by Yg /ZA , and not XM (which is by assumption equal to zero in this example). This result implies that the generic form of the 1-parameter family of surfaces in {x, y, z; ρ} can be projected, for the case (XM = 0 ), not as a projection to a 2-parameter control surface, but instead as a projection to a 1-parameter curve in the space of rescaled dimensionless variables δ and u.

Figure 5.8 Band Gap defects

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225

The roots of the “projectivized" characteristic polynomial form a curve which is similar to an imperfect (hysteretic) subcritical pitchfork bifurcation in the variables, (δ, u). Note that the domain of solution in alpha-beta space has the natural subdivision into domains where u is bounded by negative infinity and zero, zero and one, and one and plus infinity. In other words, the characteristic polynomial has been put into correspondence with the cross ratio invariants (−∞ → 0; 0 → 1; 1 → ∞) of projective geometry. In Figure 5.8 above there is a (thermodynamic energy) gap (δgap ) between the two branches indicating the fact that δ can be retraced in a negative fashion along the lower branch, but when δ < —δ gap a transition will take place to the stable upper branch. A plot of the formula {−|δ|} yields the interesting curve, which should be compared to the experimental curve the justifies the law of corresponding states:

Figure 5.9 Coexistence curve?

Figure 5.10 Shape of the Coexistence Curve

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

The shape of the curve, −|δ| vs. u, for (XM = 0 ) yields the pattern of the vapor liquid coexistence curve, which is a universal formula for two phase systems in chemistry. Is this an accidental correspondence? Compare to p. 97 in [183]. Conjecture The shape of the curve, −|δ| vs. u, for ( XM = 0 ) yields the pattern of the vapor liquid coexistence curve, which is a universal formula for two phase systems in chemistry. Is this an accidental correspondence? It may be shown that tertiary bifurcations, such as the Intermittent (Hysteretic in the language of Langford) Hopf bifurcation [132] can lead to intermittency. Such flows have non-zero Topological Torsion. An example of an exact solution to the Navier-Stokes equations that yields an intermittent transverse torsion wave packet was given in [202]. 5.5.4 Antisymmetry in the Jacobian matrix, vorticity, and torsion The interplay between the Pfaff topological dimension (based on topological antisymmetries of the 1-form of Action, A) and the Projective dimension (based on the Jacobian (correlation) matrix of the 1-form of Action, A) is still an active area of research. In 4D the algebra can be very tedious, and the general theory is far from complete (April 2006). The simpler problem of 3D can also be tedious, but some insight can be obtained by means of examples. The purpose of this subsection is to demonstrate the effects of rotation and vorticity† on the thermodynamic state of a dynamical system, which includes effects of projective correlations and collineations. The appearance of antisymmetric parts in the Jacobian correlation matrix is a sign that self-duality is lost. In the language of projective geometry, the correlation is not a polarity. Consider a 3 x 3 Jacobian matrix , [Jkm ] = [∂(Ak )/∂xm ] , constructed from the map φ : {xm } ⇒ {Ak }. The Jacobian matrix may be symmetric, in which case all of its eigenvalues are real. In such cases the cubic Cayley-Hamilton polynomial will have 3 real roots. From a thermodynamic point of view, the dynamics of such symmetric systems is always below the critical isotherm. The Cardano function will be negative. Recall that complex roots occur only when the Cardano function is positive. The only way for the system to be in a configuration above the critical isotherm will be if the Jacobian matrix will have antisymmetric parts, such that the Cardano function will be greater than zero. The method also points to the weaknesses of metric theory, where everything depends upon the (symmetric) metric correlations. Such systems, in a thermodynamic sense, can not be above the critical isotherm; they are not in the thermodynamic state of a pure gas. The eigenspectra associated with such symmetric systems does not include spinors. Systems that admit macroscopic complex spinors (of null magnitude) imply the thermodynamics is in a pure gas phase, above the critical isotherm. †

In hydrodynamics, the square of the vorticity vector, curl V, has been called the Enstrophy.

Universal Cayley-Hamilton polynomials and 3D Thermodynamics

227

Example 1 Consider three 3-component vector direction fields: G = [α, β, γ], S = [sx , sy , sz ], Ω = [ωx , ωy , ω z ].

(5.157) (5.158) (5.159)

Use these functions to represent the elements of a Jacobian matrix, h i h i T T [J] = [J] + [J] /2 + [J] − [J] /2,

= [Symmetric] + [Antisymmetric] , ⎤ ⎡ ⎤ ⎡ α sz sy 0 ωz −ωy 0 ωx ⎦ . = ⎣ sz β sx ⎦ + ⎣ −ω z sy sx γ ωy −ω x 0

(5.160) (5.161) (5.162)

The Cayley-Hamilton 3D characteristic (phase) function and its similarity invariants become: ρ3 − ZA ρ2 + YG ρ − ZA , div(G) = α + β + γ, (αβ + βγ + γα) − S · S + Ω · Ω, +(−ω 2x + s2z )α + (−ω 2y + s2y )β + (−ω2z + s2x )γ −αβγ − 2(sx sy sz + sx ω y ωz + ω x sy ω z + ωx ω y sz ). ”Enstrophy" = Ω · Ω = ω 2x + ω2y + ω 2z . Θ = XM = YG = ZA =

(5.163) (5.164) (5.165) (5.166) (5.167) (5.168)

When Ω = [ω x , ω y , ω z ] ⇒ 0, the Jacobian correlation matrix is symmetric. Hence all eigenvalues are real, and the Cardano function must be negative. However, consider the simple example where α = β = γ = 1, sx = sy = sz = 0, ⎤ ⎡ 1 ω z −ωy 1 ωx ⎦ . [J] = ⎣ −ω z (5.169) ω y −ω x 1 This Jacobian can be derived from the 1-form of Action given below with its Pfaff sequence, A = (x + wz y − zwy )dx + (−wz x + y + wx z)dy +(−wx y + z + wy x)dz, F = dA = −2wx dyˆdz − 2wy dzˆdx − 2wz dxˆdy, H = AˆdA = −2(xwx + ywy + zwz )(dxˆdyˆdz).

(5.170) (5.171) (5.172)

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

Note that the Pfaff topological dimension is 3 (as AˆdA 6= 0) unless (xwx + ywy + zwz ) ⇒ 0. For (xwx + ywy + zwz ) 6= 0, the equivalent thermodynamic system is not in thermodynamic equilibrium. The cubic polynomial becomes, Θ = ρ3 − 3ρ2 + 3(Ω · Ω)ρ − (1 − Ω · Ω), with α = β = γ = 1, sx = sy = sz = 0.

(5.173) (5.174)

The Cardano function can be evaluated as, Cardano = 4 · (Ω · Ω)3 ,

(5.175)

which is always positive for all values of the enstrophy. The roots of this cubic are, ρ1 = 1,

(5.176)

ρ2

(5.177)

ρ3

√ = 1 + i Ω · Ω, √ = 1 − i Ω · Ω.

(5.178)

It is readily observed that the Enstrophy (the square of the vorticity), Ω · Ω, has led to a pair of complex conjugate roots. The corresponding Cardano function is always greater than zero, and the physical system mimics a pure gas phase of a van der Waals gas. The Jacobian matrix admits two complex spinor eigendirection fields. Example 2 As another simple example, consider the matrix of the form related to a component of "vorticity" along the z-axis: ⎤ ⎡ α ω z −0 [J] = ⎣ −ωz β 0 ⎦ . (5.179) 0 0 γ

This Jacobian can be derived from the 1-form of Action given below with its Pfaff sequence, A = α · xdx + β · ydy + γ · zdz + wz · (ydx − xdy), F = dA = F := −2 · wz (dxˆdy), H = AˆdA = −2γ · z · wz (dxˆdyˆdz).

(5.180) (5.181) (5.182)

Note that the Pfaff topological dimension is 3 (as AˆdA 6= 0) unless γ ⇒ 0. For γ 6= 0, the equivalent thermodynamic system is not in thermodynamic equilibrium. The cubic polynomial becomes, Θ = ρ3 − (α + β + γ)ρ2 + (wz 2 + αβ + αγ + βγ)ρ − γ(αβ + wz 2 ).

(5.183)

Examples of the thermodynamics of Dynamical Systems

229

The roots of this cubic are: ρ1 = γ,

(5.184)

ρ2

(5.185)

ρ3

p = {(α + β) + (α − β)2 − 4 · wz2 )}/2, p = {(α + β) − (α − β)2 − 4 · wz2 )}/2.

(5.186)

If γ ⇒ 0 then one of the eigenvalues goes to zero, and, in addition, the Pfaff dimension drops to 2, representing an isolated thermodynamic system. The Cardano function can be evaluated as, Cardano = (−(α − β)2 + 4 · wz2 )[wz2 + αβ + (γ − β − α)γ]2 .

(5.187)

It is apparent that there is a critical value of enstrophy that occurs when wz2 = (α − β)2 /4 = Critical Enstrophy.

(5.188)

The roots are complex, if wz2 is greater than the critical value (the Cardano function is positive), and real, if wz 2 is less than the critical value (the Cardano function is negative). If (for some reason) large values of Enstrophy decay to small values of enstrophy, the physical system mimics a phase change that is in correspondence with the condensation of a universal van der Waals gas (from above the critical isotherm) to condensates (below the critical isotherm). In contrast, if rotational motion that induces vorticity (not circulation) is somehow increased above the critical enstrophy, a change of phase takes place. The bottom line is that it is the antisymmetric components of dA, the analogues of vorticity in 3D, and the analogues of E and B fields in 4D, that can lead to complex eigenvalues and macroscopic spinor solutions‡ for the eigendirection fields of the Jacobian correlation matrix constructed from the 1-form , A. If there is 1 pair of complex eigenvalues, then the thermodynamic system is in the pure gas phase region above the critical isotherm. For non-equilibrium systems, the Pfaff dimension is 3, or more, which means the topological torsion 3-form exists. Remark 44 It must be restated that the theory of projective dimension and how it refines the thermodynamic Pfaff topological dimension is far from complete, but the key idea is that non-equilibrium thermodynamics is the domain of complex eigenfunctions, and macroscopic spinors. 5.6

Examples of the thermodynamics of Dynamical Systems

In this section, certain non-equilibrium examples will be worked out using the suggestions made above. In particular, the Jacobian matrix of the projective dual 1-form ‡

Macroscopic spinors are complex null eigendirection fields of the antisymmetric components of the Jacobian correlation.

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

(constructed from the components of the dynamical system) will be evaluated. The non-zero similarity invariants for the autonomous systems are in effect the similarity invariants of the 3 × 3 Jacobian matrix of the 3-component velocity field, and lead to a cubic equation that can be related to the van der Waals gas: Φ0 = ρ3 − XM ρ2 + YG ρ − ZA ⇒ 0.

(5.189)

It is possible to solve for the Thermodynamic Critical Point as the intersection of the implicit surfaces: Critical Point: (XM − 3 = 0) ∩ (YG − 3 = 0) ∩ (ZA − 1) 6= 0.

(5.190)

The thermodynamic critical point is where the molar density, ρ, represents 3 degenerate eigenvalues, all equal to unity. The common intersection may not exist for the given dynamical system, or may be defined only parametrically. The critical isotherm, and other features of a universal van der Waals gas can be constructed: 3 Critical Isotherm: ZA − XM = 0.

(5.191)

Local Stability: (XM ≤ 0) ∩ (ZA ≤ 0) ∩ (YG ≥ 0).

(5.192)

Hopf critical point: (XM = 0) ∩ (ZA = 0) ∩ (YG = 0).

(5.193)

The local stability of a dynamic system can be determined if the eigenvalues of the Jacobian matrix have no real parts. Hence local stability requires that,

The point where local stability is lost, but with the creation of oscillations of frequency, ω, (representing the imaginary parts of the eigenvalues) will be defined as the Hopf critical point,

Beyond the critical point will be the case of Hopf Oscillations such that, Hopf Oscillations : (XM = 0) ∩ (ZA = 0) ∩ (YG = ω2 > 0).

(5.194)

In this case two of the eigenvalues are pure imaginary and have no real part, and the third eigenvalue is zero. A second possibility would be the creation of Falaco Solitons. The critical point criteria is almost the same as for the Hopf problem, but now it would be required that TK = 0. The Falaco effect requires: Falaco Rotations : (XM = 0) ∩ (ZA = 0) ∩ (YG = ω 2 − 3b2 /4 < 0).

(5.195)

The algebra for the examples can be formidable, hence a Maple program was constructed (see Volume 6, "Maple programs for Non-Equilibrium systems" ) to perform the computations and find the similarity invariants for each problem. In many

Examples of the thermodynamics of Dynamical Systems

231

instances, the resulting algebraic results are too long to be printed, and the statement will be made that the invariant is not equal to zero. The Similarity Invariants and the 3D Brand invariants are displayed for each of the examples given below. The results of creating the projective dual 1-form (5.20) are also displayed for each example. In each case, a dual 1-form of Action, A, is created (to within a factor m) in the format: A = {Ak dxk − λdt}, with λ = (V m V m ), Ak = V k .

(5.196) (5.197)

The formulas of the preceding sections are used to compute the dA, AˆdA, and dAˆdA, in order to determine the Pfaff dimension of the given problem. When the equivalent of K = 2(E ◦ B)Ω4 = (dAˆdA) is not zero, the dual dynamical system 1-form is of Pfaff dimension 4 (and defines a symplectic manifold and abstractly an "open" thermodynamic system). Evolution in the direction of the topological Torsion vector is thermodynamically irreversible. If the coefficient of K, which equals 2(E ◦ B), is zero, then the Pfaff dimension is 3 (and defines a contact manifold and abstractly a "closed" thermodynamic system). A necessary and sufficient condition that the system be of Pfaff dimension 2 is given by the 3-Vector equation, E × A + Bλ = 0. A necessary condition is that the helicity must vanish, A ◦ B = 0. A maple program for computing these functions is given in Volume 6, "Maple programs for Non-Equilibrium systems". The description of these techniques as displayed below is very incomplete. The variation of the several different parameters and their bifurcation types has not been worked out in detail. The emphasis is on the similarity invariants and the Projective dimension. Remark 45 It is a problem of current research to understand the topological differences between general collineations, which are associated with thermodynamic processes, and general correlations, which are associated with the thermodynamic system. Of special interest is the analysis of those parameters that are, or are not, chiral sensitive. Even this introductory display requires intensive algebra. Without a symbolic math program such as Maple, the required attention span is not available to most, especially to oldfolks, like me. The Eulerian Rotator Dynamical System dx/dt = Vx = ayz, dy/dt = Vy = βzx, dz/dt = Vz = γxy.

(5.198) (5.199) (5.200)

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

The abstract dynamical system is the equivalent of the Eulerian rigid body rotator, where L = [Lx , Ly , Lz ], dLx /dt = (Iy − Iz )Ly Lz /(Ix Iy Iz ), dLy /dt = (Iz − Ix )Lz Lx /(Ix Iy Iz ), dLz /dt = (Ix − Iy )Lx Lyx /(Ix Iy Iz ).

(5.201) (5.202) (5.203)

are the three components of angular momentum and coefficients (Ix Iy Iz ) are moments of inertia. The Similarity Invariants for A = Vk dxk − V k Vk dt, XM Yg ZA TK

= = = =

0, −{βγx2 + γay 2 + aβz 2 }, 2aβγxyz, 0.

(5.204) (5.205) (5.206) (5.207)

The Hopf conditions are satisfied when one of the coefficients vanishes, say γ = 0, for then both XM = 0, and ZA = 0. Then Y g = −{αβz 2 }. For Hopf oscillations, Y g > 0 which requires that αβ < 0. Helicity and Brand Invariants 3D are, Helicity = 0, Enstrophy = (γ − β)2 x2 + (β − α)2 y 2 + (α − γ)2 z 2 , Stretch = −{γ 2 (α + β) + β 2 (α + γ) +α2 (β + γ)}xyz, Brand = {(γ − β)αyz}2 + {(β − α)γxy}2 + {(α − γ)βzx}2 .

(5.208) (5.209) (5.210) (5.211)

The Implicit (Pfaff dimension 3) surface of Zero Bulk dissipation (Zero Topological Parity) 2E ◦ B = K ⇒ 0 is given by the expression, K = 4{+γ 2 (β − α)x2 y 2 + α2 (γ − β)y 2 z 2 + β 2 (α − γ)x2 z 2 }.

(5.212)

The function, K, is either positive or negative. When K 6= 0, the projective dual 1form, A, defines a 4D symplectic manifold, and, abstractly, an "open" thermodynamic system. Suppose there is a symmetry about the z-axis such that γ = 0, and β = −α; then the function K reduces to, K = 4α3 z 2 (x2 + y 2 ) ⇒ 4(Ix − Iz )3 L2z (L2x + L2y ), Ix = Iy ,

(5.213) (5.214)

and the Hopf condition becomes Y g = +(Ix − Iz )2 L2z > 0.

(5.215)

Examples of the thermodynamics of Dynamical Systems

233

Transforming to the Euler spinning body notation, the result implies that the sign of the parity function K depends on whether the spinning body is oblate or prolate. Note that d{x2 + y 2 + z 2 }/dt = 0 which implies that the total angular momentum is constant. However, the system can tumble when K is not zero. This problem had application to the fact that the early satellites were supposedly spin stabilized about the prolate axis, and quickly entered into a tumbling mode, to the embarrassment of NASA. The Henon (the ABC flow) Dynamical System dx/dt = Vx = α cos(y) + β sin(z), dy/dt = Vy = β cos(z) + γ sin(x), dz/dt = Vz = γ cos(x) + α sin(y).

(5.216) (5.217) (5.218)

The Similarity Invariants for A = Vk dxk − V k Vk dt are, XM = 0, Y g = αβ cos(y) sin(z) + βγ cos(z) sin(z) +γα cos(x) sin(y), ZA = αβγ(cos(x) cos(y) cos(z) − sin(x) sin(y) sin(z)), TK = 0.

(5.219) (5.220) (5.221) (5.222)

The fact that XM is zero implies that the mean curvature of the implicit phase function vanishes. The result is a minimal hypersurface. The flow is Trkalian as the vorticity and the velocity are parallel. The Helicity and Brand Invariants 3D are, Helicity = (α2 + β 2 + γ 2 ) + 2(αβ cos(y) sin(z) + βγ cos(z) sin(x) +αγ cos(x) sin(y)), Enstrophy = Helicity the vorticity ≈ velocity. Stretch 6= 0. Brand 6= 0.

(5.223) (5.224) (5.225) (5.226)

The Implicit (Pfaff dimension 3) surface of Zero Bulk dissipation (Zero Topological Parity) 2E ◦ B = K ⇒ 0 is given by the expression, K = +12(αβγ{sin(x) sin(y) sin(z) − cos(x) cos(y) cos(z)} +4α2 sin(y) cos(y){γ sin(x) − β cos(z)} +4β 2 sin(z) cos(z){α sin(y) − γ cos(x)} +4γ 2 sin(x) cos(x){γ sin(z) − α cos(y)}.

(5.227)

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

When K 6= 0, the projective dual 1-form, A, defines a 4D symplectic manifold, and, abstractly, an "open" thermodynamic system. The K = 0 surface defines a contact manifold of Pfaff dimension 3 which is equivalent to a "closed" thermodynamic system.

The early Moffatt STF Dynamical System

dx/dt = Vx = αx + Ωz, dy/dt = Vy = βy + Gx2 + F xy, dz/dt = Vz = γz − Ωx.

(5.228) (5.229) (5.230)

This dynamical system was suggested early on and extended by Moffatt (19851990) [155] to represent the Stretch-Twist-Fold mechanism that might represent the evolution of vortex lines in certain fluid regimes. Later developments extended the dynamical system to variants of the Saddle-Node-Hopf system defined earlier by Langford (1983) [132], and discussed in more detail in examples which follow. The Similarity Invariants for A = Vk dxk − V k Vk dt are,

XM Yg ZA TK

= = = =

α + β + F x, +Ω2 + F yΩ + (αβ + βγ + γα) + (α + γ)F x, Ω2 (β + F x) + Ω(F (βy − 2Gx2 ) + αβγ + αγF x, 0.

(5.231) (5.232) (5.233) (5.234)

Examples of the thermodynamics of Dynamical Systems

235

The Helicity and Brand Invariants 3D are: Helicity = Ω(2βy + F x(y − z) +F (γzy + βy 2 − Gx2 y − αx2 ) +2γGxz, Enstrophy = 4Ω2 + Ω(F y) + F 2 (x2 + 2y 2 ) +4Gx(Gx + F y), Stretch = 4(β + F x)Ω2 + 4F (βy − 2Gx2 )Ω +{(β + γ)y 2 + (α − 2Gy)x2 }F 2 +{4γGxy}F + 4γG2 x2 , Brand = {(4G2 + 5F 2 )x2 + F 2 y 2 + F x(8β + 4Gy) + 4β 2 }Ω2 −2F {(F (α + γ − 2β)xy +2G(2F x + γ + 2β + α)x2 − 2β 2 y)}Ω +{α2 − 4βyG + 4G2 x2 )x2 +(γ 2 + β 2 )y 2 }F 2 +{4γ 2 Gxy}F + 4γ 2 G2 x2 .

(5.235) (5.236)

(5.237)

(5.238)

The Implicit (Pfaff dimension 3) surface of Zero Bulk dissipation (Zero Topological Parity) 2E ◦ B = K ⇒ 0 is given by the expression, K = −4{F (x2 + zy) − 2Gzx}Ω2 + − 4{3F 2 + (2Gx3 + (α − γ + 4β)xy − (γ + α)xz)F +2β 2 y + 2G(α − γ + β)x2 }Ω +{4(γy − βy + 2Gx2 )xy}F 2 +{4(2γGx2 − γ 2 z − β 2 y)y + 4(α2 + βGy + 2G2 x2 )x2 }F +{−8γ 2 Gxz}.

(5.239)

When K 6= 0, the projective dual 1-form, A, defines a 4D symplectic manifold, and, abstractly, an "open" thermodynamic system. The K = 0 surface defines a contact manifold of Pfaff dimension 3 which is equivalent to a "closed" thermodynamic system. It should be noted that the hypersurface K = 0 is sensitive to the sign of the rotation, Ω. That is, there are two such hypersurfaces of different polarization. The Lorenz Dynamical System dx/dt = Vx = α(y − x), dy/dt = Vy = −xz + Rx − y, dz/dt = Vz = xy − βz.

(5.240) (5.241) (5.242)

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

The Lorenz system is of the most famous chaotic dynamical systems, often remembered as the "butterfly effect". That is, the flap of the wings of a butterfly in South America can influence the global weather patterns. The Similarity Invariants for A = Vk dxk − V k Vk dt are, XM Yg ZA TK

= = = =

(−α − 1 − β), α(β + 1 + z − R) + β + x2 , −α{(β(1 + z − R) + x(x + y)}, 0.

(5.243) (5.244) (5.245) (5.246)

The criteria for Hopf oscillations requires that XM = 0, and ZA = 0. When these constraints are inserted into the formula for YG they yield YG(hopf ) . The criteria for oscillations is that YG(hopf ) > 0. There are actually two sets of Hopf Oscillations, one occurs when YG(hopf ) = (x2 − 1) > 0, with oscillationpfrequencies (the imaginary parts of the eigenvalues of the Jacobian matrix) ω = ± (x2 − 1). The second YG(hopf ) condition is a lengthy expression that is cubic in the parameter R. The Helicity and Brand Invariants 3D are: Helicity Enstrophy Stretch Brand

= = = =

α(xy − 2x2 ) + y 2 + βz(z − R + αz), R2 − 2(z + α)R + 4x2 + y 2 + (z + α)2 , −(z − R + α)2 β − 4αx2 − 4αxy − y 2 , (z − R + α)2 β 2 + 2xy(z − R + α)β +R2 x2 + R(2x(−xz + xα + y)) +(5α2 + y 2 + z 2 − 2αz)x2 +(α2 + 1)y 2 + 2(2α2 − z + α)xy.

(5.247) (5.248) (5.249)

(5.250)

The Lorenz equations have several bifurcation parameters. The implicit surface function K = 0 is doubly sensitive to all of the parameters, implying that the surface has two sheets depending upon of the sign of the bifurcation parameter. Herein the emphasis is arbitrarily placed upon the parameter R. The Implicit (Pfaff dimension 3) surface of Zero Bulk dissipation (Zero Topological Parity) 2E ◦ B = K ⇒ 0 is given by the expression, K a1 a2 a3

= = = =

a1 R2 + a2 R + a3 , {−4}, {−4β 2 z + 2(2z − α)x2 + 4βxy}, 4{+(αz − 2x2 − z 2 − y 2 )x2 + (α2 + 1)y 2 +α(α − β + 1)xy + β 2 (z 2 + zα)}.

(5.251) (5.252) (5.253) (5.254)

The K = 0 surface defines a contact manifold of Pfaff dimension 3 which is equivalent to a "closed" thermodynamic system, and can support stationary states.

Examples of the thermodynamics of Dynamical Systems

237

The Magnetic Dynamo Dynamical System dx/dt = Vx = −βx + γzy + Ωy, dy/dt = Vy = −βy + γzx − Ωx, dz/dt = Vz = 1 − xy + Dz 2 .

(5.255) (5.256) (5.257)

The Similarity Invariants for A = Vk dxk − V k Vk dt are, XM = −2β + 2Dz, Y g = +Ω2 + γ(x2 + y 2 − γz 2 ) + β 2 − 4βDz, ZA = +{2Dz}Ω2 + 2β 2 Dz − γβ(x2 + y 2 ) −2γ 2 Dz 3 − 2γ 2 yzx, TK = 0.

(5.258) (5.259) (5.260) (5.261)

The similarity invariants are also chiral invariants relative to the sign of the rotation parameter, Ω. The Helicity and Brand Invariants 3D are, Helicity Enstrophy Stretch Brand

= = = =

−2Ω(1 + Dz 2 + γyx) + β(x2 − y 2 )(γ + 1), +4Ω2 + (x2 + y 2 )(1 + 2γ + γ 2 ), +{8Dz}Ω2 − (γ + 1)2 (2γzyx + β(x2 + y 2 )), {(γ − 1)2 (x2 + y 2 ) + 16D2 z 2 }Ω2 + {2γz(γ − 1)(γ + 1)(x − y)(x + y)}Ω +(γ + 1)2 {(γ 2 (y 2 + x2 )z 2 + 4βγxyz + β 2 (x2 + y 2 )}.

(5.262) (5.263) (5.264)

(5.265)

The third Brand invariant is not chiral invariant relative to the sign of the rotation parameter, Ω. The Implicit (Pfaff dimension 3) surface of Zero Bulk dissipation (Zero Topological Parity) 2E ◦ B = K ⇒ 0 is given by the expression, K = a1 Ω2 + a2 Ω + a3 , a1 = −4(x − y)(x + y)(γ − 1), a2 = 8(+2D2 z 2 − γ(y 2 + x2 ) + 2Dz)z −16(βγ + Dz)xy, a3 = (4(γ + 1)(γ 2 z 2 + β 2 )(x − y)(x + y).

(5.266) (5.267) (5.268) (5.269)

The implicit hypersurface K = 0 defines the boundary between 4D volume expansions and contractions. In the Dynamo case the hypersurface is sensitive (relative to the sign of the rotation Ω) yielding 4 different rotation frequencies for which the

238

The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

system is a non-equilibrium system (contact manifold) of Pfaff dimension 3. For other rotational frequencies the system is dissipative in an irreversible sense. The surface is of Pfaff dimension 3, but can support "stationary" states far from equilibrium. If time is increasing, then a 4-volume contraction would require that the spatial 3-volume is contracting. If time is increasing, the spatial volume could remain unchanged, and still the 4-volume would be expanding. The K 6= 0 surface defines a symplectic manifold of Pfaff dimension 4 which is equivalent to an "open" thermodynamic system. Rossler Attractor Dynamical System dx/dt = Vx = W y − z, dy/dt = Vy = W x − αy, dz/dt = Vz = β + xz − γy + T z.

(5.270) (5.271) (5.272)

The Similarity Invariants for A = Vk dxk − V k Vk dt are, XM Yg ZA TK

= = = =

−α + (x + T ), −W 2 − α(x + T ) + z, −W 2 (x + T ) + W γ − αz, 0.

(5.273) (5.274) (5.275) (5.276)

The Helicity and Brand Invariants 3D are, Helicity Enstrophy Stretch Brand

= = = =

−W (γy + x + xz) + α(y + yz) + γz, γ 2 + (1 + z)2 , (2W γ − α(1 + z))(1 + z), W 2 {(1 + z)2 + γ 2 } − W {2γα(1 + z)} +α2 (1 + z)2 + γ 2 .

(5.277) (5.278) (5.279) (5.280)

The stretch and fold Brand invariants are chiral sensitive to the sign of the parameter W. The Implicit (Pfaff dimension 3) surface of Zero Bulk dissipation (Zero Topological Parity) 2E ◦ B = K ⇒ 0 is given by the expression, K a1 a2 a3

= = = =

a1 W 2 + a2 W + a3 , 4(y + yz + γx), −4(α(γy + x + xz) + z 2 + z), +4α2 (y + yz) + 4γ(γy − β − zx − T z).

(5.281) (5.282) (5.283) (5.284)

Again the implicit hypersurface K = 0 is sensitive with respect to the sign of the parameter, W. There are two distinct chiral implicit hypersurfaces (contact manifolds)

Examples of the thermodynamics of Dynamical Systems

239

representing domains of Pfaff dimension 3. When K 6= 0, the projective dual 1-form, A, defines a 4D symplectic manifold, and, abstractly, an "open" thermodynamic system. The Brusselator Dynamical System dx/dt = Vx = α − βx + γx2 y − Dx, dy/dt = Vy = +βx − γx2 y, dz/dt = 0.

(5.285) (5.286) (5.287)

A more convenient parametric notation is given by the expression where Ω = β, and ΩG = γ, dx/dt = Vx = α − Ω(x − Gx2 y) − Dx, dy/dt = Vy = Ω(x − Gx2 y), dz/dt = 0.

(5.288) (5.289) (5.290)

The Similarity Invariants for A = Vk dxk − V k Vk dt are, XM Yg ZA TK

= = = =

−(1 − Gx)2 + D)Ω − D, {DGx2 }Ω, 0, 0.

(5.291) (5.292) (5.293) (5.294)

The Brusselator is spatially a 2D system, hence it is natural to expect that ZA = 0 and Tk = 0. The criteria for Hopf oscillations requires that XM = 0, and ZA = 0. When these constraints are inserted into the formula for YG they yield YG(hopf ) . The criteria for oscillations is that YG(hopf ) > 0, Hopf condition YG(hopf ) = Ω2 G(−1 + 2Gxy − Gx2 )x2 > 0, p Oscillation frequencies ω = ± YG(hopf ) .

(5.295) (5.296)

The Helicity and Brand Invariants 3D are, Helicity Enstrophy Stretch Brand

= = = =

0, Ω2 {Gx2 + 2Gxy − 1}2 , 0, 0.

(5.297) (5.298) (5.299) (5.300)

Although the problem is spatially two dimensional, it is still possible that the system is a non-equilibrium system for the Topological Torsion vector need not be zero.

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

The projective dual 1-form, A, is of Pfaff dimension 3 and defines abstractly a nonequilibrium closed thermodynamic system as a contact manifold. In such circumstances the Pfaff dimension 2 surface, Φ = 0, is an implicit surface in 2D+1 spacetime that can be computed. The Φ ⇒ 0 (Pfaff dimension 2) Implicit surface is: Φ = (a1 Ω2 + a2 Ω + a3 )Ω, a1 = −6x2 (Gxy − 1)2 ({Gx2 + 2Gxy − 1), a2 = 2x((Gxy − 1)(4γDx3 + 5GDx2 y −4αG(x2 + xy) − 3Dx + 2α), a3 = (α − Dx)(GD(3x3 + 4x2 y) −αG(3x2 − 2xy) − 3Dx + α).

(5.301) (5.302) (5.303) (5.304)

The zero set of the function Φ defines the surface of Pfaff dimension 2 in the domain of Pfaff dimension 3. As the 1-form of Action is constructed in terms of only two spatial variables and one time variable, the top Pfaff dimension is at most 3. Indeed the Brusselator is a non-equilibrium system except on the zero set of the function Φ. It is sensitive to the sign of the rotation parameter, Ω, which indicates that there are two states of polarization. On the zero set of Φ the Pfaff dimension has been reduced to 2, and the system is no longer in a non-equilibrium state, but represents, abstractly, an isolated thermodynamic system. The Chiral Separation (Kondepudi) Dynamical System dx/dt = Vx = βx + 2γxy, dy/dt = Vy = D + βy − γ(x2 + y 2 ) − E(y 2 − x2 ), dz/dt = 0.

(5.305) (5.306) (5.307)

The Similarity Invariants for A = Vk dxk − V k Vk dt are, XM Yg ZA TK

= = = =

2(β − Ey), −2βEy + β 2 − 4γE(x2 + y 2 ) + 4γ 2 (x2 − y 2 ), 0, 0.

(5.308) (5.309) (5.310) (5.311)

The criteria for Hopf oscillations requires that XM = 0, and ZA = 0. When these constraints are inserted into the formula for YG they yield YG(hopf ) . The criteria for oscillations is that YG(hopf ) > 0, Hopf condition YG(hopf ) = +4γ(γ − E)x2 − E 2 y 2 −4γ(γ + E)y 2 > 0, p Oscillation frequencies ω = ± YG(hopf ) .

(5.312) (5.313)

Examples of the thermodynamics of Dynamical Systems

241

The criteria for XM = 0 corresponds to those points where the system locally becomes unstable. If Y g > 0 the system exhibits oscillations, and if Y g < 0 the system exhibits a minimal surface Hopf condition (related to soliton formation). Kondepudi (p. 431-436 [180]) has demonstrated that this dynamical system could correspond to a (fictitious) chemical reaction which produces chiral separation. He states: "To date, no chemical reaction has produced chiral asymmetry in this simple manner. However symmetry breaking does occur in the crystallization of NaClO3 ." The Helicity and Brand Invariants 3D are, Helicity Enstrophy Stretch Brand

= = = =

0, 4x2 (−2γ + E), 0, 0.

(5.314) (5.315) (5.316) (5.317)

The system has a non-zero 3-form of topological torsion of one component (even though the helicity is zero) which implies the thermodynamic system is a closed nonequilibrium system, except for domains where the 3-form AˆF of the dual projective 1-form vanishes. The 3-form reduces to a single component Φ given by the equation, Φ = β 2 {(E − 4γ)x2 + 5y 2 E}2x +β{(3γ 2 − 2E 2 )(y 2 − x2 ) − 2γEx2 + 2ED − γD)}8xy +γ 3 {−3y 4 − 8x2 y 2 − 4x4 }8x +γ 2 {(8Dx2 − 17y 4 E + 16y 2 D + 11x4 E + 14x2 y 2 E)}2x +γ{(12E 2 y 2 x2 − 14DEx2 + 6DEy 2 − 4D2 − 10E 2 x4 − 2E 2 y 4 )}2x (5.318) +{3E 3 (y 2 − x2 )2 + 3D2 E + 6E 2 D(x2 − y 2 )}2x. On the zero set of Φ the Pfaff dimension has been reduced to 2, and the system is no longer in a non-equilibrium state, but represents abstractly a isolated thermodynamic system. The Belousov-Zhabotinsky Dynamical System dx/dt = Vx = αy + βx − γxy − 2Dx2 , dy/dt = Vy = −αy − γxy + (M/2)Ez, dz/dt = 2βx − Ez.

(5.319) (5.320) (5.321)

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

The Similarity Invariants for A = Vk dxk − V k Vk dt are: XM = β − γ(y + x) − 4Dx − α − E, Y g = {α − β + γ(x + y) + 4Dx}E + 4D(αx + γx2 ) +2γαy − β(α − γx), ZA = −E{Mβ(γx − α) + 4D(αx + γx2 ) −βγx − βα + 2αγy}, TK = 0.

(5.322) (5.323) (5.324) (5.325)

The criteria for Hopf oscillations requires that XM = 0, and ZA = 0. When these constraints are inserted into the formula for YG they yield YG(hopf ) . The criteria for oscillations is that YG(hopf ) > 0,

Hopf Constraint :

Oscillation frequencies

YG(hopf ) = +β(3γy + 8Dx − β) −E(2γy + 4Dx − β + E) −2γ 2 (y 2 − yx) − 4Dx(4Dx + 3γy), p : ω = ± YG(hopf ) .

(5.326) (5.327)

The Helicity and Brand Invariants 3D:

Helicity = ME(−1/2αy − 1/2βx + 1/2γxy + Dx2 − βz) +2β(α(y − x) + x2 γ) + Ez(γ(y − x) + α), Enstrophy = 1/4M 2 E 2 + 4β 2 + γ 2 (y − x)2 +2γα(y − x) + α2 , Stretch = M 2 E 2 (1/4β − 1/4γy − Dx) +ME(β(γ(y − 3x) + 3α)) +E(γ(y − x) + α)2 − 4β 2 (α + γx), Brand 6= 0 = {...}F 2 + {...}F + {...}.

(5.328) (5.329)

(5.330) (5.331)

The Implicit (Pfaff dimension 3) surface of Zero Bulk dissipation (Zero Topological

2D-Limit Cycles and 3D-Limit Surfaces

243

Parity) 2E ◦ B = K ⇒ 0 is given by the expression,

K = a1 M 2 + a2 M + a3 , a1 = {E 2 (α − γx)z}, a2 = −2E{(2zEβ − 5β 2 x − αyβ − 8D2 x3 +α2 y + 2βxγy − γ 2 xy 2 + 4αyDx −6γx2 yD + 2βαz + 2βγxz + 6βDx2 −γ 2 x2 y + αy 2 γ)}, a3 = {16βα2 y − 16βαDx2 + 16βγ 2 x2 y +16βγx3 D − 8Exβα − 8Exβγy +4E 2 zγy − 4E 2 zγx + 8αβ 2 x −8x2 γβ 2 + 8Ex2 βγ + 4E 2 zα}.

(5.332) (5.333)

(5.334)

(5.335)

The K = 0 hypersurface is chiral sensitive with respect the sign of the parameter, M. Most of the examples above have more than 1 family parameter (meaning not x,y,z, or t). If all but one family parameters are held fixed, then there are six classical bifurcations [132] as that family parameter is varied. These can be defined in terms of normal forms for the bifurcation parameter, ξ. The bifurcations occur when there are null eigenvalues of the Jacobian matrix, or when two eigenvalues are pure imaginary. The classical analysis looks at the stationary points of the vector field, and discusses what happens at such points as the bifurcation parameter changes. Hence the similarity invariants of the Jacobian matrix carry information about the number and existence of null eigenvalues. For example, if the determinant of the Jacobian is zero then there is at least 1 null eigenvector, indicating a point where there is a topological singularity. If two parameters are allowed to vary, solutions can also have topological singularities which occur when the birfurcations of each family parameter interact. For example there can be an interaction between the Hopf bifurcation and one of the two possible extremal fields on the 3D contact structures that occur as topological defects in the 4D symplectic manifold [203]. An example of the Hopf reduction and its relationship to a minimal surface is given in section 3.5. Other examples are to follow. 5.7

2D-Limit Cycles and 3D-Limit Surfaces

5.7.1 Closed 2D Limit Curves This example is related to the Eliashberg analysis whereby a 3D contact manifold (which in general is of odd dimension 2n+1) has a local representation defined by a non-zero 1-form, A, with a Pfaff Topological dimension equal to 3. The 1-form can have several formats, producing slightly different Pfaff sequences, and a non-zero 3-form, AˆdA:

244

The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

Standard "tight" Contact Manifold Translational A dA AˆdA dAˆdA

= = = =

ydx − dz, −dxˆdy, dxˆdyˆdz, 0.

(5.336) (5.337) (5.338) (5.339)

Standard "tight" Contact Manifold Rotational A dA AˆdA dAˆdA

= = = =

(ydx − xdy) − dz, −2dxˆdy, 2dxˆdyˆdz, 0.

(5.340) (5.341) (5.342) (5.343)

The point is to note that the 2n+1=3-volume element, Ω = dxˆdyˆdz is global. There are no singularities for the "tight" contact manifold, where AˆdA =(non-zerofunction)·Ω. These structures have been discussed before in relation to the classic thermodynamic potentials. However, there is another class of 1-forms which are said to be "overtwisted". The fact that a 3-form, AˆdA, is overtwisted implies the existence of a limit cycle [69]. As a simple example consider the Pfaff sequence defined by: Standard "overtwisted" Contact Manifold Rotational A dA AˆdA dAˆdA

= = = =

(ydx − xdy) − (f (x, y))dz, −2dxˆdy − (∂f /∂xdx + ∂f /∂ydy)ˆdz, (+2f (z) − x∂f /∂x − y∂f/∂y)dxˆdyˆdz, 0.

(5.344) (5.345) (5.346) (5.347)

In each of the overtwisted cases, the coefficient of the 3D volume form can have singularities, depending upon the form of the function f (x, y). In the rotational case, in domains where the function f (x, y) is homogeneous of degree 2 in (x, y), the volume element has singular sets which can form limit cycles. For purposes herein an "overtwisted" contact submanifold of a symplectic manifold has a periodic orbit. A crucial feature of limit cycles of the Van der Pol type is that the mechanical energy of the system is time dependent, decaying from any set of initial conditions to a "breathing" system, with the power being positive on certain parts of the cycle and negative on others. As such, the limit cycle orbits are not tractable in terms of an extremal Hamiltonian analysis. The objective of this

2D-Limit Cycles and 3D-Limit Surfaces

245

section is to demonstrate that the Van der Pol oscillator can be considered as an example of a symplectic system on a four-dimensional variety. First the concept of an "overtwisted" contact submanifold should be made clear. Consider the example of a hyperboloid of revolution. By looking at the surface, no concept of whether the hyperboloid is "twisted" is evident. However, the hyperboloid my be constructed by the envelope of an array of straight lines from one lower base circle to another upper base circle. If the threads connecting the points on one boundary circle to the other are orthogonal to the two circles, the resulting envelope of threads is a cylinder. If, with the threads attached, the upper circle is rotated by an angle of less than pi with respect to the bottom circle, the envelope of the threads forms a hyperboloid of revolution as a ruled surface with an discernible twist. If the relative rotation of the upper circle is pi, the envelope is a twisted cone. If the relative rotation is more that pi degrees, it is apparent that the twist is "tight". The fact that the hyperboloid is twisted is an artifact that the twisted surface is constrained to a two-dimensional surface in a non-integrable three-dimensional "contact submanifold". It is also apparent that on the twisted hyperboloid with a relative rotation of less that pi, there is a closed path around the enveloping hyperboloid which is a minimum in length. It is the circle of constriction. The circle of constriction is the epitome of the "limit cycle". A orbit with initial conditions on the surface will follow some "minimal" trajectory winding around the hyperboloid and being gradually attracted to the limit cycle. Consider the Lagrangian, L(t, x, v) = −1/2kx2 + 1/2mv 2 − β 2 (v3 /3c2 − v)x + m0 c2 , on a space of variables {t; x, v, p}. β 2 , c, m0 } are constants) is,

(5.348)

The Action 1-form (presuming that {κ, σ =

A = L(x, v, t)dt + p(dx − vdt).

(5.349)

Explicit evaluation of the functions and 1-forms defined above yield the following equations: H (h/2π)k ∆p dΘ

= . = = =

1/2kx2 + pv − 1/2mv 2 − m0 c2 + β 2 x(v 3 /3 − v), p − ∂L/∂v = p − mv + β 2 (v2 − 1)x 6= 0, dp − ∂L/∂xdt = dp + [κx + β 2 (v3 /3c2 − v)]dt ⇒ 0, [p − mv + β 2 (v2 /c2 − 1)x]{dv − adt} 6= 0.

(5.350) (5.351) (5.352) (5.353)

Note there is a critical velocity, v2 /c2 − 1 = 0. The criteria that (h/2π)k 6= 0 insures that the 4D space is symplectic in terms of the 2-form, dA; the 2-form is of maximal (symplectic) rank 4. The non-zero function k has a covector gradient that is never zero, dk = 1dp − mdv − vdm + β 2 (v2 /c2 − 1)dx + 2xβ 2 v/c2 dv. (5.354)

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

This Fomenko symplectic system requires that the Lagrange-Euler constraint be satisfied, hence the equations of motion are given by a Pfaffian equation which acts as a differential constraint on the 4D space, ∆p = dp + [κx + β 2 (v3 /3c2 − v)]dt = 0.

(5.355)

The criteria for a subspace upon which the Action 1-form is completely integrable is given by the mass equation, dp = [{p(p − ∂L/∂v)}/L}]dv ⇒ mdv.

(5.356)

On the surfaces of constant "mass" in the 4D domain, m(x, v, p) = [{p(p − β 2 (v 2 /c2 − 1)x)}/{L}], ⇒ m0 ,

(5.357) (5.358)

the Pfaffian equation of motion becomes, mdv + [κx + β 2 (v 3 /3c2 − v)]dt = 0.

(5.359)

Adjoining to this 1-form the hypothesis of null differential fluctuations of position, (dx − vdt = 0), leads to the second degree differential system of the Van der Pol equation on the 2D integrable subspace of the 4D symplectic system, m(d2 x/dt2 ) + β 2 [(dx/dt)2 /3c2 − 1](dx/dt) + κx = 0.

(5.360)

Note that the mass constraint implies that the momenta, p, is not canonically defined, and depends on both velocity and coordinate for constant mass. Starting from any initial condition {0, q0 , v0 , p0 ≡ φ(m, q0 , v0 }, the power either decays or increases always being attracted to the limit cycle or its whiskers. When the limit cycle is reached the power P ower = v • dp/dt becomes cyclic. The constant mass function, m, defines a surface in the three-dimensional subspace of (x, v, p). Orbits of the evolutionary constant mass process reside on this surface. The usual velocity-position display of the Van der Pol Oscillator is not in phase space, but is a projection from an orbit on the constant mass surface to the v-x plane. 5.7.2

Closed 3D Limit Bounding Surfaces

Closed, re-entrant, secondary flows with coherent topological structures may be constructed and embedded in environmental surroundings of different topology. The classic example is the Hill spherical vortex given in Figure 5.11.

2D-Limit Cycles and 3D-Limit Surfaces

247

Figure 5.11 Topological defects: Vorticity Bubble in Streamline flow The flow is confined by a bounding implicit surface, where the implicit surface function in 3D space, Φ(x, y, z), is a "conformal" invariant of the velocity field that defines the process, L(V ) Φ = λΦ, or i(V )dΦ = V ◦ grad Φ = λΦ.

(5.361) (5.362)

The velocity field is bounded by the implicit surface, and resides on the surface. That is, the flow is never across the surface, but is tangential to it. It should be noticed that if, (x, y, z) ⇒ (U, V, W ), and Φ(x, y, z) ⇐ Ψ(U, V, W ),

(5.363) (5.364)

then the "conformal" constraint becomes equivalent to Euler’s equation for a homogeneous function of degree D in the variables (U, V, W ) (see Chapter 8): L(V ) Φ = λΦ, L(V ) Ψ(V ) = DΨ(V ).

(5.365) (5.366)

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

Of particular interest to me are those flows with compact boundary which have domains with topological torsion and a Pfaff dimension > 2. These domains have finite helicity density, and their streamlines have Frenet torsion. The streamlines may be either ergodic or closed within the bounded domain. These structures, which are essentially topological defects, have received little exploitation in the hydrodynamic literature. Such long lived topological structures can appear in Tertiary Hopf bifurcations, which will be discussed in more detail in the next section. A most remarkable feature of the Tertiary Hopf bifurcations is that they can be shown to be exact solutions to the Navier-Stokes equations in rotating coordinates [207]. A generalization of the Saddle Node Hopf (SNH) solution to the Navier-Stokes equations which exhibits compact domains is given by the vector field, V = [u, v, w], where, SNH : Generalized solution u = C x ∂f (z)/∂z − ωy, v = C y ∂f (z)/∂z + ωx, w = +α f (z) − T h(r2 ).

(5.367) (5.368) (5.369) (5.370)

The terms involving Ω represent the rotation or swirl of the fluid. The constraint of incompressibility requires that 2C = α. The polynomial functions, f (z) and h(r), are arbitrary, but polynomial expressions of the quartic variety are of interest to the problem of wakes. The specialization that fits the classic saddle-node Hopf solution, and which is studied in detail below, is given by the quadratic polynomial formulas, f (z) and h(r): SNH : Classic polynomial format αf (z) = F − D z 2 and h(r) = r2 + 10.

(5.371) (5.372)

The simpler quadratic polynomial solutions are of interest to single defect structures, and it is easy to show that for all quadratic flows the Navier-Stokes viscous dissipation term is representable (locally) by a gradient field, grad div V − curl curl V ⇒ grad Ψ.

(5.373)

It follows that the pressure and viscous forces associated with such flows do not produce cyclic work, hence the systems are conservative. Energy of rotation induced by viscous torques is recoverable! It is also true that the vorticity of a quadratic flow is globally harmonic, curl curl curl V = 0,

div curl curl V = 0.

(5.374)

Recall that harmonic vorticity implies conservation of angular momentum, the closure criteria required to produce harmonic vorticity does not imply necessarily that a

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gradient representation for the viscous force term is globally valid. Also a harmonic vector field is known to generate a Minimal Surface [165]. It is interesting to note that the degree of the polynomials, f and h, is twice the number of distinct defects that the flow represents. As the quadratic velocity field does not produce irreversible dissipation in a Stokes fluid (the Stokes term is a gradient field) then it becomes apparent that the number of defects must exceed two if entropy is to increase. In all cases the bounding "ellipsoidal" surface to the secondary flow generated by the Saddle Node Hopf equations will be given by the zero set of the function, Φ(x,y,z), such that V· grad Φ = λ Φ(x, y, z). (5.375) Note that this requirement forces the velocity field to be "tangent" to the bounding surface, Φ(x, y, z) ⇒ 0. The bounding function, Φ, is a conformal invariant of the flow, V, with conformality factor, λ. When the conformality factor vanishes, the bounding ellipsoid is an absolute invariant of the flow. These two cases correspond to (asymptotic) continuity and uniform continuity, respectively. Further note that the "renormalized" velocity field, Φ · V, satisfies the "no-slip" condition on the bounding surface. This important result states that the streamlines of a no-slip flow and the streamlines of a flow that leaves the bounding surface a conformal invariant are the same. This correspondence between Eulerian flow and no-slip flow around obstacles is a curiosity that has appeared in the literature before, but the association to a conformally invariant bounding surface is apparently novel. The representation of a no-slip surface by the conformal constraint may be utilized to show that the Euler characteristic of the flow manifold is completely determined by the zeros of V that are simultaneously "singular" points of the bounding surface. In the general polynomial case, if the function h(r) is written as a polynomial, X h(r) = hn rn , (5.376) then the bounding "ellipsoid" may be constructed as the zero set of, Φ(r, z) = G(r) + D F (z), where, G(r) = It follows that,

X

T D hn rn /(D − nγ).

V· grad Φ = {D ∂f (z)/∂z } Φ(x, y, z).

(5.377) (5.378) (5.379)

All such bounded compact flows are solutions to the Navier-Stokes equations in a rotating frame of reference. An example of the bounding ellipsoid is given in Figure 5.16. Remark 46 The bounding surface is therefore a zero-form that is homogeneous of degree {D ∂f (z)/∂z }. This implies that the bounding surface can be fractal.

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The Saddle Node Hopf Solution

The simple Saddle Node Hopf flow field as given by the equations above will be studied as a function of the mean flow parameter, F . The study as presented in the figures below will demonstrate that a torsion bubble defect, containing a bounded secondary reverse flow, can be produced as a result of a parametrically induced Hopf bifurcation. The topological creation and persistence of a large scale structure in a Navier-Stokes flow is thereby demonstrated by this example. The bounded secondary flow can be embedded in a closed flow representing the environment.

Figure 5.12 Bubble defects and a Saddle Node bifurcation

A specialization of the two polynomials to the quadratic forms (5.371) leads to a flow with two fixed points of the saddle node Hopf variety. In the vicinity of {z, r} = {+((F − 10T )/D)1/2 , 0} the fixed point has two positive and one negative Lyapunov exponents {+, + ,-}. while the fixed point at {z, r} = {−((F − 10T )/D)1/2 , 0} has two negative and one positive exponents, {-,-, +}. Both regions are locally unstable, but the flow in the neighborhood of one fixed point is coupled to the flow in the neighborhood of the second fixed point to produce a globally stable structure.

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Figure 5.13 Topological evolution and Torsion bursts Expressions for the vorticity and the helicity density of the SNH flow may be analytically computed. In particular it is to be noted that the vorticity is independent of the parameter F , but the helicity density depends on F . The topological transition of the surface of null helicity density is described qualitatively as a function of the flow parameter, F , in the following Figure 5.14. For low flow rates (F = 2) the surface of zero helicity density is a hyperbola of revolution consisting of one sheet. The regions of negative helicity density are connected, as are the regions of positive helicity density. For the flow described, the value F = 10 is a critical value, where above the critical value the helicity density is a disconnected surface of two sheets. It is also possible to compute the flow lines for various values of the flow rate F . For the parameters chosen, and such that divV = 0 (T = 1, D = C), when the flow parameter F = 10, another critical point is reached, for now the paired Lyapunov exponents can become complex. A locus of Hopf points forms a circle of radius, r = {+((F − 10T )/T )1/2 } in the z = 0 plane. At this critical parameter point, the surface of zero helicity density becomes a cone. Except for those streamlines in the vicinity of the origin, the swirl component has been suppressed in the figures; only the envelope of the swirling streamlines is displayed, for reasons of visual clarity.

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Figure 5.14 Emergence of a Bubble topological defect When the flow parameter F exceeds 10, a defect torsion bubble will form with an axial dimension given by z = +[(F − 10T )/D]1/2 and with the radius of the bounding surface given by r = {+(2(F − 10T )/T )1/2 . The Hopf circle is always contained by the bounding ellipsoid. Note however that the surface of null helicity density becomes a hyperbola of two sheets, and a "helicity gap" develops (along with the torsion bubble) separating domains of negative helicity density. For F = 18, the streamlines are presented in the lower portion of the above Figure 5.14. Again, only the envelope of the streamlines outside the defect are plotted, while the swirl is detailed for trajectories within the defect bubble. Note that the secondary flow has a reverse velocity component within the bubble defect, and that all streamlines are tangent to the bounding surface. Recall that although the visual defect appears to be associated with a "vortex burst", such is not the case. During this parametric creation (only the mean flow parameter F has been changed) of a large scale structure in a viscous media, the lines of vorticity are the same for all values of the parameter F. The vorticity of the saddle node Hopf flow is an absolute invariant of the flow itself, and its value is independent of the mean flow parameter, F . It is suggested that this visual effect is better described by the words, "torsion burst", than by the words "vortex burst". To make a more direct contact with the experimental exposes of the "torsion bursting" problem, a slight modification to the SNH solution is presented in subsequent section. A small cubic addition to the f (z) polynomial changes topological features of the solution. 5.7.4 The Hysteretic Hopf Solution If the polynomial, f (z), is modified from the SNH quadratic to include a term cubic in z (add 0.05z 3 to the SNH formula for the z-component of velocity), SNH : hysteretic polynomial format h(r) = r2 + 10, αf (z) = F − D z 2 + 0.05z 3 and

(5.380) (5.381)

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then the streamlines for low values of F will appear as given in subsequent Figure 5.15. The near central streamline flows in the direction of the single real fixed point, and the diverges in a spiral fashion around the surface generated by the cubic polynomial. If the parameter F is increased to 26, then again a torsion bubble is formed upstream from and before the spiral instability is encountered. The streamlines give a visual appearance that may be compared to the experimental data presented for the "vortex bursting" problem [143].

Figure 5.15 Emergence of a Torsion Burst In the numerical evaluations of the streamlines, if a small div V 6= 0 term is permitted, then the central streamlines will enter into the torsion bubble in the vicinity of the downstream "fixed" point. The streamline remains trapped within the "bubble" for long periods of computing time. This result mimics the experimental dye results presented in [143], and is suggestive of the Arnold diffusion process in phase space when the KAM (bounding) surface becomes non-compact and has non-Isolated boundary points or holes through which the phase space trajectories can "diffuse". 5.7.5 Anisotropic Modifications of the SNH flow In the following example the function f (z) will be constrained to the format of a quadratic form, A f (z) = (F − D z 2 ), (5.382)

but the polynomial h(r), as well as the coefficients, C, will be modified to permit anisotropy in the x-y plane, h(r) ⇒ H(x, y) = S x2 + T y 2 .

(5.383)

The modified saddle node Hopf solution becomes the flow V = [u, v, w], where, u = (C − R) x ∂f (z)/∂z − ωy, v = (C − R) y ∂f (z)/∂z + ωx, w = +D f (z) + S x2 + T y 2 .

(5.384) (5.385) (5.386)

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R is a measure of the "asymmetry" of the bounding ellipsoid in the z = 0 plane and will be defined as the "perturbation parameter". The solution has axial symmetry when R = 0. The "order parameter" is defined as the mean radial "size" of the bounding ellipsoid. The bounding ellipsoidal surface is given by the expression, Φ(x, y, z) = D f (z) + {SD/(D − 2C − 2R)}x2 + {T D/(D − 2C)}y 2 .

(5.387)

It is possible to choose R, S, and T such that the bounding ellipsoid is an ellipsoid of revolution, or even a sphere. The example flow is remarkable in that the two fixed points along the z-axis form the polar points of the bounding ellipsoid, and are adjoined by a Hopf ellipse in the z = 0 plane. For various values of R < 2D, the Hopf ellipse is confined within the bounding ellipsoid and the resulting streamlines of motion lie on nested tori with a guiding center dictated by the Hopf ellipse. As demonstrated above, the streamlines are confined to toroidal surfaces of Euler characteristic zero. At larger values of R, 20 < R < 40, the Hopf ellipse penetrates the bounding ellipsoid, and the resulting streamlines are confined to a surface that has a non-zero Euler characteristic, and the appearance of a topological button. A topological phase transformation takes place when R = 2D. The streamlines of the flow in the torus phase exhibit Poincare sections that are indicative of periodic (section dimension 0) or doubly periodic (section dimension 1) motion. The streamlines of the flow in the button phase exhibit Poincare sections that appear to be chaotic (section dimension > 1). It should be remarked that (5.384) represent an exact solution of the Navier-Stokes equations in a rotating frame of reference, when S = T . When S 6= T Navier-Stokes theory requires an anisotropic external force. 5.7.6 Phase Transitions in Dynamical Systems As suggested by me at the IUTAM conference on Topological Hydro Mechanics, a dynamical system should be expected to exhibit thermodynamic phase transitions induced by parametric variations. Recall that phase transitions imply a change in topology. If the Saddle Node Hopf solution to the Navier-Stokes equations is modified slightly to include an anisotropic perturbation parameter, R, then a study of the parametric change of topology in the bounding surface of the secondary flow can be made analytically. In brief, for the example equations presented above, the ellipsoidal bounding surface of the modified SNH flow admits two isolated singular points for small values of the anisotropy parameter. The radial "size" of the bounding ellipsoid is a measure of the topological coherence in the defect, and will be defined as an order parameter for the flow. Confined within the interior of the bounding ellipsoidal surface is a closed curve of Hopf singular points in the form of an ellipse. As the perturbation parameter is increased, the Hopf ellipse grows until the Hopf ellipse penetrates the bounding ellipsoidal surface, such that the bounding surface now admits six singular points.

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Figure 5.16 Hopf Z-plane ellipsoidal singularity The radial size of the bounding ellipsoid, which acts as an "order" parameter, decreases with increasing anisotropic perturbations. The interior solution streamline trajectories are always confined by the bounding surface, and are rotationally guided by "or attracted to" the Hopf ellipse.

Figure 5.17 Chaotic flow lines and a topological defect

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The trajectories, representing evolving streamlines, sweep out paths that are confined within a torus for small values of the perturbation parameter. As the perturbation parameter is changed, a topological phase transition takes place when the Hopf ellipse penetrates the bounding ellipsoid, and thereafter the stream lines are confined to a surface with the appearance of a topological button with 2 handles. At a certain point, further increase of the perturbation parameter, R, destroys the dynamical stability of the system, and the order parameter goes to zero (or becomes imaginary). The variation of the order parameter is associated with a change of topology, and a corresponding change of phase. The streamlines in the torus phase numerically seem to be non-chaotic (in that the topological handle along the z-axis maintains its "size"). However, the streamlines in the button phase exhibit "Lagrangian" chaos. The throat size of each of the two Hopf induced handles varies intermittently, and the Poincare return map fills large domains of the return map surface of section. A portion of a single stream line is presented above as a chaotic streamline, demonstrating the topological structures of the flow solution in the button phase.

Figure 5.18

Topological Phase change in modified Saddle Node Hopf fluid flow

As mentioned above, an example of a parametrically induced phase transition in a dynamical system was presented by me at the Cambridge IUTAM conference. At the same conference, I was stimulated by the numerical work of Bajer, et. al. [155]], which was presented for a particular (and ingeniously chosen) stretch-twist-fold vector field. After the presentation I realized that the stretch-twist-fold vector field of Moffatt was a special case of the recently discovered Saddle Node Hopf solution to the Navier-Stokes equations, and should exhibit a topological phase transition. The problem was formulated analytically with results, as presented above, that extend the Bayer numerical results. The detailed equations for the analytical modified SNH flow are given in the previous section.

The Falaco Soliton - A Topological String in a Swimming Pool

5.7.7

257

Phase transitions and topological evolution

In the work described above, the theory of continuous topological evolution was utilized to explain the creation of large scale structures in continuous media. Such creation processes involve topological change. Recall that conservative classical Hamiltonian mechanics is not applicable to problems that involve topological change. Topological evolution implies that the transition from an initial to a final state is NOT a homeomorphism. It follows that the transformation representing topological change is either 1.) NOT continuous, or 2.) irreversible, or both. Otherwise, topological properties would be invariant, as a continuous reversible transformation is a homeomorphism that preserves all topological properties. In the development of these ideas (which have been limited to continuous processes) the third order tensor field of topological torsion has played a dominant role. The divergence of this tensor field leads to a fundamental equation of continuity with source, where the source term is the fourth rank tensor field of topological parity. These topological concepts are independent from a metric or other geometrical constraints (such as a connection) that may be imposed on the variety, {x,y,z,t}. The intuitive idea is that a large scale structure is a topological defect in an otherwise homogeneous domain.

5.8 5.8.1

The Falaco Soliton - A Topological String in a Swimming Pool The Hopf bifurcation

Although of importance to the cosmological concept of a universe expressible as a low density (non-equilibrium) van der Waals gas near its critical point, the reduction of the Jacobian characteristic polynomial into a cubic factor is not the only cosmological possibility. Of particular interest is the factorization that leads to a Hopf bifurcation (where the factorization will contain quadratic factors). In this case the Characteristic Polynomial (always) vanishes, the Adjoint cubic factor of eigenvalues vanishes, ZA = 0, the mean linear average of eigenvalues vanishes, XM = 0, (indicating a possible minimal surface), but the Gauss quadratic coefficient of eigenvalues is positive, YG > 0. The two pairs (or 4 different) eigenvalues of the characteristic polynomial are pure imaginary conjugates. For the Hopf bifurcation, the system is locally stable, as all real components of the eigenvalues of the Jacobian matrix are negative. Of particular interest are the coherent, topologically stable, component pairs, the Falaco Solitons, that can be created as long lived topological defects in a swimming pool.

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Figure 5.19

Falaco Solitons in a Swimming Pool

The Hopf bifurcation leads to a pair of "minimal" surfaces. However, in the real domain, a minimal surface XM = 0 does not have YG > 0. In the case that TK = 0, there is only one conjugate pair of imaginary eigenvalues. If TK 6= 0, the eigenvectors of the matrix that generated to quartic polynomial phase function are null isotropic vectors. That is, they are Spinors, and can be combined to represent spiral structures. 5.8.2 A Visual Topologically Coherent Defect in a Fluid During March of 1986, while visiting an old MIT friend in Rio de Janeiro, Brazil, I became aware of a significant topological event involving visual solitons that can be replicated experimentally by almost everyone with access to a swimming pool. Study the photo which was taken by David Radabaugh, in the late afternoon, Houston, TX 1986. The extraordinary photo is an image of 3 pairs of what are now called Falaco Solitons, a few minutes after their creation. Each Falaco Soliton consists of a pair of globally stabilized rotational indentations in the water-air discontinuity surface of the swimming pool. The dimple shape is as if a conical pencil point was pushed into

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a rubber sheet causing a deformation, but the indentation is dominated by dynamic rotation, not translation. Unseen in the photograph, each pair of contra-rotating dimples are connected by a singular thread in the form of a circular arc extending from the vertex of one dimple to the vertex of the other dimple of the pair. The "thread" can be made visible by injecting drops of dye into the fluid near the rotation axis of one of the dimples. These Solitons are apparently long-lived states of matter far from thermodynamic equilibrium. They will persist for many minutes in a still pool of water, maintaining their topological coherence so as to permit their inclusion into the class of objects called Solitons. The Falaco Solitons are extraordinary, not only due to the fact that they are so easily created in a macroscopic dynamical systems environment, but also because they offer real life, easily observed, evidence for the continuous evolution and creation of topological defects. The long lifetime, and the topological stability, of the Falaco Solitons in a dissipative fluid media is not only remarkable but also is a matter of applied theoretical interest. The equilibrium discontinuity surface of the fluid in the "uniform" g field is flat, and has both zero mean curvature and zero Gauss curvature. The shape of the observed discontinuity surface defect of a Falaco Soliton dimple indicates that the surface mean curvature is zero, but the Gauss curvature is not zero. In Euclidean spaces, such real surfaces are minimal surfaces of negative Gauss curvature. Such surfaces are locally unstable, so it has been presumed that the pair of defect structures that make up the Falaco Soliton must be globally stabilized. It has been conjectured that the connecting string is under tension in order to maintain the shape of the pair of dimpled indentations. This conjecture is justified by the observation that if the singular thread is abruptly "severed" (by experimental chopping motions under the surface of the fluid), the dimpled endcaps disappear in a rapid, non-diffusive, manner. The dimpled surface pairs of the Falaco Soliton are most easily observed in terms of the dramatic black discs that they create by projection of the solar rays to the bottom of the pool. The optics of this effect will be described below. Careful examination of the photo of Figure 5.19 will indicate, by accidents of noticeable contrast and reflection, the region of the dimpled surface of circular rotation. The dimples appear as (deformed) artifacts to the left of each black spot, and elevated above the horizontal plane by about 25 degrees (as the photo was taken in late afternoon). Also, notice that the vestiges of caustic spiral arms in the surface structures around each pair of rotation axes can be seen. These surface spiral arms can be visually enhanced by spreading chalk dust on the free surface of the pool. The bulk fluid motion is a local (non-rigid body) rotational motion about the interconnecting circular thread. In the photos of Figure 5.19 and Figure 5.20, the depth of each of the actual indentations of the free surface is, at most, of a few millimeters in extent. A better photo, also taken by D. Radabaugh, but in the year 2004 in a swimming pool in Mazan, France, demonstrates more clearly the dimpled surface defects, and the Snell refraction. The sun is to the left and at an elevation of about 30

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degrees.

Figure 5.20. Surface Indentations of a Falaco Soliton The photo is in effect a single frame of a digital movie that demonstrates the creation and evolutionary motions of the Falaco Solitons. The experimental details of creating the Falaco Solitons are described below, but the movie explains their creation and dynamics far better than words. The digital movie may be downloaded from [247]. Remark 47 The bottom line is that it is possible to produce, hydrodynamically, in a viscous fluid with a surface of discontinuity, a long lived topologically coherent structure that consists of a set of macroscopic topological defects. The Falaco Solitons are representative of non-equilibrium long lived structures, or "stationary states", far from equilibrium. These observations were first reported at the 1987 Dynamics Days conference in Austin, Texas [197] and subsequently in many other places, mostly in the hydrodynamic literature [202], [203], [210], [215], as well as several APS meetings. More detail is presented in [274]. 5.8.3 Falaco Surface dimples are of zero mean curvature From a mathematical point of view, the Falaco Soliton is interpreted as a connected pair of two-dimensional topological defects connected by a one-dimensional topological defect or thread. The surface defects of the Falaco Soliton are observed dramatically due the formation of circular black discs on the bottom of the swimming pool. The very dark black discs are emphasized in contrast by a bright ring or halo of focused light surrounding the black disc. All of these visual effects can be explained by means of the unique optics of Snell refraction from a surface of zero mean curvature.

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Figure 5.21. Snell Refraction of a Falaco Soliton surface defect Remark 48 This explanation of the optics was reached about 30 minutes after I first became aware of the Soliton structures, while standing in the pristine white marble swimming pool of an old MIT roommate, Jose Haraldo Falçao, under the brilliant Brazilian sunshine in Rio de Janeiro. At MIT, Haraldo was always called Falaco, after he scored 2 goals in a MIT soccer match, and the local newspapers misprinted his name. Hence I dubbed the topological defect structures, Falaco Solitons. Haraldo will get his place in history. I knew that finally I had found a visual, easily reproduced, experiment that could be used to show people the importance and utility of Topological Defects in the physical sciences, and could be used to promote my ideas of Continuous Topological Evolution. The observations were highly motivating. The experimental observation of the Falaco Solitons greatly stimulated me to continue research in applied topology, involving topological defects, and the topological evolution of such defects which can be associated with phase changes and thermodynamically irreversible and turbulent phenomena. When colleagues in the physical and engineering sciences would ask “What is a topological defect?" it was possible for me to point to something that they could replicate and understand visually at a macroscopic level. During the initial few seconds of decay to the metastable soliton state, each large black disk is decorated with spiral arm caustics, remindful of spiral arm galaxies. The spiral arm caustics contract around the large black disk during the stabilization process, and ultimately disappear when the "topological steady" soliton state is achieved. The spiral caustics appear to be swallowed up by the black "hole". It

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should be noted that if chalk dust is sprinkled on the surface of the pool during the formative stages of the Falaco Soliton, then the topological signature of the familiar Mushroom Spiral pattern is exposed. Notice that the black spots on the bottom of the pool in the photo are circular and not distorted ellipses, even though the solar elevation is less than 30 degrees. The important experimental fact deduced from the optics of Snell refraction is that each dimpled surface appears to be a surface of zero mean curvature. This conclusion is justified by the fact that the Snell projection to the floor of the pool is almost conformal, preserving the circular appearance of the black disc, independent from the angle of solar incidence. This conformal projection property of preserving the circular shape is a property of normal projection from minimal surfaces of zero mean curvature [251]. As mentioned above, a feature of the Falaco Soliton that is not immediately obvious is that it consists of a pair of two-dimensional topological defects, in a surface of fluid discontinuity, which are connected by means of a topological singular thread. Dye injection near an axis of rotation during the formative stages of the Falaco Soliton indicates that there is a unseen thread, or one-dimensional string singularity, in the form of a circular arc that connects each of the two-dimensional surface singularities or dimples. Transverse Torsional waves made visible by dye streaks (caused by dye drops injected near one of the surface rotation axes) can be observed to propagate, back and forth, from one dimple vertex to the other dimple vertex, guided by the "string" singularity. The effect is remindful of the whistler propagation of electrons along the guiding center of the earth’s pole to pole magnetic field lines.

Figure 5.22. Falaco Topological Defects with connecting thread However, as a soliton, the topological system retains its coherence for remarkably long time - more than 15 minutes in a still pool. The long lifetime of the Falaco

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Soliton is conjectured to be due to this global stabilization of the connecting string singularity, even though a real surface of zero mean curvature is locally unstable. The Falaco Soliton starts out from a non-equilibrium thermodynamic state of Pfaff topological dimension 4, which quickly and irreversibly decays to a "topologically stationary" state, still far from equilibrium, but with a long dynamic lifetime [273] [274]. 5.8.4 Falaco Surfaces are related to Harmonic vector fields The long life of the soliton state in the presence of a viscous media indicates that the flow vector field describing the dynamics is probably harmonic. This result is in agreement with the assumption that the fluid can be represented by a Navier-Stokes equation where the viscous dissipation is dominated by affine shear viscosity times the vector Laplacian of the velocity field. If the velocity field is harmonic, the vector Laplacian vanishes, and the shear dissipation term goes to zero - no matter what is the magnitude of the shear viscosity term. Hence a palatable argument is offered in terms of harmonic velocity fields for the existence of the long lifetime of the Falaco Solitons (as well as the production of wakes in fluid dynamics [275]). Moreover it is known in the theory of minimal surfaces [165] that surfaces of zero mean curvature are generated by harmonic vector fields. Remark 49 The bottom line is that the idea of a long lifetime in a dissipative media is to be associated with Harmonic vector fields and surfaces of zero mean curvature. Initially, I thought that the surface configuration, immediately after creation, was in the form of a Rankine vortex (of positive mean curvature, and positive Gauss curvature in a 3D Euclidean space), which then decayed into a classic minimal surface of zero mean curvature, but negative Gauss curvature. Such an evolutionary process can be found in Langford bifurcations [274] which can be shown to be solutions to the Navier-Stokes equations in a rotating frame of reference. However, such a dynamics seems to require that the connection (the string) between the Falaco pairs has an open throat (like a Wheeler worm hole). Note that for a "stationary" Euclidean soap film between two boundary rings (Figure 5.23a), the system is stable only if the separation of the boundary rings is less than (approximately) 2.65 times the minimal throat diameter. Experimentally the stationary non-rotating soap film between two boundary rings will break apart if the soap film is stretched too far. The single component catenoid (with zero mean curvature and negative Gauss curvature, and with real equal and opposite principle curvatures) will bifurcate into two flat components, one on each ring, and each of zero Gauss curvature as well as zero mean curvature. The process has been demonstrated in fluid flow in a rotating frame, where the zero helicity function of the fluid flow has the appearance of a minimal surface. As the bulk flow increases, the helicity function changes sign, and therefore represents a change in topology from a connected set to a disconnected set. With the change in sign, a torsion bubble (or a torsion burst) appears in the flow pattern [203] [275].

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Admittedly, the extended catenoid of Figure 5.23b (a deformed Wheeler Wormhole with an open throat ?) has the some of the features and appearance of the Falaco Solitons, but the extended singular thread (without an open throat) between vertex singularities does not appear to be replicated.

Figure 5.23a Soap film between rings

Figure 5.23b Deformed Wheeler Wormhole

It remains difficult to utilize the minimal surface soap film conjecture of a decaying Rankine vortex to support the idea of topological evolution to a structure of two dimpled surfaces of zero mean curvature connected by a one-dimensional thread. The question arises as how to explain the creation and existence of the Falaco Solitons. The idea that the Falaco Solitons are related to strings connecting branes led to the thought that perhaps the modern advances in topology and string theory could yield a theoretical explanation. According, challenges and requests for help were sent out to many of the string theorists, asking for theoretical help to describe this "real life string connecting branes"; the lack of response indicates that none of the string gurus seemed to think the effort was worthwhile. However, the theoretical work of Dzhunushaliev [65] seems to have many correspondences with the experimental facts of the Falaco Solitons. In Euclidean space, the real minimal surface defects of zero mean curvature are of negative (or zero) Gauss curvature, and are, therefore, locally unstable. However, stationary non-rotating soap films can be stabilized by certain boundary conditions. As mentioned above the experimental equilibrium state of the fluid discontinuity surface is a surface of both zero Gauss curvature and zero Mean curvature (both principle surface curvatures are zero). From the optics of Snell refraction, a Falaco endcap is obviously a surface of zero mean curvature, and if equivalent to a stationary soap film, it should be locally unstable. However, it was conjectured that the local instability could be overcome globally by a string whose tension globally stabilizes the locally unstable endcaps. Could the tension be related to a rotationally induced positive contribution to the otherwise negative Gauss curvature? These conjectures originally were explained (partially) in terms of a bifurcation process and solutions

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to the Navier-Stokes equations in a rotating frame of reference [274]. A summary of such analysis is presented below. More recently, it was determined that an alternative, and perhaps better, description can be given in terms of a fluid with a surface discontinuity that has zero mean curvature relative to a Minkowski metric; the Minkowski surface has a Gauss curvature which is positive. A topological birfurcation process from a Rankine vortex to Falaco Solitons would then be such as to change the 3D Euclidean signature into a 3D Minkowski signature. Such surfaces of zero mean curvature embedded in Minkowski space have been called maximal surfaces by the differential geometers, and have conical singular points [72] . It is now believed that the Falaco thread is attached to the conical singular points of a pair of such maximal surfaces. Alternatively, the Euclidean metric can be maintained, and a result similar to the immersion of the 2D surface into Minkowski space can be attributed to the fact that infinitesimal rotations admit Spinor complex isotropic eigendirection fields with non-zero, pure imaginary eigenvalues. The Gauss curvature of such systems is positive, even though the eigendirection fields are complex Spinors, not vectors in the diffeomorphic sense. A discussion of the Hopf map as applied to this idea will be found below. 5.8.5 Spinors and zero mean curvature surfaces The theory of minimal surfaces (of zero mean curvature) are intertwined with the concept of complex isotropic direction fields, defined as pure Spinors by Eli Cartan. The Weierstrass formulas of minimal surface theory [165] consider a holomorphic complex velocity field in 3D, which upon integration leads to conjugate pairs of minimal surfaces defined by the real and imaginary components of the position vector formed by complex integration. The key feature of this holomorphic "velocity" field, so useful to minimal surface theory, is that it is a complex isotropic collection of components, whose Euclidean sums of squares is zero. Such isotropic complex direction fields of zero quadratic form (length) were defined as Spinors by E. Cartan [44]. In addition, E. Cartan demonstrated that infinitesimal rotations are generated by antisymmetric matrices. It is rather remarkable (and was only fully appreciated by me in February, 2005) that there is a large class of direction fields (still given the symbol ρV4 ) that do not behave as diffeomorphic vectors. For more detail about Spinors, see Chapter 2.3. Remark 50 Falaco Solitons appear to be physical artifacts of non-tensorial (Spinor) properties of physical systems where rotations produce surfaces of zero mean curvature, but with positive, not negative, Gauss curvature. 5.8.6 Topological Universality independent from scales The reader must remember that the Falaco Soliton is a topological object that can and will appear at all scales, from the microscopic, to the macroscopic, from the sub-submicroscopic world of strings connection branes, to the cosmological level of

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spiral arm galaxies connected by threads. At the microscopic level, the method offers a view of forming spin pairs that is different from Cooper pairs and could offer insight into Hi-TC Superconductivity. At the level of Cosmology, the concept of Falaco Solitons could lead to explanations of the formation of flat spiral arm galaxies. At the submicroscopic level, the Falaco Solitons mimic quark pairs confined by a string. At the microscopic level, the Falaco Solitons appear as the dimpled vortex structures in rotating Bose-Einstein condensates. They also model the concepts of a Photon as being the singular thread attached to dimples on two expanding light cone shells. At the macroscopic level, similar topological features of the Falaco Solitons can be found in solutions to the Navier-Stokes equations in a rotating frame of reference. Under deformation of the discontinuity surface to a flattened ball, the visual correspondence to hurricane structures between the earth surface and the tropopause is remarkable. In short, as a topological defect, the concept of Falaco Solitons is a universal phenomenon, which can appear at all scales. 5.8.7 The Experiment The Falaco Soliton phenomenon is easily reproduced by placing a large circular disc, such as dinner plate, vertically into the swimming pool until the plate is half submerged and its oblate axis resides in the water-air free surface. Then move the plate slowly in the direction of its oblate axis. At the end of the stroke, smoothly extract the plate (with reasonable speed) from the water, imparting kinetic energy and distributed angular momentum to the fluid. The dynamical system undergoes a short period (a few seconds) of stabilization, followed by a longer period (many minutes) of a "topologically stationary" state. It is this topologically stationary state that is defined as the Falaco Soliton. Thermodynamically, the system starts in an initial state of Pfaff topological dimension 4 and decays by continuous topological evolution to a "stationary" state of Pfaff topological dimension 3. According to the theory of non-equilibrium thermodynamics [273], the processes during the initial stabilization period are thermodynamically irreversible, but once the Pfaff dimension 3 configuration is reached, the evolutionary processes preserving topological features can be described in a Hamiltonian manner. Both the initial and the "stationary" soliton states are thermodynamic states far from equilibrium. At first it was thought that the initial deformed surface state could be related to a Rankine vortex structure (which has regions of both positive and negative Gauss curvature). Recall that a Rankine vortex has a core that is equivalent to rigid body rotation. This description of the formative state of stabilization is too naive, for observations indicate that the sharp edge of the plate described above generates instability patterns [275] as it is stroked through the fluid. After the initial injection of energy and angular momentum, the fluid spends a few seconds during a process of stabilization, during which a surface of zero mean curvature is formed transiently, producing the easily visible large black spots formed by Snell refraction. Associated with the evolution to a "stationary" Soliton state, is a visible set of spiral arm caustics on the pool surface around each dimples rotation axis. As the stabilization proceeds,

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the spiral caustics appear to grow tighter around the black spot, and are almost gone when the Soliton becomes stable. In a few tries you will become an expert experimentalist at stroking the plate and creating Falaco Solitons. The drifting black spots are easily created and, surprisingly, will persist for many minutes in a still pool. The dimpled depressions are typically of the order of a few millimeters in depth, but the zone of circulation around each rotation axis typically extends over a disc of some 10 to 30 centimeters radius, depending on the plate diameter. The "stationary" configuration, or coherent topological defect structure, has been defined as the Falaco Soliton. For purposes of illustration , the vertical depression has been greatly exaggerated in Figures 5.21 and 5.22. If a thin broom handle or a rod is placed vertically in the pool, and the Falaco Soliton pair is directed in its translation motion to intercept the rod symmetrically, as the soliton pair comes within range of the scattering center, or rod, (the range is approximately the separation distance of the two rotation centers) the large black spots at first shimmer and then disappear. Then a short time later, after the soliton has passed beyond the interaction range of the scattering center, the large black spots coherently reappear, mimicking the numerical simulations of soliton coherent scattering. For hydrodynamics, this observation firmly cements the idea that these objects are truly coherent "Soliton" structures. This experiment is the only (known to me) macroscopic visual experiment that demonstrates these coherence features of soliton scattering. If the string connecting the two endcaps is sharply "severed", the confined, two-dimensional endcap singularities do not diffuse away, but instead disappear almost explosively. The process of "severing" can be accomplished by moving your hand (held under the water approximately above the circular arc or "string" connecting the two dimple vertices) in a karate chop motion. It is this observation that leads to the statement that the Falaco Soliton is the macroscopic topological equivalent of the illusive hadron in elementary particle theory. The 2, two dimensional, surface defects (the quarks) are bound together by a string of confinement, and cannot be isolated. The dynamics of such a coherent structure is extraordinary, for it is a system that is globally stabilized by the presence of the connecting one-dimensional string. For a movie of the process, see [247]. 5.9

Bifurcation Processes and the Production of Topological Defects

5.9.1 Lessons from the bifurcation to Hopf Solitons Local Stability - a review of the similarity invariants Consider a thermodynamic system that can be encoded (to within a factor, 1/λ) on the variety of independent variables {x, y, z, t} in terms of a 1-form of Action, A = {Ak (x, y, z, t)dxk − φ(x, y, z, t)dt}/λ(x, y, z, t).

(5.388)

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Then construct the Jacobian matrix of the (covariant) coefficient functions, £ ¤ [Jkj (A)] = ∂(Ak /λ)/∂xj .

(5.389)

This Jacobian matrix can be interpreted as a projective correlation mapping of "points" (contravariant vectors) into "hyperplanes" (covariant £ k ¤vectors). The cork relation mapping is the dual of a collineation mapping, Jm (V ) , which takes points into points. Correlations are related to the thermodynamic system, and collineations are related to thermodynamic processes. Linear (local) stability occurs at points where the (possibly complex) eigenvalues of the Jacobian matrix are such that the real parts are not positive. The eigenvalues, ξ k , are determined by solutions to the Cayley-Hamilton characteristic polynomial of the Jacobian matrix, [Jkj (A)], Θ(x, y, z, t; ξ) = ξ 4 − XM ξ 3 + YG ξ 2 − ZA ξ + TK ⇒ 0.

(5.390)

The Cayley-Hamilton polynomial equation defines a family of implicit functions in the space of variables, XM (x, y, z, t), YG (x, y, z, t), ZA (x, y, z, t), TK (x, y, z, t). The functions XM , YG , ZA , TK are defined as the similarity invariants of the Jacobian matrix. If the eigenvalues, ξ k , are distinct, then the similarity invariants are given by the expressions, XM YG ZA TK

= ξ 1 + ξ 2 + ξ 3 + ξ 4 = T race [Jkj (A)] , = ξ 1ξ 2 + ξ 2ξ 3 + ξ 3ξ 1 + ξ 4ξ 1 + ξ 4ξ 2 + ξ 4ξ3, = ξ 1ξ 2ξ 3 + ξ 4ξ 1ξ 2 + ξ 4ξ 2ξ3 + ξ4ξ 3ξ 1, = ξ 1 ξ 2 ξ 3 ξ 4 = det [Jkj (A)] .

(5.391) (5.392) (5.393) (5.394)

In the differential geometry of three-dimensional space, {x, y, z}, when the scaling coefficient is chosen to be the quadratic isotropic Holder norm of index 1 (the Gauss map), then the determinant of the 3x3 Jacobian matrix vanishes, and the resulting similarity invariants become related to the mean curvature and the Gauss curvature of the Shape matrix. Bifurcation and singularity theory involves the zero sets of the similarity invariants, and the algebraic intersections of the implicit hypersurfaces so generated by these zero sets. Recall that the theory of linear (local) stability requires that the eigenvalues of the Jacobian matrix have real parts which are not greater than zero. For a 4th order polynomial, either all 4 eigenvalues are real; or, two eigenvalues are real, and two eigenvalues are complex conjugate pairs; or there are two distinct complex conjugate pairs. Local stability therefore requires, Local Stability Odd XM ≤ 0, Even YG ≥ 0,

Odd ZA ≤ 0, Even TK ≥ 0.

(5.395) (5.396)

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Useful properties of the Hopf Map (see Chapter 3.5) The Hopf map is a rather remarkable projective map from 4 to 3 (real or complex) dimensions that has interesting and useful topological properties related to links and braids and other forms of entanglement. The Hopf map satisfies the criteria of Local Stability, and yet is not an integrable system, and admits irreversible dissipation. The map can be written as {x, y, z, s = ct} ⇒ {x1, x2, x3}, |H1i = [x1, x2, x3]T = [2(xz + ys), 2(xs − yz), (x2 + y 2 ) − (z 2 + s2 )]T . (5.397) A remarkable feature of this map is that, Hopf Map

hH1| · |H1i = (x1)2 + (x2)2 + (x3)2 = (x2 + y 2 + z 2 + s2 )2 .

(5.398)

Hence a real (and imaginary) four-dimensional sphere maps to a real three-dimensional sphere. The Hopf map can also be represented in terms of complex functions by a map from C2 to R3, as given by the formulas, H1 = [x1, x2, x3] = [α · β ∗ + β · α∗ , i(α · β ∗ − β · α∗ ), α · α∗ − β · β ∗ ].

(5.399)

By permuting the formulas it is possible to construct 3 linearly independent Hopf vectors, all of which have same Euclidean norm. Note that it is possible to construct complex isotropic spinors by forming a vector valued complex number composed of the Hopf vectors and their permutations, Spinor

|σ1i = |H2i + i |H3i ,

hσ1| ◦ |σ1i = 0 .

(5.400)

It should be expected that there is a connection to surfaces of zero mean curvature and Spinors. For H1, the Hopf map generates 3 exact differential 1-forms, and a 4th nonintegrable 1-form, A, of Pfaff topological dimension 4. The formula for the 1-form, AHopf , in section 3.5 can be generalized to include constant coefficients (not necessarily unity) of polarization, R, and chirality, L, to read, AHopf = {R(−yd(x) + xd(y)) + L(−sd(z) + zd(s))}.

(5.401)

ZA = 0, Odd XM = 0, 2 2 Even YG = R + L ≥ 0, TK = R2 L2 ≥ 0.

(5.402) (5.403)

The 2-form F = dAHopf has only Spinor eigendirection fields (e1, e2, e3, e4) with pure imaginary eigenvalues (+iR, −iR, +iL, −iL). The similarity invariants are,

Hence the canonical Hopf 1-form, AHopf , is locally stable. The similarity curvatures are pure imaginary, but the Gauss curvature is positive! For the simple case where

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L = 0, the Hopf map describes an minimal surface with imaginary individual curvatures. The classic real minimal surface has a Gauss curvature YG which is negative, and for which the individual curvatures are real. As dAHopf ˆdAHopf 6= 0, the Hopf 1-form AHopf is of Pfaff dimension 4. The associated Topological Torsion 4-vector is proportional to the ray vector from the origin to a point in the space, T4 = −2RL[x, y, z, t] .

(5.404)

Any process that evolves with a component in the direction of T4 is thermodynamically irreversible, as, L(T4 )A = −8RL · A = Q, and QˆdQ 6= 0.

(5.405) (5.406)

Consider the evolution of the Falaco Solitons to be represented by a dynamical system topologically equivalent to an exterior differential system of 1-forms, ω k = dxk − Vk (x, y, z, t)dt.

(5.407)

When all three 1-forms vanish, ω k ⇒ 0, imposing the existence of a topological limit structure on the base manifold of four dimensions, {x, y, z, t}, the result is equivalent to a 1D solution manifold defined as a kinematic system. The solution manifold to the dynamical system is in effect a parametrization of the parameter t to the space curve Cparametric in 4D space, where for Kinematic Perfection, [Vk , 1] is a tangent vector to the curve Cparametric . Off the kinematic solution submanifold, the non-zero values for the 1-forms, ω k 6= 0, can be interpreted as topological fluctuations from "Kinematic Perfection". If "Kinematic Perfection" is not exact, then the three 1-forms ωk are not precisely zero, and have a finite triple exterior product that defines a 3-form in the 4D space. From the theory of exterior differential forms it is the intersection of the zero sets of these three hypersurfaces ωk that creates an implicit curve Cimplicit in 4D space, Cimplicit = ω x ˆωy ˆωz = dxˆdyˆdz − Vx dyˆdzˆdt + Vy dxˆdzˆdt − Vz dxˆdyˆdt = −i([V, 1])Ω4 .

(5.408) (5.409) (5.410)

The discussion brings to mind the dualism between points (rays) and hypersurfaces (hyperplanes) in projective geometry. If a ray (a point in the projective 3-space of four dimensions) is specified by the 4 components of the 4D vector, [V, 1], multiplied by any non-zero factor, κ, (such

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that [V, 1] ≈ κ[V, 1]), then the equation of a dual projective hyperplane is given by the expression [A, −φ] ≈ γ[A, −φ] such that, hγ[A, −φ]| ◦ |κ[V, 1]i = 0.

(5.411)

The principle of projective duality [168] implies that (independent from the factors γ and κ), φ = A ◦ V.

(5.412)

A particularly easy choice is to assume that (to within a factor), Ak =Vk , and

φ = V ◦ V,

A = Vk dxk − Vk V k dt. k

Vk (x, y, z, t) ≈ V (x, y, z, t), the 3 functions of a dynamical system.

(5.413) (5.414) (5.415)

It should be remembered that not all dynamic features are captured by the similarity invariants of a dynamic system. The antisymmetric features of the dynamics are better encoded in terms of Cartan’s magic formula. Cartan’s formula expresses the evolution of a 1-form of Action, A, in terms of the Lie differential with respect to a vector field, V, acting on the 1-form that encodes the properties of the physical system. For example, consider the 1-form of Action (the canonical form of a Hopf system) given by the equation, AHopf = R(ydx − xdy) + L(tdz − zdt).

(5.416)

The Jacobian matrix of this Action 1-form has eigenvalues which are solutions of the characteristic equation, Θ(x, y, z, t; ξ)Hopf = (ξ 2 + R2 )(ξ 2 + L2 ) ⇒ 0.

(5.417)

The eigenvalues are two conjugate pairs of pure imaginary numbers, {±iR, ±iL} and are interpreted as "oscillation" frequencies. The similarity invariants are XM = 0, YG = R2 + L2 > 0, ZA = 0, TK = R2 L2 > 0. The Hopf eigenvalues have no real parts that are positive, and so the Jacobian matrix is locally stable. The criteria for a double Hopf oscillation frequency requires that the algebraically odd similarity invariants vanish and the algebraically even similarity invariants are positive definite. The stability critical point of the Hopf bifurcation occurs when all similarity invariants vanish. In such a case the oscillation frequencies are zero. This Hopf critical point is NOT necessarily the same as the thermodynamic critical point, as exhibited by a van der Waals gas. The oscillation frequencies have led the Hopf solution to be described as a "breather". The Hopf system is a locally stable system in four dimensions. Each of the pure imaginary frequencies can be associated with a "minimal" hypersurface.

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Suppose that L = 0. Then the resulting characteristic equation represents a "minimal surface" as XM = 0, but with a Gauss curvature which is positive definite, YG = R2 > 0. The curvatures of the implicit surface are imaginary. In differential geometry, where the eigenfunctions can be put into correspondence with curvatures, the Hopf condition, XM = 0, for a single Hopf frequency would be interpreted as "strange" minimal surface (attractor). The surface would be strange for the condition YG(hopf ) = R2 > 0 implies that the Gauss curvature for such a minimal surface is positive. A real minimal surface has curvatures which are real and opposite in sign, such that the Gauss curvature is negative. As a real minimal surface has eigenvalues with one positive and one negative real number, the criteria for local stability is not satisfied for real minimal surfaces. Yet experience indicates that soap films can occur as "stationary states" when stabilized by certain boundary conditions. The implication is that soap films can be globally stabilized, even though they are locally unstable. As developed in the next section, the Falaco critical point and the Hopf critical point are the same, all similarity invariants vanish. For the autonomous examples it is possible to find an implicit surface, YG(hopf ) = YG(f alaco) = 0, in terms of the variables {x, y, z; a, b, C...} where a, b, C... are the parameters of the dynamical system. Recall that the classic (real) minimal surface has real curvatures with a sum equal to zero, but with a Gauss curvature which is negative (XM = 0, YG < 0). Such a system is not locally stable, for there exist eigenvalues of the Jacobian matrix with positive real parts. Yet persistent minimal soap films between boundaries can exist under such conditions and are apparently stable macroscopically (globally). This experimental evidence can be interpreted as an example of global stability overcoming local instability. 5.9.2 The bifurcation to a Falaco Soliton Similar to, and guided by experience with, the Hopf bifurcation, the bifurcation that leads to a Falaco Soliton must agree with the experimental observation that the endcaps have zero mean curvature, and are in rotation. The stability of the Falaco Soliton pair of topological defects is global, experimentally. If the singular thread connecting the vertices of the two dimples is cut, the system decays nondiffusively. Hence the bifurcation to a Falaco Soliton can not imply local stability. This experimental result is related to the theoretical confinement problem in the theory of quarks. To analyze the problem consider the case where the TK term in the Cayley-Hamilton polynomial vanishes (implying that one eigenvalue of the 4D Jacobian matrix is zero). Experience with the Hopf bifurcation suggests that the Falaco Soliton may be related to another form of the characteristic polynomial, where XM = 0, ZA = 0, YG < 0. This bifurcation is not equivalent to the Hopf bifurcation, but has the same critical point, in the sense that all similarity invariants vanish at the critical point. Similar to the Hopf bifurcation this new bifurcation scheme can be of Pfaff topological dimension 4, which implies that the abstract thermodynamic

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system generated by the 1-form (which is the projective dual to the dynamical system) is an open, non-equilibrium thermodynamic system. The odd similarity invariants of the 4D Jacobian matrix must vanish. However there are substantial differences between the bifurcation that lead to Hopf solitons (breathers) and Falaco Solitons. Experimentally, the Falaco Soliton appears to have a projective cusp at the critical point (the vertex of the dimple) and that differs from the Hopf bifurcation which would be expected to have a projective parabola at the critical point. When TK = 0, the resulting cubic factor of the characteristic polynomial will have 1 real eigenvalue, B, one eigenvalue equal to zero, and possibly 1 pair of complex conjugate eigenvalues, (σ + iΩ), (σ − iΩ). To be stable globally it is presumed that, Global Stability

Odd XM = B + 2σ ≤ 0, Odd ZA = B(σ2 + Ω2 ) ≤ 0, Even YG = σ 2 + Ω2 + 2Bσ undetermined, Even TK = 0.

(5.418) (5.419)

If all real coefficients are negative then YG > 0, and the system is locally stable. Such is the situation for the Hopf bifurcation. However, the Falaco Soliton experimentally requires that YG < 0. By choosing B ≤ 0, in order to satisfy ZA ≤ 0, leads to the constraint that σ = −B/2 > 0, such that the real part of the complex solution is positive, and represents an expansion, not a contraction. Substitution into the formula for YG leads to the condition for generation of a Falaco Soliton, YG(f alaco) = Ω2 − 3B 2 /4 < 0.

(5.420)

It is apparent that local stability is lost for the complex eigenvalues of the Jacobian matrix can have positive real parts, σ > 0. Furthermore it follows that YG < 0 (leading to negative Gauss curvature) if the square of the rotation speed, Ω, is smaller than the 3/4 of the square of the real (negative) eigenvalue, B. This result implies that the "forces" of tension overcomes the inertial forces of rotation. In such a situation, a real minimal surface is produced (as visually required by the Falaco Soliton). The result is extraordinary for it demonstrates a global stabilization is possible for a system with one contracting direction, and two expanding directions coupled with rotation. The contracting coefficient B (similar to a spring constant) is related to the surface tension in the "string" that connects the two global endcaps of negative Gaussian curvature. The critical point occurs when Ω2 = 3B 2 /4. It is conjectured that if the coefficient B is in some sense a measure of of a reciprocal length (such that B ≈ 1/R, a curvature), then there are three interesting formulas comparing angular velocity (orbital period) and length (orbital radius),

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Falaco : Ω2 R2 = constant, Kepler : Ω2 R3 = constant, Planck : Ω2 R4 = constant.

Figure 5.24.

(5.421) (5.422) (5.423)

Spiral galaxy mass distributions

The bifurcations to Hopf Solitons suggest oscillations of expansions and contractions of imaginary minimal surfaces (or Soliton concentration breathers) and have been exhibited in certain chemical reactions. On the other hand, the bifurcations to Falaco Solitons suggest the creation of spiral concentrations, or density waves, on real rotating minimal surfaces. The molal density distributions (or order parameters) are complex. The visual bifurcation structures of the Falaco Solitons in the swimming pool would appear to offer an explanation as to the origin of (≈ flat) spiral arm galaxies at a cosmological level, and would suggest that the spiral arm galaxies come in pairs connected by a topological string. Moreover, the kinetic energy of the stars far from the galactic center would not vary as the radius of the "orbit" became very large. This result is counter to the Keplerian result that the kinetic energy of the stars should decrease as 1/R. If is assumed that the density distribution of star mass is more or less constant over the central region of the spiral arm flat disc-like structures, then over this region, the Newtonian gravitation force would lead to a "rigid body" result, Ω2 R2 = R. Figure 5.24 demonstrates the various options.

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5.9.3

Falaco Solitons as Landau Ginsburg structures in micro, macroscopic and cosmological systems Landau Ginsburg structures (Bose Einstein Condensates)

Figure 5.25. Landau Ginsburg theory and Falaco Solitons The Falaco experiments lead to the idea that such topological defects are available at all scales. The Falaco Solitons consist of spiral "vortex defect" structures (analogous to CGL theory) on a two-dimensional minimal surface, one at each end of a one-dimensional "vortex line" or thread (analogous to GPG theory). Remarkably, in Euclidean space, the topological defect surface structure of the Falaco Soliton is locally unstable, as it would be a surface of negative Gauss curvature. Yet the pair of locally unstable 2D surfaces is globally stabilized by the 1D line defect attached to the "vertex" points of the minimal surfaces. The connecting string is unseen in the swimming pool experiments, unless there is dirt or bubbles or dye drops in the water. The rotational motion of the impurities makes the connecting "string" visible. The "string" that connects the two vertices acts like a spring with tension, "holding" the dimpled depressions in a state far from equilibrium. For some specific physical systems it can be demonstrated that period (circulation) integrals of the 1-form of Action potentials, A, lead to the concept of "vortex defect lines". The idea is extendable to "twisted vortex defect lines" in three dimensions. The "twisted vortex defects" become the spiral vortices of a Complex Ginsburg Landau (CGL) theory, while the "untwisted vortex lines" become the defects of Ginzburg-Pitaevskii-Gross (GPG) theory [260].

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Cosmological strings between Galaxies

Figure 5.26.

Cosmic Strings from the Hubble Telescope

Now jump from the microscopic coherent structures of the quantum world to the possibility of seeing universal Falaco Soliton defects in the cosmological world. In the macroscopic domain, the experiments visually indicate "almost flat" spiral arm structures during the formative stages of the Falaco Solitons. In the cosmological domain, it is suggested that these universal topological defects represent the ubiquitous "almost flat" spiral arm galaxies. Based on the experimental creation of Falaco Solitons in a swimming pool, it has been conjectured that M31 and the Milky Way galaxies could be connected by a topological defect thread [197]. Only recently has photographic evidence appeared suggesting that galaxies may be connected by strings. The quark confinement problem At the other extreme, the rotational minimal surfaces of zero mean curvature which form the two endcaps of the Falaco Soliton, like quarks, apparently are confined by the string. If the string (whose "tension" induces global stability of the unstable endcaps) is severed, the endcaps (like unconfined quarks in the elementary particle

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domain) disappear (in a non-diffusive manner). In the microscopic electromagnetic domain, the Falaco Soliton structure offers an alternate, topological, pairing mechanism on a Fermi surface, that could serve as an alternate to the Cooper pairing in superconductors. 5.9.4 Wheeler Wormholes and Falaco Strings between Branes It is extraordinary, but the Falaco Solitons appear to be another form of a zero mean curvature surface structure, either related to macroscopic realizations of the Wheeler wormhole (with a very narrow throat), or to Spinor surfaces generated by complex eigendirection fields of infinitesimal rotations. The Wheeler wormhole structure was presented early on by Wheeler (1955), but was considered to be unattainable in a practical sense. To quote Zeldovich p. 126 [298], "The throat or "wormhole" (in a Kruskal metric) as Wheeler calls it, connects regions of the same physical space which are extremely remote from each other. (Zeldovich then gives a sketch that topologically is similar to the Falaco Soliton.) Such a topology implies the existence of ’truly geometrodynamic objects’ which are unknown to physics. Wheeler suggests that such objects have a bearing on the nature of elementary particles and antiparticles and the relationships between them. However, this idea has not yet borne fruit; and there are no macroscopic"geometrodynamic objects" in nature that we know of. Thus we shall not consider such a possibility." This quotation dates back to the period 1967-1971. Now the experimental evidence justifies (again) Wheeler’s intuition. Both the Wheeler wormhole and the Falaco Soliton are related to surface structures of zero mean curvature. The catenoidal surface of zero mean curvature, and negative Gauss curvature, in a 3D Euclidean space is a Wheeler Wormhole (with an open throat), while the conical surface of zero mean curvature, and positive Gauss curvature, and its conical singular point in a 3D Minkowski space is a part of the rotationally induced Falaco Soliton. Remark 51 The bottom line is that the remarkable feature of creating a stable surface of zero mean curvature and positive Gauss curvature (the Falaco Soliton) is explained either by assuming that the usual 3D Euclidean Signature is rotationally dependent and can topologically evolve into a 3D Minkowski Signature (forming what is called a maximal, rather that a minimal, surface). A second possibility is that Falaco Soliton endcap dimples (which are presumed to be surfaces of zero mean curvature and positive Gauss curvature) are related to Spinor eigendirection fields associated with antisymmetric matrices representing Symplectic spaces (see Figures 5.27 and 5.28 on the next page).

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Figure 5.27.

Rotational Surfaces of Zero mean curvature in Euclidean and Minkowski 3-space

If the "maximal" surfaces appear as deformations in disconnected hypersurfaces of discontinuity, the topological structure has the appearance of "strings connecting branes", a concept touted by the string theorists.

Figure 5.28. Falaco Solitons as connected dimples between Branes

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The new feature is that the "brane" surface of discontinuity is deformed by the maximal surface dimple. This deformation has been called a space-time foam by the string theorists [65]. This Falaco structure between branes motivated a model for the photon [227]. The idea also can be related to the rotational structures found in rotating Bose-Einstein condensates [272]. In the next section, it is assumed that a thermodynamic (electromagnetic) system can be encoded by a 1-form of Action potentials, A, which leads, by exterior differentiation, to a 2-form of field intensities, F = dA. The null eigenvectors of the antisymmetric matrix representation of F will form 3D expanding spherical surfaces of propagating field discontinuities (related to the spatial portions of the Minkowski lightcone where F ˆF = 0). In addition, the isotropic Spinor eigenvectors of F will form surfaces of zero mean curvature as defect structures on the spherical spatial portions of the lightcone. The result is a Falaco Soliton pair (with AˆF 6= 0) between the two bounding cycles of a spherical shell. The claim is that this concept serves as a model for the Photon [227]. More details of the Falaco Solitons, including their generation as tertiary bifurcations of dynamical systems that are found among the family of solutions to the Navier - Stokes equations, and are described in the next section, as well as in Volume 2 of this series in Adventures in Applied Topology [274]. In Volume 2, the Falaco endcap is described as a surface of zero mean curvature in a 3D Minkowski space. 5.9.5 A Cosmological Conjecture The objective of this section is to examine topological aspects and defects of thermodynamic physical systems and their possible continuous topological evolution, creation, and destruction on a cosmological scale. The creation and evolution of stars and galaxies will be interpreted herein in terms of the creation of topological defects and evolutionary phase changes in a very dilute turbulent, non-equilibrium, thermodynamic system of maximal Pfaff topological dimension 4. The cosmology so constructed is opposite in viewpoint to those efforts which hope to describe the universe in terms of properties inherent in the quantum world of Bose-Einstein condensates, super conductors, and superfluids [272]. Both approaches utilize the ideas of topological defects, but thermodynamically the approaches are opposite in the sense that the quantum method involves, essentially, equilibrium systems, while the approach presented herein is based upon non-equilibrium systems. Based upon the single assumption that the universe is a non-equilibrium thermodynamic system of Pfaff topological dimension 4 leads to a cosmology where the universe, at present, can be approximated in terms of the non-equilibrium states of a very dilute van der Waals gas near its critical point. The stars and the galaxies are the topological defects and coherent - but not equilibrium - structures of Pfaff topological dimension 3 in this non-equilibrium system of Pfaff topological dimension 4. The topological theory of the ubiquitous van der Waals gas leads to the concepts of negative pressure, string tension, and a Higgs potential as natural consequences of a topological point of view applied to thermodynamics. Perhaps of more importance is the fact that

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

these concepts do not depend explicitly upon the geometric constraints of metric or connection, and yield a different perspective on the concept of gravity. The original motivation for this conjecture is based on the classical theory of correlations of fluctuations presented in the Landau-Lifshitz volume on statistical mechanics [128]. However, the methods used herein are not statistical, not quantum mechanical, and instead are based on Cartan’s methods of exterior differential forms and their application to the topology of thermodynamic systems and their continuous topological evolution [204]. Landau and Lifshitz emphasized that real thermodynamic substances, near the thermodynamic critical point, exhibit extraordinary large fluctuations of density and entropy. In fact, these authors demonstrate that for an almost perfect gas near the critical point, the correlations of the fluctuations can be interpreted as a 1/r potential giving a 1/r2 force law of attraction. Hence, as a cosmological model, the almost perfect gas - such as a very dilute van der Waals gas - near the critical point yields a reason for both the granularity of the night sky and for the 1/r2 force law ascribed to gravitational forces between massive aggregates. A topological (and non-statistical) thermodynamic approach can be used to demonstrate how a four-dimensional variety can support a turbulent, non-equilibrium, physical system with universal properties that are homeomorphic (deformable) to a van der Waals gas [223]. The method leads to the necessary conditions required for the existence, creation or destruction of topological defect structures in such a non-equilibrium system. For those physical systems that admit description in terms of an exterior differential 1-form of Action potentials of maximal rank, a Jacobian matrix can be generated in terms of the partial derivatives of the coefficient functions that define the 1-form of Action. When expressed in terms of intrinsic variables, known as the similarity invariants, the Cayley-Hamilton four-dimensional characteristic polynomial of the Jacobian matrix generates a universal phase equation. Certain topological defect structures can be put into correspondence with constraints placed upon those (curvature) similarity invariants generated by the Cayley-Hamilton fourdimensional characteristic polynomial. These constraints define equivalence classes of topological properties. The characteristic polynomial, or Phase function, can be viewed as representing a family of implicit hypersurfaces. The hypersurface has an envelope which, when constrained to a minimal hypersurface, is related to a swallowtail bifurcation set. The swallowtail defect structure is homeomorphic to the Gibbs surface of a van der Waals gas. Another possible defect structure corresponds to the implicit hypersurface surface defined by a zero determinant condition imposed upon the Jacobian matrix. On a four-dimensional variety (space-time), this non-degenerate hypersurface constraint leads to a cubic polynomial that always can be put into correspondence with a set of non-equilibrium thermodynamic states whose kernel is a van der Waals gas. Hence this universal topological method for creating a low density turbulent non-equilibrium media leads to the setting examined statistically by Landau and Lifshitz in terms of

Bifurcation Processes and the Production of Topological Defects

281

classical fluctuations about the critical point. The conjecture presented herein is that non-equilibrium topological defects in a non-equilibrium four-dimensional medium represent the stars and galaxies, which are gravitationally attracted singularities (correlations of fluctuations of density fluctuations) of a real gas near its critical point. Note that the Cartan methods do not impose (a priori) a constraint of a metric, connection, or gauge, but do utilize the topological properties associated with constraints placed on the similarity invariants of the universal phase function. Based upon the single assumption that the universe is a non-equilibrium thermodynamic system of Pfaff topological dimension 4 leads to a cosmology where the universe, at present, can be approximated in terms of the non-equilibrium states of a very dilute van der Waals gas near its critical point. The stars and the galaxies are the topological defects and coherent (but not equilibrium) self-organizing structures of Pfaff topological dimension 3 formed by irreversible topological evolution in this non-equilibrium system of Pfaff topological dimension 4. The turbulent non-equilibrium thermodynamic cosmology of a real gas near its critical point yields an explanation for: 1. The granularity of the night sky as exhibited by stars and galaxies. 2. The Newtonian law of gravitational attraction proportional to 1/r2 . 3. The expansion of the universe (4th order curvature effects). 4. The possibility of domains of negative pressure (explaining what has recently been called dark energy) due to a classical Higgs mechanism for aggregates below the critical temperature (3rd order curvature effects). 5. The possibility of domains where gravitational effects (2nd order Gauss curvature effects) appear to be related to entropy and temperature properties of the thermodynamic system. 6. The possibility of cohesion properties (explaining what has recently been called dark matter) due to string or surface tension (1st order Mean curvature effects). 7. Black Holes (generated by Petrov Type D solutions in gravitational theory [47]) are to be related to Minimal Surface solutions to the Universal thermodynamic 4th order Phase function. A somewhat more detailed description can be found at [223], [274].

282

5.10

The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

Details of the Langford tertiary Hopf Bifurcations

The idea that multiple parameter Dynamical Systems can produce tertiary bifurcations was studied by Langford [132]. His developments were organized about certain non-linear equations in polar coordinates, with multiple parameters (r, θ, z, t; a, b, , ..), dr/dt = rg(r, z,a, G, C), dθ/dt = 1, dz/dt = f (r, z,a, b, D).

(5.424) (5.425) (5.426)

It is remarkable that these tertiary bifurcations can be demonstrated to be solutions of the Navier-Stokes equations in a rotating frame of reference [207]. Langford was interested in how these "normal" forms of dynamical systems could cause bifurcations to Hopf breather-solitons. Herein, it is also of interest to determine how and where these dynamical systems can cause bifurcations to Falaco rotational solitons. The properties of the Langford tertiary bifurcations in dynamical systems are detailed below for several examples. It is of some pedagogical utility to transform the Langford equations to {x, y, z} coordinates with parameters, a, b, .... In polar coordinates, a map between the variables {x, y, z} ⇒ {r, θ, z} leads to the following expressions, p p x2 + y 2 , dr = (xdx + ydy)/ x2 + y 2 , (5.427) r = (5.428) θ = tan(y/x), dθ = ±(ydx − xdy)/(x2 + y 2 ), z = z, dz = dz. (5.429) Substitution of the differentiable map into the polar equations yields the system of 1-forms, ω1 = dr − rg(r, z,a, b...)dt p = (xdx + ydy)/ x2 + y 2 − rg(r, z,a, b...)dt, ω2 = dθ − Ωdt = ±(ydx − xdy)/(x2 + y 2 ) − ωdt, ω3 = dz − f (r, z,a, b)dt.

(5.430) (5.431) (5.432)

The 3-form C composed from the three 1-forms becomes to within an arbitrary factor, ω1 ˆω2 ˆω3 = i(ρV4 )Ω4 = i(ρV4 )dxˆdyˆdzˆdt, = −{Vx dyˆdzˆdt − Vy dxˆdzˆdt +Vz dxˆdyˆdt − 1dxˆdyˆdt}, where V4 = [V3 , 1],

(5.433)

(5.434)

with the components of V3 given by, Vx = {∓Ωy + (xg(r, z,a, b...}, Vy = {±Ωx + (yg(r, z,a, b...)}, Vz = f (r, z,λ, α).

(5.435) (5.436) (5.437)

Details of the Langford tertiary Hopf Bifurcations

283

The rotation speed (angular velocity) is represented by Ω. The Langford examples are specializations of the functions f (r, z,a, b...) and g(r, z,a, b...). The following examples yield solutions to the similarity invariants for three of the Langford examples that he described as the Saddle Node Hopf bifurcation, the Hysteresis Hopf bifurcation, and the Transcritical Hopf bifurcation. The similarity invariants are computed for the projective dual 1-form of Action, Projective dual 1-form A = Vk dxk − Vk V k dt.

(5.438)

It can be shown that the Pfaff topological dimension of the projective dual 1-form of each of the examples below is 4. This fact implies that the abstract thermodynamic system is an open system far from thermodynamic equilibrium. Thermodynamic systems of Pfaff dimension 4 are inherently dissipative, and admit processes which are thermodynamically irreversible. Such irreversible processes are easily computable, and are proportional to the 3-form of Topological Torsion, AˆdA. If, for i(T4 )Ω4 = AˆdA, such that hT4 | ◦ |V4 i = 0,

(5.439) (5.440)

then the dynamical system is not representative of an irreversible process. For a physical system represented by the projective dual 1-form, and a process defined by the direction field of the dynamical system, the process is irreversible only if it has a component in the direction of the Topological Torsion vector. Direct computation indicates that for the projective dual 1-form composed of the components of an autonomous dynamical system, then a necessary condition for reversibility in an abstract thermodynamic sense is:

Theorem 7 Autonomous V (such that ∂V/∂t = 0) are reversible when grad(V ◦ V) ◦ curl V = 0. Pitchfork-Hopf Dynamical System

f g dx/dt dy/dt dz/dt

= = = = =

z{a + bz 2 + D(x2 + y 2 )}, {ea − G + Cz 2 + T (x2 + y 2 )}, Vx = x{ea − G + Cz 2 + T (x2 + y 2 )} ∓ Ωy, Vy = y{ea − G + Cz 2 + T (x2 + y 2 )} ± Ωx, Vz = z{a + bz 2 + D(x2 + y 2 )}.

(5.441) (5.442) (5.443) (5.444) (5.445)

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

The Similarity Invariants for A = Vk dxk − V k Vk dt are, XM = 2T (x + y) + a + 3bz 2 + D(x2 + y 2 ), YG = +Ω2 + {2T (y − x)}Ω + 2(y + x){T D(x2 + y 2 ) +aT + (3T b − 2DC)z 2 }, ZA = {β}Ω2 +2z(y − x){α}Ω, TK = 0, where α = (T D((x2 + y 2 ) − 2DCz 2 + T (3bz 2 + a), and β = (D(x2 + y 2 ) + a + 3bz 2 ).

(5.446) (5.447) (5.448) (5.449) (5.450) (5.451)

The criteria for Hopf oscillations requires that XM = 0, and ZA = 0. When these constraints are inserted into the formula for YG they yield YG(hopf ) . The criteria for oscillations is that YG(hopf ) > 0, Hopf Condition YG(hopf ) = Ω2 + Ω4T (x2 + y 2 )/(y − x), p Oscillation frequencies : ω = ± YG(hopf ) .

(5.452)

Helicity = {2z(a + bz 2 + C(x + y))}Ω +2z(x − y)(+eaD − aT − GD + (DC − T b)z 2 ), Enstrophy = +4Ω2 + {8T (y − x)}Ω + 4T 2 (x − y)2 +4z 2 (D2 (x2 + y 2 ) − 2CD(x + y) + 2C 2 ), Stretch = 4{βΩ2 + {8(x − y)(α)}Ω + {4T (x − y)2 (α)}, Brand = {...}Ω2 + {8(x − y)αβ)}Ω + {4(x − y)2 α2 }, where α = (T D((x2 + y 2 ) − 2DCz 2 + T (3bz 2 + a), and β = (D(x2 + y 2 ) + a + 3bz 2 ).

(5.454)

(5.453)

The Helicity and Brand Invariants in 3D are,

(5.455) (5.456) (5.457) (5.458) (5.459)

In this example it is noted that the Brand Invariants are not Polarization invariants. The Implicit (Pfaff dimension 3) surface of Zero Bulk dissipation (Zero Topological Parity), 2E ◦ B = K ⇒ 0, is given by the expression, K = {b1 Ω2 + b2 Ω + b3 }, b1 = 8z{C(y − x)}, b2 = −8z{−β 2 − e(Da(x + y) + 2aC) − 2bz 2 (D(x2 + y 2 ) − a) +(GD − DT (y 2 + x2 ))(x + y) − Cz 2 (Dx + 2C + Dy) −2CT (y 2 + x2 ) − 2 + 2GC}, b3 = 8z{(y − x)βα}.

(5.460) (5.461)

(5.462) (5.463)

Details of the Langford tertiary Hopf Bifurcations

285

For this example the function K is Polarization sensitive and depends upon the sign of the rotation parameter, Ω. When K 6= 0, the projective dual 1-form, A, defines a 4D symplectic manifold, and, abstractly, an "open" thermodynamic system. The K = 0 surface defines a contact manifold of Pfaff dimension 3 which is equivalent to a "closed" thermodynamic system. Saddle-Node Hopf Dynamical System f g dx/dt dy/dt dz/dt

= = = = =

a + bz 2 + D(x2 + y 2 ), (G + Cz), Vx = x(G + Cz) ∓ Ωy, Vy = y(G + Cz) ± Ωx, Vz = a + bz 2 + D(x2 + y 2 ).

(5.464) (5.465) (5.466) (5.467) (5.468)

The Similarity Invariants for A = Vk dxk − V k Vk dt are, XM = 2(G + Cz + bz), YG = +Ω2 − 2CD(x2 + y 2 ) + G2 + 2G(C + 2b)z +C 2 (1 + 4b)z 2 , ZA = +{2bz}Ω2 +2(G + Cz)(Gbz + Cbz 2 − DC(x2 + y 2 ), TK = 0.

(5.469) (5.470) (5.471) (5.472)

The similarity invariants are also chiral invariants with respect to the sign of the rotation parameter, Ω. The criteria for Hopf oscillations requires that XM = 0, and ZA = 0. When these constraints are inserted into the formula for YG they yield YG(hopf ) . The criteria for oscillations is that YG(hopf ) > 0, where, The Hopf Constraint YG(hopf ) = +3Ω2 − b2 z 2 > 0, p yields oscillation frequencies : ω = ± YG(hopf ) .

(5.473)

YG(f alaco) = Ω2 − 3B 2 /4 < 0, Falaco tension B 2 = 4b2 z 2 /9.

(5.475) (5.476)

(5.474)

Note that YG(hopf ) is a quadratic form in terms of the rotation parameter. When YG(hopf ) < 0, it is defined as YG(f alaco) . It is therefore easy to identify the tension parameter, B, for the Falaco Soliton by evaluating the Falaco formula,

The coefficient b can be interpreted as the Hooke’s law force (tension) associated with a linear spring extended in the z direction, with a spring constant equal to 2/3B.

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

Indeed, computer solutions to the Saddle node Hopf system indicate the trajectories can be confined internally to a sphere, and that Falaco surfaces of negative Gauss curvature are formed at the North and South poles by the solution trajectories. The Helicity is given by the expressions, Helicity = V ◦ curlV, Helicity = −(C(x2 + y 2 ) + 2a + 2bz 2 )Ω.

(5.477)

If the process described by the dynamical system is to be reversible in a thermodynamic sense, then the Helicity must vanish. This constraint fixes the value of the rotation frequency Ω in the autonomous system for reversible bifurcations. The Hopf-Falaco critical point in similarity coordinates can be mapped to an implicit surface in xyz coordinates, eliminating the rotation parameter, Ω, YG(hopf −critical) = YG(f alaco−critical) = −(3DC(x2 + y 2 ) + 4b2 z 2 ) ⇒ 0.

(5.478)

Depending on the values assigned to the parameters, and especially the signs of C and D, the critical surface is either open or closed. When the critical surface function is positive, the Hopf-Falaco bifurcation leads to Hopf Solitons (breathers), and if the critical surface function is negative, the bifurcation leads to Falaco Solitons. The Brand Invariants in 3D are, Enstrophy = +4Ω2 + (2D − C)2 (x2 + y 2 ), Stretch = {8bz}Ω2 + (G + Cz)(2D − C)2 (x2 + y, Brand = {(C − 2D)2 (x2 + y 2 ) + 16b2 z 2 }Ω2 +{(G + Cz)2 (C − 2D)2 }(x2 + y 2 ).

(5.479) (5.480) (5.481)

The Saddle Node Hopf system will be studied in greater detail in later sections, with examples of phase transitions. Recall that the tertiary Hopf bifurcations represented by these examples are solutions to the Navier Stokes equations in a rotating frame of reference. From a hydrodynamic point of view, Enstrophy is a measure of the square of the vorticity and intuitively is related to "twisting". Stretch can be related to the stretching of vortex lines, and it is a conjecture that the Brand(fold) invariant is related to folds. Note that all three Brand invariants are also chiral invariant relative to the sign of the rotation parameter, Ω. The Implicit (Pfaff dimension 3) surface of Zero Bulk dissipation (Zero Topological Parity), 2E ◦ B = K ⇒ 0, is given by the expression, K = 8Ω{2bz(D(x2 + y 2 ) + bz 2 + a) + C(x2 + y 2 )(G + Cz) ⇒ 0.

(5.482)

The Saddle-Node Hopf example also appears in the next section, where it can be related to the phenomenon of Torsion bursting (incorrectly called vortex bursting) in

Details of the Langford tertiary Hopf Bifurcations

287

the hydrodynamics literature. The dissipation function K is polarization sensitive and linear in the rotation parameter Ω, but this polarization sensitivity does not produce two polarization implicit functions. In the Pitchfork case, there were 4 distinct critical speeds determined by the polarization sensitivity and the quadratic form of the implicit hypersurface equation. By analog, the Saddle Node Hopf is related to the case of EM signal propagation in a Lorentz vacuum. All four speeds of propagation (two polarizations and two propagation directions) are the same. Hysteresis-Hopf Dynamical System This flow was introduced in the previous section where numeric solutions to certain trajectories indicated the onset of a torsion burst. The fundamental formulas are given by the expressions,

f g dx/dt dy/dt dz/dt

= = = = =

a + bz + Ez 3 + D(x2 + y 2 ), (−G + Cz), Vx = x(−G + Cz) ∓ Ωy, Vy = y(−G + Cz) ± Ωx, Vz = a + bz + Ez 3 + D(x2 + y 2 ).

(5.483) (5.484) (5.485) (5.486) (5.487)

The Similarity Invariants for A = Vk dxk − V k Vk dt are, XM = 2(Cz + G) + (b + 3Ez 2 ), Y g = Ω2 + {G2 − 2Gb − 2DC(x2 + y 2 )} + {2G(b − C)}z +{C 2 − 6GE}z 2 + {6CE}z 3 , ZA = {b + 3Ez 2 }Ω2 + {2GCD(x2 + y 2 ) + G2 b} + {−2C 2 D(x2 + y 2 ) − 2GCb}z + {3G2 E + C 2 b}z 2 +{−6GCE}z 3 + {3C 2 E}z 4 , TK = 0.

(5.488) (5.489)

(5.490) (5.491)

The criteria for Hopf oscillations requires that XM = 0, and ZA = 0. When these constraints are inserted into the formula for YG they yield YG(hopf ) . The condition for oscillations is that YG(hopf ) > 0, where, The Hopf Condition : YG(hopf ) = 3Ω2 − 9/4E 2 z 4 −3/2bEz 2 − 1/4b2 > 0, p yields oscillation frequencies : ω = ± YG(hopf ) .

(5.492) (5.493)

Note that (like the Saddle Node Hopf case) YG(hopf ) is a quadratic form in terms of the rotation parameter. It is therefore easy to identify the tension parameter for the

288

The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

Falaco Soliton by evaluating the Falaco formula: YG(f alaco) = Ω2 − 3b2 /4 < 0, Falaco tension b2 = (9E 2 z 4 + 6bEz 2 + b2 )/9.

(5.494) (5.495)

In this case the tension is not that of a linear spring, but instead can be interpreted as a non-linear spring constant for extensions in the z direction. Indeed, computer solutions to the Hysteresis - Hopf - Falaco system indicate the trajectories can be confined internally to a sphere-like surface, and that Falaco minimal surfaces are visually formed at the North and South poles [132]. The Helicity becomes,

Helicity = V ◦ curlV, Helicity = −{C(x2 + y 2 ) + 2a + 2z(b + Ez 2 )}Ω.

(5.496)

If the process described by the dynamical system is to be reversible in a thermodynamic sense, then the Helicity must vanish. This constraint fixes the value of the rotation frequency Ω in the autonomous system for reversible bifurcations. The Hopf-Falaco critical point in similarity coordinates can be mapped to an implicit surface in xyz coordinates, eliminating the rotation parameter, Ω, YG(hopf −critical) = YG(f alaco−critical) = −(3DC(x2 + y 2 ) + (3EZ 2 + b)2 ) ⇒ 0. (5.497) Depending on the values assigned to the parameters, and especially the signs of C and D, the critical surface is either open or closed. When the critical surface function is positive, the Hopf-Falaco bifurcation leads to Hopf Solitons (breathers), and if the critical surface function is negative, the bifurcation leads to Falaco Solitons. Note that if E = 0, DC < 0, then there is a circular limit cycle in the x,y plane. Direct integration of the differential equations demonstrates the decay to this attractor. The Brand Invariants in 3D are, Enstrophy = +4Ω2 + (2D − C)2 (x2 + y 2 ), Stretch = 4{b + 3Ez 2 }Ω2 +(zC − G)(2D − C)2 (x2 + y 2 ), Brand = {(2D + C)2 (x2 + y 2 ) + 4(3Ez 2 + b)2 }Ω2 +(2D − C)2 (zC − G)2 (x2 + y 2 ).

(5.498) (5.499) (5.500)

The Implicit (Pfaff dimension 3) surface of Zero Bulk dissipation (Zero Topological Parity), 2E ◦ B = K ⇒ 0, is given by the expression, K = 8Ω{Ez 2 (3D(x2 + y 2 ) + 4bz + 3Ez 3 + 3a) +(bD − GC + C 2 z)(x2 + y 2 ) + b2 z + ab} ⇒ 0.

(5.501)

Details of the Langford tertiary Hopf Bifurcations

289

Again, like the Saddle Node Hopf case, the Implicit hypersurface K = 0 has only one solution depending upon the rotation parameter Ω. When K 6= 0, the projective dual 1-form, A, defines a 4D symplectic manifold, and, abstractly, an "open" thermodynamic system. The K = 0 surface defines a contact manifold of Pfaff dimension 3 which is equivalent to a "closed" thermodynamic system. Transcritical Hopf Dynamical System f g dx/dt dy/dt dz/dt

= = = = =

az + bz 2 + D(x2 + y 2 ), a − G + Cz, Vx = x(a − G + Cz) ∓ Ωy, Vy = y(a − G + Cz) ± Ωx, Vz = az + bz 2 + D(x2 + y 2 ).

(5.502) (5.503) (5.504) (5.505) (5.506)

The Similarity Invariants for A = Vk dxk − V k Vk dt are, XM = 3a − 2G + 2(C + b)z, Y g = +Ω2 − 2CD(x2 + y 2 ) + (4Cb + C 2 )z 2 + 2z(2b(a − G) +C(2C − G) + (G2 + 3a2 − 4Ga), ZA = +{a + 2bz}Ω2 + a3 + 2a2 bz − 2Ga2 − 4aGbz + 2Cza2 +4aCz 2 b − 2aCy 2 D + G2 a + 2G2 bz − 2GCza − 4GCz 2 b +2GCy 2 D + C 2 z 2 a + 2C 2 z 3 b − 2C 2 zy 2 D +2Dx2 C(G − a − Cz), TK = 0.

(5.507) (5.508)

(5.509) (5.510)

The similarity invariants are chiral invariants relative to the rotation parameter Ω. The criteria for Hopf oscillations requires that XM = 0, and ZA = 0. When these constraints are inserted into the formula for YG they yield YG(hopf ) . The criteria for (breather) oscillations is that YG(hopf ) > 0, where, The Hopf Condition yields oscillation frequencies :

: YG(hopf ) = 3Ω2 abz − b2 z 2 − 1/4a2 > 0,(5.511) p ω = ± −YG(hopf ) . (5.512)

Note that (again) YG(hopf ) is a quadratic form in terms of the rotation parameter. It is therefore easy to identify the tension parameter for the Falaco Soliton by evaluating the Falaco formula, YG(f alaco) = Ω2 − 3b2 /4. Falaco tension B 2 = (4b2 z 2 + a2 )/9abz).

(5.513) (5.514)

290

The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

In this case the tension is again to be associated with a non-linear spring with extensions in the z direction. The Helicity is given by the expressions, Helicity = V ◦ curlV, Hbif urcation = −{C(x2 + y 2 ) + 2z(a + bz)}Ω. If the process described by the dynamical system is to be reversible in a thermodynamic sense, then the Helicity must vanish. This constraint fixes the value of the rotation frequency Ω in the autonomous system for reversible bifurcations. The Hopf-Falaco critical point in similarity coordinates can be mapped to an implicit surface in xyz coordinates, eliminating the rotation parameter, Ω, YG(hopf −critical) = YG(f alaco−critical) = −(3DC(x2 + y 2 ) + (2bz + a)2 ) ⇒ 0.

(5.515)

Depending on the values assigned to the parameters, and especially the signs of C and D, the critical surface is either open or closed. When the critical surface function is positive, the Hopf-Falaco bifurcation leads to Hopf Solitons (breathers), and if the critical surface function is negative, the bifurcation leads to Falaco Solitons. Note that if b = 0, DC < 0, then there is a circular limit cycle in the x,y plane. Direct integration of the differential equations demonstrates the decay to this attractor. The Brand Invariants in 3D are, Enstrophy = +4Ω2 + (2D − C)2 (x2 + y 2 ), Stretch = +{4a + 8bz}Ω2 +(x2 + y 2 )(2D − C)2 (a − G + Cz), Brand = {4(D + C)2 (x2 + y 2 ) + 4(a + 4bz)2 }Ω2 +(x2 + y 2 )(2D − C)2 (a − G + Cz).

(5.516) (5.517) (5.518)

The Brand invariants are polarization invariants, as they do not depend upon the sign of the rotation. They involve only the even powers of the rotation, Ω. The Implicit (Pfaff dimension 3) surface of Zero Bulk dissipation (Zero Topological Parity), 2E ◦ B = K ⇒ 0 is given by the expression, K = −8Ω{(bz(−2D(x2 + y 2 ) − 3az − 2bz 2 ) +(C(G − Cz) − a(C + D))(x2 + y 2 ) − a2 z}.

(5.519)

When K 6= 0, the projective dual 1-form, A, defines a 4D symplectic manifold, and, abstractly, an "open" thermodynamic system. The K = 0 surface defines a contact manifold of Pfaff dimension 3 which is equivalent to a "closed" thermodynamic system.

Details of the Langford tertiary Hopf Bifurcations

291

Minimal Surface Hopf and Falaco Bifurcations The utility of Maple (see Volume 6, "Maple programs for Non-Equilibrium systems" ) becomes evident when generalizations of the Langford systems can be studied. The dynamical system is, f g dx/dt dy/dt dz/dt

= = = = =

a + bz + F z 2 + Ez 3 + D(x2 + y 2 ), G + Cz, Vx = x(G + Cz) ∓ Ωy, Vy = y(G + Cz) ± Ωx, Vz = a + bz + F z 2 + Ez 3 + D(x2 + y 2 ).

(5.520) (5.521) (5.522) (5.523) (5.524)

This system can be studied with about as much ease as all of the preceding examples. An especially interesting case is given by the system, f g dx/dt dy/dt dz/dt

= = = = =

a + P sinh(αz) + D(x2 + y 2 ), G + Cz, Vx = x(G + Cz) ∓ Ωy, Vy = y(G + Cz) ± Ωx, Vz = a + P sinh(αz) + D(x2 + y 2 ).

(5.525) (5.526) (5.527) (5.528) (5.529)

The Similarity Invariants for A = Vk dxk − V k Vk dt are, XM = 2(G + Cz) + αP cosh(αz), Y g = +Ω2 − 2CD(x2 + y 2 ) + (G + Cz)2 +2(G + Cz)P α cosh(αz), ZA = (+Ω2 + (G + Cz)2 )P α cosh(αz) −2CD(G + Cz)(x2 + y 2 ), TK = 0.

(5.530) (5.531) (5.532) (5.533)

The similarity invariants are chiral invariants relative to the rotation parameter Ω. The criteria for Hopf oscillations requires that XM = 0, and ZA = 0. When these constraints are inserted into the formula for YG they yield YG(hopf ) . The criteria for (breather) oscillations is that YG(hopf ) > 0, where, YG(hopf ) = 3Ω2 − 1/4α2 P 2 (cosh(αz))2 > (5.534) 0, p yields oscillation frequencies : ω = ± −YG(hopf ) . (5.535) the Hopf Condition:

Note that (again) YG(hopf ) is a quadratic form in terms of the rotation parameter. It is therefore easy to identify the tension parameter for the Falaco Soliton by evaluating the Falaco formula, YG(f alaco) = Ω2 − 3b2 /4, Falaco tension b2 = (α2 P 2 (cosh(αz))2 )/3).

(5.536) (5.537)

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

In this case the tension is again to be associated with a non-linear spring with extensions in the z direction. The Helicity is given by the expressions, Helicity = V ◦ curlV, Hbif urcation = −{C(x2 + y 2 ) + 2(a + P sinh(αz))}Ω.

(5.538)

If the process described by the dynamical system is to be reversible in a thermodynamic sense, then the Helicity must vanish. This constraint fixes the value of the rotation frequency Ω in the autonomous system for reversible bifurcations. The Hopf-Falaco critical point in similarity coordinates can be mapped to an implicit surface in xyz coordinates, eliminating the rotation parameter, Ω, YG(hopf _critical) = YG(f alaco_critical) ⇒ 0, = −{3DC(x2 + y 2 ) + α2 P 2 (cosh(αz))2 }.

(5.539) (5.540)

When the parameters DC have a product which is negative, then the critical surface is the catenoid — A Minimal Surface. That is the Hopf critical surface is an implicit surface as given by the equation, (x2 + y 2 ) = {(α2 P 2 )/(3|DC|)}(cosh(αz))2 . The throat diameter of the catenoid is proportional to the coefficient, p Diam = 2 (α2 P 2 )/(3|DC|).

Figure 5.30 Hopf Minimal surface

(5.541)

(5.542)

Emergence of Contact structures from Symplectic domains

5.11

293

Emergence of Contact structures from Symplectic domains

In this section the physical systems will be limited to four dimensions and the generalized Hamiltonian format will be used. The unconstrained action 1-form is presumed to be of Pfaff dimension 4. To review, the 1-form of Action representing the physical system will be written in terms of the generalized Hamiltonian format, A = P dq − Φdt.

(5.543)

The total energy function Φ will be defined in terms of Hamiltonian independent variables (or coordinates):

Φ with dΦ with Φq ΦP

= = = =

Total energy function Φ(q, t, P, Mc2 ), Φq dq + Φt dt + ΦP dP + ΦMc2 d(Mc2 ), ∂Φ/∂q, Φt = ∂Φ/∂t, ∂Φ/∂P, ΦMc2 = ∂Φ/∂(Mc2 ).

(5.544) (5.545) (5.546) (5.547)

The natural 4-Volume element in terms of the independent variables is given by the expression, Ω4 = d(Mc2 )ˆdP ˆdqˆdt. (5.548) The Pfaff sequence has the (topological) terms: Action A = P dq − Φdt, Vorticity dA = dP ˆdq − {Φq dq + ΦP dP + ΦMc2 d(Mc2 )}ˆdt, Torsion AˆdA = {P ΦP − Φ}dP ˆdqˆdt +{P ΦMc2 }d(Mc2 )ˆdqˆdt, Parity dAˆdA = −2{ΦMc2 }d(Mc2 )ˆdP ˆdqˆdt = −2d(Φ)ˆdP ˆdqˆdt.

(5.549) (5.550) (5.551) (5.552) (5.553)

5.11.1

Effect of Constraints on the Reduction of the Pfaff Topological Dimension from 4 to 3 The objective is to examine various constraints that will reduce the Pfaff dimension of the Action from 4 to 3. Constraint 1: Φ = constant It is apparent that if the total energy function was a universal constant then the 4-form dAˆdA vanishes and the Pfaff topological dimension is reduced. This global constraint will be defined as the strong constraint, and called the conservation of total energy.

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

Conservation of total energy

dΦ = 0,

Φ(q, t, P, Mc2 ) = constant.

(5.554)

This strong constraint reduces the Pfaff dimension from 4 to 3, and insures the Contact structure is of the form, AˆdA = {−Φ}dP ˆdqˆdt.

(with Φ = constant),

(5.555)

The 3-form will have singularities if the total energy function has zeros. Hence, if the total energy function is never zero, then the 3-form is, in the language of Eliashberg, tight without limit cycles. Constraint 2: ΦMc2 = ∂Φ/∂(Mc2 ) = 0 It is also apparent that the weaker constraint where ∂Φ/∂(Mc2 ) = 0 also reduces the Pfaff dimension from 4 to 3, for if valid the 4-form dAˆdA ⇒ 0. The Contact structure becomes, (with ΦMc2 = 0),

AˆdA = {P ΦP − Φ}dP ˆdqˆdt.

(5.556)

The constraint permits limit cycles even for cases where the function Φ does not have zeros. Constraint 3: ΦMc2 d(Mc2 ) + Φt dt = 0 This constraint reduces the differential d(Φ) to the non-zero 1-form, d(Φ) = Φq dq + ΦP dP.

(5.557)

When this structure is inserted into the 4-form dAˆdA, the 4-form vanishes, and the 3-form of topological torsion yields the same Contact structure as the Constraint 2, (with ΦMc2 d(Mc2 ) + Φt dt = 0),

AˆdA = {P ΦP − Φ}dP ˆdqˆdt.

(5.558)

The interpretation is now more interesting for it corresponds to the case where the total energy function contains the variable Mc2 but now the "Mass" energy changes with time, d(Mc2 )/dt = Φt /ΦMc2 . (5.559) and the total energy is not necessarily a constant, d(Φ) 6= 0.

Emergence of Contact structures from Symplectic domains

295

Constraint 4: ΦMc2 d(Mc2 ) + ΦP dP = 0 This constraint reduces the differential d(Φ) to the non-zero 1-form, d(Φ) = Φq dq + Φt dt.

(5.560)

This 1-form is of the type that corresponds to kinematic velocity. When this structure is inserted into the 4-form dAˆdA, the 4-form vanishes, and the 3-form of topological torsion yields the same Contact structure as for Constraint 1, (with ΦMc2 d(Mc2 ) + ΦP dP = 0),

AˆdA = {−Φ}dP ˆdqˆdt.

(5.561)

Constraint 5: ΦMc2 d(Mc2 ) + Φq dq = 0 This constraint reduces the differential d(Φ) to the non-zero 1-form, d(Φ) = ΦP dP + Φt dt.

(5.562)

This 1-form is of the type that corresponds to Newtonian force. When this structure is inserted into the 4-form dAˆdA, the 4-form vanishes, and the 3-form of topological torsion yields the same Contact structure as for Constraint 2, (with ΦMc2 d(Mc2 ) + Φq dq = 0),

AˆdA = {P ΦP − Φ}dP ˆdqˆdt.

(5.563)

Constraint 6: ΦMc2 d(Mc2 ) + Φq dq + Φt dt = 0 This constraint reduces the differential d(Φ) to the non-zero 1-form, d(Φ) = ΦP dP.

(5.564)

When this structure is inserted into the 4-form dAˆdA, the 4-form vanishes, and the 3-form of topological torsion yields the same Contact structure as for Constraint 2, (with ΦMc2 d(Mc2 ) + Φq dq + Φt dt = 0),

AˆdA = {P ΦP − Φ}dP ˆdqˆdt. (5.565)

Constraint 7: ΦMc2 d(Mc2 ) + ΦP dP + Φt dt = 0 This constraint reduces the differential d(Φ) to the non-zero 1-form, (5.566)

d(Φ) = Φq dq.

When this structure is inserted into the 4-form dAˆdA, the 4-form vanishes, and the 3-form of topological torsion yields the same Contact structure as for Constraint 1, (with ΦMc2 d(Mc2 ) + ΦP dP + Φt dt = 0),

AˆdA = {−Φ}dP ˆdqˆdt.

(5.567)

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

Constraint 8: ΦMc2 d(Mc2 ) + Φq dq + ΦP dP = 0 This constraint reduces the differential d(Φ) to the non-zero 1-form, (5.568)

d(Φ) = Φt dt.

When this structure is inserted into the 4-form dAˆdA, the 4-form vanishes, and the 3-form of topological torsion yields the same Contact structure as for Constraint 1, (with ΦMc2 d(Mc2 ) + Φq dq + ΦP dP = 0),

AˆdA = {−Φ}dP ˆdqˆdt.

(5.569)

It should be noted that in cases where the 3-form admits limit cycles, a further global change in Pfaff topological dimension occurs when the total energy is homogeneous of degree 1 in the momenta, P. In all but the first case, the total energy is not a global constant. It also should be noted that a similar reduction in Pfaff dimension can be obtained by constraining the momenta, P. Constraint 9. ΦP dP + Φq dq + Φt dt = 0 The constraint is relevant to second order differential equations. It follows that d(Φ) = ΦMc2 d(Mc2 ). Substitution leads to Pfaff reduction, as dAˆdA ⇒ 0. However, the 3-form of topological torsion becomes, (with ΦP dP + Φq dq + Φt dt = 0),

AˆdA = {P ΦMc2 }d(Mc2 )ˆdqˆdt.

(5.570)

5.11.2 Emergence of Lagrangian submanifolds from Symplectic manifolds Consider the Symplectic manifold of dimension 4, generated by the exact 2-form, dA. The Pfaff topological dimension is 4 on the symplectic space, where dAˆdA 6= 0. The Lagrangian submanifold is a two-dimensional subspace upon which the 2-form, dA, when restricted to the submanifold, is zero. This means that in terms of the twodimensional coordinates of the submanifold, the 2-form, dA, vanishes. Hence, the Pfaff dimension on the Lagrangian submanifold is 1, and the system 1-form represents an equilibrium system. In Section 2.10, the phase function for a 4th order system was examined, and an example was given whereby two of the eigenvalues of the Jacobian matrix were zero, indicating that there existed a 2D submanifold. In fact it was demonstrated that the mean curvature of the example was zero, and yet the Gauss curvature was positive. This fact meant that the two equal and opposite eigenvalues were pure imaginary pairs. Such a result is called a Hopf bifurcation in the literature of Dynamical Systems. From a topological viewpoint of thermodynamics the construction of a minimal surface is in fact the constraint that the 2-form dA should vanish. That is, there exists a complex vector of 3-complex-components (the 6-components of the 2-form dA on the 4D symplectic space), and that this (complex) vector has a square which is zero. Such vectors are called isotropic null vectors, or Spinors. The minimal surface construction is then developed in terms of a spinor.

Summary

297

Such a state is an isotropic state, and has the features of an equilibrium surface. Fluctuations may cause such states to be not perfect equilibrium states, but they are presumed to be very close to equilibrium states. If the minimal surface is regular, such that the Hermitean square is non-zero, then there are no interior topological defects. In electromagnetic parlance, one would say that the two Poincare 4-forms vanish. The first reduces the domain from Pfaff dimension 4, the second reduces the structure to Pfaff dimension 2 (implying no self intersections or other defect structures). The 1-form is of Pfaff dimension 2, and therefore admits an integrating factor, such that the system can be expressed as a perfect differential. Hence, the 2-form vanishes and the system is in an equilibrium state. 5.12

Summary

The many examples given above have utilized the concepts of Pfaff topological dimension of 1-forms, and the existence of a Universal Phase function (generated from the Jacobian matrix of the 1-form of Action), to give credence to the idea that nonequilibrium thermodynamics, from the perspective of continuous topological evolution, is a common basis of many different physical and engineering disciplines. For example, it has been demonstrated that the concept of a Faraday induction law has a realization in both electromagnetism and hydrodynamics. Both systems have a Master Equation. Both systems have the equivalent to a Lorentz force law (without the imposition of special relativity).

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The Thermodynamics of Dynamical Systems of Non-Linear ODE’s

Part III Topological Aspects of Exterior Differential Forms

299

Chapter 6 CARTAN’S TOPOLOGICAL STRUCTURE 6.1

Introduction

In this chapter∗ , a topological perspective will be used to extract those properties of physical systems and their evolution that are independent from the geometrical constraints of connections and/or metrics. It is subsumed that the presence of a physical system establishes a topological structure on a (possibly geometric) base space of independent variables. This concept is different from, but similar to, the geometric perspective of general relativity, whereby the presence of a physical system is presumed to establish a metric on a base space of independent variables. Note that a given base of independent variables may support many different topological structures; hence a given base may support many different physical systems. A major success of theory is that continuous non-homeomorphic processes of topological evolution establish a logical basis for thermodynamic irreversibility and the arrow of time [221] without the use of statistics. The fundamental axioms utilized in this chapter (similar to those used in previous chapters) are: Axiom 1. Thermodynamic physical systems can be encoded in terms of a 1-form of covariant Action Potentials, Ak (x, y, z, t...), on a ≥ fourdimensional abstract variety of ordered independent variables, {x, y, z, t...}. The variety supports a differential volume element Ω4 = dxˆdyˆdzˆdt... Axiom 2. Thermodynamic processes are assumed to be encoded, to within a factor, ρ(x, y, z, t...), in terms of contravariant Vector and/or complex isotropic Spinor direction fields, V4 (x, y, z, t...). Axiom 3. Continuous topological evolution of the thermodynamic system can be encoded in terms of Cartan’s magic formula (see p. 122 in [148]). The Lie differential, when applied to a exterior differential 1∗

The basis for this chapter was presented as a talk given in August, 1991, at the Pedagogical Workshop on Topological Fluid Mechanics held at the Institute for Theoretical Physics, Santa Barbara UCSB. Part of the T4 truth table was due to Phil Baldwin. The recognition that the Cartan topology was a disconnected topology is due to the author.

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Cartan’s Topological Structure

form of Action, A = Ak dxk , is equivalent abstractly to the first law of thermodynamics.

Cartan’s Magic Formula L(ρV4 ) A First Law Inexact Heat 1-form Q Inexact Work 1-form W Internal Energy U

= : = = =

i(ρV4 )dA + d(i(ρV4 )A), W + dU = Q, W + dU = L(ρV4 ) A, i(ρV4 )dA, i(ρV4 )A.

(6.1) (6.2) (6.3) (6.4) (6.5)

Axiom 4. Equivalence classes of systems and continuous processes can be defined in terms of the Pfaff topological dimension of the 1-forms of Action, A, Work, W , and Heat, Q. In the period from 1899 to 1926, Eli Cartan developed his theory of exterior differential systems [40], [41], which included the ideas of spinor systems [44] and the differential geometry of projective spaces and spaces with torsion [42]. The theory was appreciated by only a few contemporary researchers, and made little impact on the main stream of the physical sciences until about the 1960’s. Even specialists in differential geometry (with a few notable exceptions [51] ) made little use of Cartan’s methods until the 1950’s. Even today, many physical scientists and engineers have the impression that Cartan’s theory of exterior differential forms is just another formalism of fancy. However, Cartan’s theory of exterior differential systems has several advantages over the methods of tensor analysis that were developed during the same period of time. The principle fact is that differential forms are well behaved with respect to functional substitution of C1 differentiable maps. Such maps need not be invertible even locally, yet differential forms are always deterministic in a retrodictive sense [192], by means of functional substitution. Such determinism is not to be associated with contravariant tensor fields, if the map is not a diffeomorphism. Cartan’s theory of exterior differential systems contains topological information, and admits non-diffeomorphic maps which can describe topological evolution. Although the word "topology" had not become popular when Cartan developed his ideas (topological ideas were described as part of the theory of analysis situs), there is no doubt that Cartan’s intuition was directed towards a topological development. For example, Cartan did not define what were the open sets of his topology, nor did he use, in his early works, the words "limit points or accumulation points" explicitly, but he did describe the union of a differential form and its exterior differential as the "closure" of the form. With this concept, Cartan effectively used

Introduction

303

the idea that the closure of a subset is the union of the subset with its topological limit points. What was never stated (until 1990) is the idea that the exterior differential is indeed a limit point generator relative to a Cartan topology. The union of the identity operator and the exterior differential satisfy the axioms of a Kuratowski closure operator [141], which can be used to define a topology. The other operator of the Cartan calculus, the exterior product, also has topological connotations when it is interpreted as an intersection operator. In a perhaps over simplistic comparison, it might be said that ubiquitous tensor methods are restricted to geometric applications, while Cartan’s methods can be applied directly to topological concepts as well as geometrical concepts. Cartan’s theory of exterior differential systems is a topological theory not necessarily limited by geometrical constraints and the class of diffeomorphic transformations that serve as the foundations of tensor calculus. A major objective of this chapter is to show how limit points, intersections, closed sets, continuity, connectedness and other elementary concepts of modern topology are inherent in Cartan’s theory of exterior differential systems. These ideas do not depend upon the geometrical ideas of size and shape. Hence Cartan’s theory, as are all topological theories, is renormalizable (perhaps a better choice of words is that the topological components of the theory are independent from scale). In fact the most useful of Cartan’s ideas do not depend explicitly upon the geometric ideas of a metric, nor upon the choice of a differential connection between basis frames, as in fiber bundle theories. The theme of this chapter is to explore the physical usefulness of those topological features of Cartan’s methods which are independent from the constraints and refinements imposed by a connection and/or a metric. In this chapter the Cartan topology will be constructed explicitly for an arbitrary exterior differential system, Σ. For a particular, simple, but useful, system consisting of a single 1-form of Action, A, all elements of the Cartan topology will be evaluated, and the limit points, the boundary sets and the closure of every subset will be computed abstractly. Earlier intuitive results, which utilized the notion that Cartan’s concept of the exterior product may be used as an intersection operator, and his concept of the exterior differential may be used as a limit point operator acting on differential forms, will be given formal substance in this chapter. A major result of this chapter, with important physical consequences in describing topological evolutionary processes, is the demonstration that the Cartan topology is not necessarily a connected topology, unless the property of topological torsion vanishes, and that thermodynamic irreversibility is a consequence of four dimensions or more. 6.1.1

A Point Set Topology Example

As an example of the topological ideas, consider the set of 4 elements or points, X : {a, b, c, d},

(6.6)

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Cartan’s Topological Structure

and all possible subsets: ∅, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, c, d}, {b, c, d}, {a, b, d}, {a, b, c, d} = X.

(6.7) (6.8) (6.9) (6.10) (6.11)

Select the following subset elements as a topological basis, basis selection {a}, {a, b}, {c}, {c, d},

(6.12)

and then compose a topology T 4 of open sets from all possible unions of the selected basis elements: T 4{open} : ∅, {a}, {c}, {a, b}, {c, d}, {a, c}, {a, b, c}, {a, c, d}, {a, b, c, d}.

(6.13)

The closed sets are the compliments of the open sets: T 4{closed} : {a, b, c, d}, {b, c, d}, {a, b, d}, {c, d}, {a, b}, {b, d}, {d}, {b}, ∅.

(6.14)

It is an easy exercise to demonstrate that the collections above indeed satisfy the axioms of a topology. (This is not the only topology that can be constructed over 4 elements.) This simple example of a point set topology permits explicit construction of all the topological concepts, which include limit sets, interiors, boundaries, and closures, for the all of subsets of X, relative to the topology, T 4. The standard definitions are: 1. A limit point of a subset A is a point p such that all open sets that contain p also contain a point of A not equal to p. 2. The closure of a subset A is the union of the subset and its limit points, and is the smallest closed set that contains A. 3. The interior of a subset is the largest open set contained by the subset. 4. The exterior of a subset is the interior of its compliment. 5. A boundary of a subset is the set of points not contained in the interior or exterior. 6. The closure of a subset is also equal to the union of its interior and its boundary.

Introduction

305

The results of applying these definitions to the T 4 topology of 4 points are: Table 1. A T4 Topology of 4 points X = {a, b, c, d} Basis subsets {a}, {a, b}, {c}, {c, d} T 4{open} : ∅, {a}, {c}, {a, b}, {c, d}, {a, c}, {a, b, c}, {a, c, d}, X T 4{closed} : X, {b, c, d}, {a, b, d}, {c, d}, {a, b}, {b, d}, {d}, {b}, ∅ Subset Limit Pts Interior Boundary Closure ∅ ∅ ∅ ∅ ∅ {a} {b} {a} {b} {a, b} {b} ∅ ∅ {b} {b} {c} {d} {c} {d} {c, d} {d} ∅ ∅ {d} {d} ∅ {a, b} {b} {a, b} {a, b} {b, d} {a, c} {b}, {d} X {a, c} {b, d} {a} {b} {a, b, d} {a, d} {c} {d} {b, c, d} {b, c} {b, d} ∅ ∅ {b, d} {b, d} {b, d} {c, d} {d} {c, d} {c, d} ∅ {a, b, c} {b}, {d} {a, b, d} {d} X {b, c, d} {d} {c, d} {b} {b, c, d} {a, c, d} {b}, {d} {a, c, d} {b} X {a, b, d} {b} {a, b} {d} {b, c, d} {a, b, c, d} {b}, {d} X ∅ X

(6.15)

This T 4 topology is quite interesting for many demonstrable reasons. First note that all of the singletons of the topology are not closed. This implies that the topology is NOT a metric topology, NOT a Hausdorf topology, and even does NOT satisfy the separation axioms to be a T1 topology† . Note that all closed sets contain all of their limit points. Some open sets can contain limit points, but some open sets do not contain their limit points. Some subsets have boundaries that are composed of their limit points. Some subsets have limit points which are not boundary points. Certain subsets have a boundary, but do not have limit points, and in other cases there are subsets that have limit points, but do not have a boundary. There are certain subsets with a boundary, but without an interior. There are certain subsets with an interior, but without a boundary. These situations, though topologically correct, are not always intuitive to those accustomed to metric based topological concepts, which impose a number of additional constraints on the sets of interest. Yet all of these topological ideas, including the non-intuitive ones, are easy to grasp from the simple example of the T 4 point set topology. †

For those not familiar with point set topology, chapter 5 in Schaum’s outline [141] can be useful.

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One other very important observation is that there are subsets of the T 4 topology, {a, b} and {c, d}, (other than ∅ and X) which are both open and closed. The union of these two subsets {a, b} and {c, d} is X. Topologies with this property are said to be disconnected topologies. What is important is that it is possible to construct a continuous map from a disconnected topology to a connected topology, but it is impossible to construct a continuous map from a connected topology to a disconnected topology. If the mapping process is interpreted as an evolutionary process, these facts establish a logical or topological basis for the arrow of time [221]. This idea can be exploited to explain the concept of thermodynamic irreversibility without the use of statistics. What is even more remarkable is that properties of the T 4 topology can be replicated in terms of the Pfaff sequence of exterior differential sets, Pfaff Sequence : {A, dA, AˆdA, dAˆdA...},

(6.16)

generated from any given 1-form of Action, A, on a N-dimensional geometric variety. The Pfaff sequence is readily computed, and will contain M ≤ N elements, where M is defined as the Pfaff topological dimension (or class) of the given 1-form, A. The realization of a T 4 topology in terms of exterior differential forms is herein defined as the "Cartan topology", and is detailed in the next section. The Cartan topology has far reaching consequences in applications to physical problems. 6.1.2 Algebraic and Differential Closure The concept of closure is one of the most important ideas in Cartan’s theory. His methods center on two procedures of closure, one algebraic, and one differential. Both processes are closed in the sense that when they operate on a subset of a set of exterior differential forms, the objects created are also subsets of the set of exterior differential forms. There are no surprises. Cartan utilized the exterior algebra over a variety of dimension N to construct a vector space of exterior differential forms of dimension 2N . The N subspaces of this (Grassmann) space are vector spaces of dimension equal to N things taken p at a time. The elements of the subspaces are called p-forms. In four dimensions, the subspace sets are one dimensional, N = four dimensional, N(N+1)/2 = six dimensional, N = four dimensional, and one dimensional. The elements of the subspaces are often called scalars (0-forms), vectors (1-forms), tensors (2-forms), pseudovectors (3-forms), and pseudo-scalars‡ (4-forms) in relativistic physical theories. The Exterior (Grassmann) algebra has a finite 2N basis (equal to 16 elements in a space of 4 independent variables). The concept of closure means that the operations on elements of the 2N -dimensional space yield results that are contained within the 2N -dimensional space. When the operations are applied to elements of a subspace, the results usually are not contained in the same subspace, but they are contained within the 2N -dimensional vector space of p-forms. ‡

Distinctions between differential form Scalars and differential form Densities will modify this terminology

Introduction

307

The exterior product (with symbol ˆ) takes elements of the 2N -base space and multiplies them together in a manner such that the result is contained as an element of the 2N -base space. This process of exterior multiplication is closed, for the action of the process on any subset of the 2N -base space produces another subset of the 2N -base space. However, the exterior product takes a p-form times a q-form into a (p+q)-form. The elements of the product can be from different or from the same vector subspaces, but the resultant is always a linear combination of the subspaces of the Exterior algebra. Similarly the concept of exterior differentiation (with symbol d) is defined such that the operation produces a (p+1)-form from a p-form. This process of exterior differentiation is "closed", for the action of the process on any subset of the 2N base space produces another subset of the 2N base space. A differential ideal is defined as the union of a collection of given p-forms and their exterior differentials. An "interior" product with respect to a direction field V (with symbol i(V) and of dimension N) can be defined on the Grassmann algebra of exterior differential forms. The interior product takes a p-form to a (p-1)-form, and in this sense is another operation which is closed within the Grassmann algebra. The resultant product is still an element of the 2N -base space. Where the exterior differential raises the rank of a p-form to a (p+1)-form, the inner product lowers the rank of a p-form to a (p-1)-form. (There are other useful operators that lower the rank of the exterior differential p-form, and involve integration.) By composition of the exterior differential and the inner product operators, the Lie differential operator (with symbol L(V) = i(V)d + di(V)) can be constructed, such that when the Lie differential operates on an exterior p-form, the resultant object is another p-form. For a 1-form of Action, A, the process reads, L(V) A = i(V)dA + d(i(V)A) = Q.

(6.17)

The resultant is not only closed relative to the Grassmann algebra, it also remains within the same Grassmann vector subspace. The Lie differential does not depend upon a metric nor upon a connection. When the Lie differential acting on a p-form vanishes, the p-form is said to be an invariant of the process, V. When the Lie differential of a p-form does not vanish, the topological features of the resultant p-form permit the processes, V, that produce such a result, to be put into equivalence classes, depending on the Pfaff dimension of the resultant form. For example, if in the formula given above for a 1-form, A, yields a result Q such that dQ = 0, then the process V belongs to the class of process known as Hamiltonian processes in mechanics, and to the Helmholtz class of processes that conserve vorticity in Hydrodynamics. Of particular interest to this monograph are processes where Q is of Pfaff dimension greater than 2. The Pfaff sequence constructed from Q contains three or more elements. Such processes, V, that produce Heat 1-forms, Q, which are of Pfaff topological dimension 3, are thermodynamically irreversible.

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The Lie differential will be used extensively in physical applications of Cartan’s theory, especially to the study of processes that involve topological evolution. The perhaps more familiar covariant derivative, highly constrained by connection or metric assumptions, is a special case of the Lie differential. The use of the covariant derivative leads to useful, but limited, physical theories for which the description of topological evolution is awkward, if not impossible. It has been shown in Chapter 3, the evolutionary processes that are defined by the covariant derivative are adiabatic processes. 6.1.3 The Exterior Product and Set Intersection Cartan’s theory of exterior differential systems has its foundations in the Grassmann algebra, where the two combinatorial processes are defined to produce algebraic and differential closure. The algebra is based upon the concepts of vector space addition, and an algebraic closure multiplication process now called the exterior product [73]. The Cartan calculus is defined in terms of the another closure operator now called the exterior differential § . In that which follows the operators of the exterior product and exterior differential will be applied to objects defined as exterior differential p-forms. An exterior differential p-form is a function of independent variables, xυ , and their differentials, dxµ . An exterior differential 1-form, A, is given by the expression, A = Aµ (xυ )dxµ .

(6.18)

The Cartan operations of exterior product (symbol ˆ) and exterior differential (symbol d), when operating on 1-forms, A and B, obey the rules, AˆA = 0, AˆB = −BˆA.

(6.19) (6.20)

The non-zero product, AˆB, defines an exterior differential 2-form; the product of three 1-forms defines a 3-form; etc.. For more detail consult [73] [7] [12] [139], [140]. In simple cases, a 1-form can be constructed from the differential of an ordinary function. In such cases, the coefficients of the 1-form are proportional to the gradient of the function, (6.21) A = Aµ dxµ = ∇φ · dr = (∂ φ/∂xµ )dxµ . In surface theory, the gradient is classically interpreted as vector direction field orthogonal to the implicit surface, φ(xµ ) = 0. Consider the simple case where the 1-forms A and B each have coefficients which form the components of (different) gradient fields, A = Aµ dxµ = ∇φ · dr §

B = Bµ dxµ = ∇ψ · dr.

(6.22)

Cartan originally defined the calculus operation d as the exterior derivative. Then in the later years he defined calculus operation d as the exterior differential.

Introduction

309

Do the two implicit (curved) surfaces intersect? The answer is yes if the two surfaces have points in common. The classic analysis in 3D says there is a curve of points in common defined by a non-zero value of the Gibbs cross product of the two gradient fields, Intersection of two implicit surfaces: J = ∇φ × ∇ψ 6= 0. (6.23) Note that (in three dimensions ) the exterior product of the two 1-forms has coefficients exactly equal to the Gibbs cross product, AˆB = Jz dxˆdy + Jx dyˆdz + Jy dzˆdx.

(6.24)

This result pictorially cements the notion that the exterior product (acting on 1forms) is an operator related to the concept of intersection. If the two surfaces do not intersect, the exterior product vanishes, and then the direction fields of the gradients of φ and χ are proportional to one another. The two functions, φ and χ, are not functionally independent if the exterior product vanishes. These concepts extend to 1-forms which are not representable by gradient fields, and to p-forms of higher rank. If the exterior product of two p-forms is not zero, then the p-forms have non-zero intersections. The coefficient functions are functionally independent. An exterior differential 1-form, A, is deterministic, as a predictive (or retrodictive ) invariant, with respect to all tensor diffeomorphisms. The coefficient functions, Aµ (xυ ), are presumed to behave as a covariant vector, and the differentials, dxµ , behave as a contravariant vector, with respect to tensor diffeomorphisms. The exterior differential 1-form is not a 1-tensor; instead the construction creates a diffeomorphic scalar. All p-forms discussed in this chapter are exterior differential p-form scalars. (The other type of exterior differential p-form, called exterior p-form densities, will be discussed later in Chapter 8.) However, the exterior differential 1-form, and hence all p-forms, are also well behaved with respect to a larger class of transformations, which contain the tensor diffeomorphisms as special cases. The exterior differential 1-form is deterministic in a retrodictive sense (but not in a predictive sense) with respect to C1 mappings that do not have a local or a global inverse. These C1 mappings do not preserve all topological features, where diffeomorphisms of tensor theory, are special cases of homeomorphisms, which do preserve all topological properties. These extraordinary features demonstrate that Cartan’s theory is not just another formalism of fancy, and goes well beyond the theory of tensor analysis. In fact, these features of exterior differential forms can be exploited to develop something that has slipped through the net of tensor analysis, a non-statistical theory of thermodynamic irreversibility. 6.1.4 The Exterior Differential and Limit Points The second closure operator found in Cartan’s theory of exterior differential systems is the exterior differential. The exterior differential, like the exterior product, also has topological connotations when applied to differential forms, but the results are

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sometimes surprising and unfamiliar. Where the exterior product is related to the topological concept of set intersection, the exterior differential is related to the topological idea of limit points. It will be demonstrated that: Conjecture With respect to the Cartan topology, the exterior differential is a limit point generator. The exterior differential is a differential operator which takes the p-forms into (p+1)-forms. Hence, like the exterior product, the exterior differential generates a vector in a different vector subspace of the exterior algebra, d(ωp ) ⇒ ωp+1 .

(6.25)

The exterior differential of a function (0-form) is equivalent to the total differential of a scalar function, and yields a 1-form with coefficients proportional to the gradient field. The exterior differential of a 1-form is defined as, dω1 = = = =

d(Ab dy b ) = (dAb )ˆdy b + Ab d(dy b ), (∂Ab /∂y e dy e )ˆdy b + 0, (∂Ab /∂y e − ∂Ae /∂y b ) dy e ˆ dy b , F[eb] dy [eb] = F[H] dy [H] .

(6.26) (6.27) (6.28) (6.29)

It has been assumed that the coefficients of the forms are C2, such that dd(ω p ) = 0. The collective index notation [H] = [eb] permits the formula defining exterior differentiation defined for a 1-form can be generalized to p-forms. Assume that H is a collective index, H = {i, j, k...}. (6.30) Then the p-form ωp can be written as,

ωp = Ai,j,k.. dxi ˆdxj ˆdxk ... = AH dxH ,

(6.31)

and its exterior differential becomes, dωp = d(AH dy H ) = (dAH )ˆdy H = ({∂AH /∂y e } dy e )ˆdy H = ({∂Ai,j,k .../∂y e } dy e )ˆdxi ˆdxj ˆdxk .

(6.32) (6.33) (6.34)

Other properties of the exterior differential will be exemplified by the rules for distributing the operator over a product of 1-forms, A and B, d(AˆB) = dAˆB − AˆdB.

(6.35)

It can be shown that the operator KCl = I ∪ d, where I is the identity and d is the exterior differential, acting on a system of differential forms satisfies the

The Cartan "Point Set" Topology

311

"Kuratowski closure" axioms [141], and therefore can be used to define a topology. Starting from a single 1-form, A, on a four-dimensional space, it is possible to generate the Pfaff Sequence, P faff Sequence

: {A, dA, AˆdA, dAˆdA} = {A, F, H, K}.

(6.36) (6.37)

The subsets of the Cartan topological space consist of all possible unions of the subsets that make up the Pfaff sequence. The Cartan topology will be constructed from a topological basis which consists of the odd elements of the Pfaff sequence, and their closures, the Cartan basis : {A, KCl (A), AˆdA, KCl (AˆdA)}. (6.38) When applied to the Pfaff sequence generated by a single 1-form of Action, A, on a space of four dimensions, the base elements correspond to the set, 1 − form basis : {A, A ∪ F, H, H ∪ K},

(6.39)

which should be compared to the point set example above, {a, b, c, d}. When it is realized that the exterior differential acts as a limit point generator, it becomes apparent why Cartan referred to the union of Σ and dΣ as the closure of Σ, Closure = (KCl ) ◦ Σ = (I ∪ d) ◦ Σ = Σ + dΣ = subset + limit points.

(6.40)

In the next section, the topological features of the Cartan topology, based on the Cartan topological base, will be worked out in detail. It will turn out that the Cartan topology can be put into correspondence with the T4 topology of 4 points displayed in a previous section. It will be evident, indeed, that the exterior differential is a limit point generator for any subset relative to the Cartan topology. This is a remarkable result, for as will be demonstrated below, all C2 vector fields acting through the concept of the Lie differential on a set of differential forms, with C2 coefficients¶ , generate continuous transformations with respect to the Cartan topology. Moreover, the Cartan topology is disconnected if AˆdA 6= 0. As the conditions for unique integrability of the 1-form A are given by the Frobenius theorem, which requires AˆdA = 0, it should be expected that one of the features of the disconnected Cartan topology is that if solutions exist, they are not unique. 6.2

The Cartan "Point Set" Topology

Cartan built his theory around an exterior differential system, Σ, which consists of a collection of 0-forms, 1-forms, 2-forms, etc. [43]. He defined the closure of this collection as the union of the original collection with those forms which are obtained ¶

Adiabatic processes may be C1.

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Cartan’s Topological Structure

by forming the exterior differentials of every p-form in the initial collection. In general, the collection of exterior differentials will be denoted by dΣ, and the closure of Σ by the symbol, KCl (Σ), where, Kuratowski Closure operator:

KCl (Σ) = Σ ∪ dΣ.

(6.41)

For notational simplicity in this monograph the systems of p-forms will be assumed to consist of the single 1-form, A. Then the exterior differential of A is the 2-form F = dA, and the closure of A is the union of A and F : KCl (A) = A ∪ F . The other logical operation is the concept of intersection, so that from the exterior differential it is possible to construct the set AˆF defined collectively as H : H = AˆF. The exterior differential of H produces the set defined as K = dH, and the closure of H is the union of H and K : KCl (H) = H ∪ K. This ladder process of constructing exterior differentials, and exterior products, may be continued until a null set is produced, or the largest p-form so constructed is equal to the dimension of the space under consideration. The set so generated is defined as a Pfaff sequence. The largest rank of the sequence determines the Pfaff dimension of the domain (or class of the form, [238]), which is a topological invariant. The idea is that the 1-form A (in general the exterior differential system, Σ) generates on space-time four equivalence classes of points that act as domains of support for the elements of the Pfaff sequence, A, F, H, K. The union of all such points will be denoted by X = A ∪ F ∪ H ∪ K. The fundamental equivalence classes are given specific names [202]: Topological ACTION A Topological VORTICITY dA Topological TORSION AˆdA Topological PARITY dAˆdA

: = : = : = : =

A Aµ dxµ F = dA Fµν dxµ ˆdxν H = AˆdA Hµνσ dxµ ˆdxν ˆdxσ K = dAˆdA Kµνστ dxµ ˆdxν ˆdxσ ˆdxτ .

(6.42) (6.43) (6.44) (6.45) (6.46) (6.47) (6.48) (6.49)

The Cartan topology is constructed from a basis of open sets, which are defined as follows. First consider the domain of support of A. Define this "point" by the symbol A. A is the first open set of the Cartan topology. Next construct the exterior differential, F = dA, and determine its domain of support. Next, form the closure of A by constructing the union of these two domains of support, KCl (A) = A ∪ F . A ∪ F forms the second open set of the Cartan topology. Next construct the intersection H = AˆF , and determine its domain of support. Define this "point" by the symbol H, which forms the third open set of the

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313

Cartan topology. Now follow the procedure established in the preceding paragraph. Construct the closure of H as the union of the domains of support of H and K = dH. The construction forms the fourth open set of the Cartan topology. In four dimensions, the process stops, but for N > 4, the process may be continued. Now consider the basis collection of open sets that consists of the subsets:

B = {A, KCl (A), H, KCl (H)} = {A, A ∪ F, H, H ∪ K}.

(6.50)

The collection of all possible unions of these base elements, and the null set, ∅, generate the Cartan topology of open sets:

T (open) = {X, ∅, A, H, A ∪ F, H ∪ K, A ∪ H, A ∪ H ∪ K, A ∪ F ∪ H}.

(6.51)

These nine subsets form the open sets of the Cartan topology constructed from the domains of support of the Pfaff sequence constructed from a single 1-form, A, in four dimensions. The compliments of the open sets are the closed sets of the Cartan topology:

T (closed) = {∅, X, F ∪ H ∪ K, A ∪ F ∪ K, A ∪ F, H ∪ K, F ∪ K, F, K}.

(6.52)

From the set of 4 "points" {A, F, H, K} that make up the Pfaff sequence it is possible to construct 16 subset collections by the process of union. It is possible to compute the limit points for every subset relative to the Cartan topology. The classical definition of a limit point is that a point p is a limit point of the subset Y relative to the topology T if and only if for every open set which contains p there exists another point of Y other than p [141]. The results of this and other standard definitions are presented in Table 2, and are to be compared to Table 1.

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Table 2. The Cartan T4 Topology A 1-form in 4D: A = Ak (x)dxk X = {A, F = dA, H = AˆF, K = F ˆF } Basis subsets {A, KCl (A), H, KCl (H)} = {A, A ∪ F, H, H ∪ K} T (open) = {X, ∅, A, H, A ∪ F, H ∪ K, A ∪ H, A ∪ H ∪ K, A ∪ F ∪ H} T (closed) = { ∅, X, F ∪ H ∪ K, A ∪ F ∪ K, H ∪ K, A ∪ F, F ∪ K, F, K} Subset Limit Pts Interior Boundary Closure σ dσ ∂σ σ ∪ dσ . ∅ ∅ ∅ ∅ ∅ A F A F A∪F F ∅ ∅ F F H K H K H ∪K K ∅ ∅ K K A∪F ∅ F A∪F A∪F X F ∪K A∪H A∪H F, K A∪F ∪K F ∪K A A∪K F F ∪H ∪K F ∪K H F ∪H K F ∪K F ∪K ∅ F ∪K ∅ H ∪K ∅ H ∪K H ∪K K A∪F ∪H F, K A∪F ∪K K X F ∪H ∪K K H ∪K F F ∪H ∪K A∪H ∪K F, K A∪H ∪K F X A∪F ∪K F A∪F K A∪F ∪K X F, K X ∅ X

(6.53)

By examining the set of limit points so constructed for every subset of the Cartan system, and presuming that the functions that make up the forms are C2 differentiable (such that the Poincare lemma is true, ddω = 0, any p − f orm, ω), it is easy to show that for all subsets of the Cartan topology every limit set is composed of the exterior differential of the subset thereby proving the conjecture that the exterior differential is a limit point operator relative to the Cartan topology. Theorem 52 With respect to the Cartan topology, the exterior differential is a limit point generator. For example, the open subset, A ∪ H, has the limit points that consist of F and K. The limit set consists of F ∪ K which can be derived directly by taking the exterior differentials of the elements that make up A ∪ H; that is, (F ∪ K) = d(A ∪ H) = (dA ∪ dH). Note that this open set, A ∪ H, does not contain its limit points. Similarly for the closed set, A ∪ F , the limit points are given by F which may be deduced by direct application of the exterior differential to (A ∪ F ) : (F ) = d(A ∪ F ) = (dA ∪ dF ) = (F ∪ ∅) = (F ).

Topological Torsion, Connected vs. Non-Connected Cartan topologies

6.3

315

Topological Torsion, Connected vs. Non-Connected Cartan topologies

Topological torsion of a 1-form, A, is defined as the exterior product of the 1-form and its exterior differential, H = AˆdA. Topological torsion is different from, but can be related to, the Frenet torsion of a space curve and the affine torsion of a connection. If non-zero, Topological torsion has important topological properties. The Cartan topology as given in Table 2 is composed of the union of two sub-sets which are both open and closed: (X = KCl (A) ∪ KCl (H) = {A ∪ F } ∪ {H ∪ K}),

(6.54)

a result that implies that the Cartan topology is not necessarily a connected topologyk [141]. An exception exists if the topological torsion, H = AˆdA, and hence its closure, vanishes, for then the Cartan topology is connected. This extraordinary result has broad physical consequences. The connected Cartan topology based on a vanishing topological torsion is at the basis of most physical theories of equilibrium. In mathematics, the connected Cartan topology corresponds to the Frobenius integrability condition for Pfaffian forms. In thermodynamics, the connected Cartan topology is associated with the Caratheodory concept of inaccessible thermodynamic states [96], and the existence of an equilibrium thermodynamic surface. If the non-exact 1-form, Q, of heat generates a Cartan topology of null topological torsion relative to the 1form Q, QˆdQ = ∅, then the Cartan topology built on Q is connected. Such systems are "isolated" in a topological sense, and the Heat 1-form, Q, has a representation in terms of two and only two functions, conventionally written as, Q = T dS. Note again that a fundamental physical concept, in this case the idea of equilibrium, is a topological concept independent from geometrical properties of size and shape. Processes that generate the 1-form Q such that QˆdQ = ∅ are thermodynamically reversible. If QˆdQ 6= ∅, the process that generates Q is thermodynamically irreversible. When the Cartan topology is connected, it might be said that all forces are extendible over the whole of the set, and that these forces are of "long range". Conversely when the Cartan topology is disconnected, the "forces" cannot be extended indefinitely over the whole domain of independent variables, but perhaps only over a single component. The components are not arc connected. In this sense, such forces are said to be of "short range", as they are confined to a specific component. Note that this notion of short or long range forces does not depend upon geometrical size or scale. The physical idea of short or long range forces is a topological idea of connectivity, and not a geometrical concept of "how far". In an earlier article, these ideas were formulated intuitively in order to give an explanation of the "four forces" of physics. The earlier work was based upon experience with differential geometry [188]. The features of the Pfaff sequence were k

It should be noted that disconnected is not the same as separated, for the disconnected component boundaries could be in contact.

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used to establish equivalence classes for 1-forms constructed from known example metric field solutions, gµυ , to the Einstein field equations. The original ideas, based upon experience with systems in differential geometry, can now be given credence based upon differential topology. The construction of a 1-form, A = gµ4 dxµ , whose coefficients are the space-time components of a metric tensor, will divide the topology into equivalence classes depending upon the number of non-zero elements of its Pfaff sequence. This number has been defined above as the Pfaff topological dimension. Long range parity preserving forces due to gravity (Pfaff dimension 1) and electromagnetism (Pfaff dimension 2) are to be associated with a Cartan Topology that is connected (H = AˆF = AˆdA = 0). Both the strong force (Pfaff dimension 3) and the weak force (Pfaff dimension 4) are "short" range (H 6= 0) and are to be associated with a disconnected Cartan topology. The strong force is parity preserving (K = 0) and the weak force is not (K 6= 0). The fact that the Cartan topology is not necessarily connected is the topological (not metrical) basis that may be used to distinguish between short and long range forces. The methods also have applicability to the theory of black holes. In much of our physical experience with nature, it appears that the disconnected domains of Pfaff dimension 3 or more are often isolated as nuclei, while the surrounding connected domains of Pfaff dimension 2 or less appears as fields of charged or non-charged molecules and atoms. However, part of the thrust of this monograph is to demonstrate that such disconnected topological phenomena are not confined to microscopic systems, but also appear in a such mundane phenomena as the flow of a turbulent fluid. Physical examples of the existence of topological torsion (and hence a non-connected Cartan topology) are given by the experimental appearance of what appear to be coherent structures in a turbulent fluid flow. To prove that a turbulent flow must be a consequence of a Cartan topology that is not connected, consider the following argument. First consider a fluid at rest and from a global set of unique, synchronous, initial conditions generate a vector field of flow. Such flows must satisfy the Frobenius complete integrability theorem, which requires that H = AˆdA = 0. The Cartan topology for such systems is connected, and the Pfaff dimension of the domain is 2 or less. Such domains do not support topological torsion (the 3-form H = 0). Such globally laminar flows are to be distinguished from flows that reside on surfaces, but do not admit a unique set of connected sychronizable initial conditions. Next consider turbulent flows which, as the antithesis of laminar flows, can not be integrable in the sense of Frobenius; such turbulent domains support topological torsion (H = AˆdA 6= 0), and therefore a disconnected Cartan topology [190]. The components of the disconnected Cartan topology can be defined as the (topologically) coherent structures induced by the turbulent flow. Note that a domain can support a homogeneous topology of one component and then undergo continuous topological evolution to a domain with some interior holes. The domain is simply connected in the initial state, and multiply connected

Applications of Cartan’s Topological Structure

317

in the final state, but still connected. However, consider the dual point of view where the originally connected domain consists of a homogeneous space that becomes separated into multiple components. The evolution to a topological space of multiple components is not continuous. It follows that the case of a transition from an initial laminar state (H = AˆdA = 0) to the turbulent state (H = AˆdA 6= 0) is a transition from a connected topology to a disconnected topology. therefore the transition to turbulence is NOT continuous. However, note that the decay of turbulence can be described by a continuous transformation from a disconnected topology to a connected topology. Condensation is continuous, gasification is not. It is demonstrated below that relative to the Cartan topology all C2 differentiable, V, acting on C2 p-forms by means of the Lie differential are continuous. The conclusion is reached that the transition to turbulence must involve transformations that are not C2, hence can occur only in the presence of shocks or tangential discontinuities. 6.4

Applications of Cartan’s Topological Structure

6.4.1 Continuous Processes A topological structure is defined to be enough information to decide whether a transformation is continuous or not [79]. The classical definition of continuity depends upon the idea that every open set in the range must have an inverse image in the domain. This means that topologies must be defined on both the initial and final state, and that somehow an inverse image must be defined. Note that the open sets of the final state may be different from the open sets of the initial state, because the topologies of the two states can be different. There is another definition of continuity that is more useful for it depends only on the transformation, and not its inverse, explicitly. A transformation is continuous if and only if the image of the closure of every subset is included in the closure of the image. This means that the concept of closure and the concept of transformation must commute for continuous processes. Suppose the forward image of a 1-form A is Q, and the forward image of the set F = dA is Z. Then if the closure, KCl (A) = A∪F is included in the closure of KCl (Q) = Q ∪ dQ, for all subsets, the transformation is defined to be continuous. The idea of continuity becomes equivalent to the concept that the forward image Z of the limit points, dA, is an element of the closure of Q [97]: Definition A function that produces an image f [A] = Q is continuous iff for every subset A of the Cartan topology, Z = f [dA] ⊂ KCl (Q) = (Q ∪ dQ). The Cartan theory of exterior differential systems can now be interpreted as a topological structure, for every subset of the topology can be tested to see if the process of closure commutes with the process of transformation. For the Cartan

318

Cartan’s Topological Structure

topology, this emphasis on limit points rather than on open sets is a more convenient method for determining continuity. A simple evolutionary process, X ⇒ Y , is defined by a map Φ. The map, Φ, may be viewed as a propagator that takes the initial state, X, into the final state, Y . For more general physical situations the evolutionary processes are generated by vector fields of flow, V. The trajectories defined by the vector fields may be viewed as propagators that carry domains into ranges in the manner of a convective fluid flow. The evolutionary propagator of interest to this monograph is the Lie differential with respect to a vector field, V, acting on differential forms, Σ [21]. The Lie differential has a number of interesting and useful properties. 1. The Lie differential does not depend upon a metric or a connection. 2. The Lie differential has a simple action on differential forms producing a resultant form that is decomposed into a transversal and an exact part, L(V) ω = i(V )dω + di(V )ω.

(6.55)

This formula is known as "Cartan’s magic formula". For those vector fields V which are "associated" with the form ω, such that i(V )ω = 0, the Lie differential becomes equivalent to the covariant differential of tensor analysis. Otherwise the two differential concepts are distinct. 3. The Lie differential may be used to describe deformations and topological evolution. Note that the action of the Lie differential on a 0-form (scalar function) is the same as the directional or convective differential of ordinary calculus, L(V) Φ = i(V )dΦ + di(V )Φ = i(V )dΦ + 0 = V · gradΦ. It may be demonstrated that the action of the Lie differential on a 1-form will generate equations of motion of the hydrodynamic type. In fact Arnold calls the Lie differential the "convective" or "Fisherman’s" differential. 4. With respect to vector fields and forms constructed over C2 functions, the Lie differential commutes with the closure operator. Hence, the Lie differential (restricted to C2 functions) generates transformations on differential forms which are continuous with respect to the Cartan topology. The last statement requires a formal proof: Proof First construct the closure, {Σ ∪ dΣ} . Next propagate Σ and dΣ by means of the Lie differential to produce the decremental forms, say Q and Z, L(V) Σ = Q and L(V) dΣ = Z. (6.56) Now compute the contributions to the closure of the final state as given by {Q ∪ dQ}. If Z = dQ, then the closure of the initial state is propagated

Applications of Cartan’s Topological Structure

319

into the closure of the final state, and the evolutionary process defined by V is continuous. However, dQ = dL(V) Σ = di(V )dΣ + dd(i(V )Σ) = di(V )dΣ ,

(6.57)

as dd(i(V )Σ) = 0 for C2 functions; but, Z = L(V) dΣ = d(i(V )dΣ) + i(V )ddΣ = di(V )dΣ,

(6.58)

as i(V )ddΣ = 0 for C2 p-forms. It follows that Z = dQ, and therefore V generates a continuous evolutionary process relative to the Cartan topology. QED It is to be noticed that this concept of a topological structure is developed in terms of the action of the Lie differential acting on a 1-form. The method does not depend upon metric or connection. Certain special cases arise for those subsets of vector fields that satisfy the equations, d(i(V)Σ) = 0. In these cases, only the functions that make up the p-form, Σ, need be C2 differentiable, and the vector field need only be C1. Such processes will be of interest to symplectic processes, with Bernoulli-Casimir invariants, and to the analysis of tangential discontinuities. By suitable choice of the 1-form of Action it is possible to show that the action of the Lie differential on the 1-form of Action can generate the Navier Stokes partial differential equations. See section 3.4.2 or [186], [209]. The analysis above indicates that C2 differentiable solutions to the Navier-Stokes equations can not be used to describe the transition to turbulence. The C2 solutions can, however, describe the irreversible decay of turbulence to the globally laminar state. 6.4.2 Uniform Continuity, Frozen - in Fields, the Master Equation A starting point for many discussions of the magnetic dynamo and allied problems in hydrodynamics starts with what has been called the "master equation" [115], Master equation Curl(V × B) = ∂B/∂t.

(6.59)

Using the Cartan methods it may be shown that this equation is equivalent to the constraint of "uniform" continuity relative to the Cartan topology. Moreover, it is easy to show these constraints generate symplectic processes which include Hamiltonian evolutionary systems, such as Euler flows, as well as a number of other evolutionary processes which are continuous, but not homeomorphic. In addition a criteria can be formulated to develop an extension of the "helicity" conservation law to a more general setting. The proof of these results produces a nice exercise in use of the Cartan theory. Consider a 1-form A that satisfies the exterior differential system, F − dA = 0,

(6.60)

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Cartan’s Topological Structure

where A is a 1-form of Action, with twice differentiable coefficients (potentials proportional to momenta) which induce a 2-form, F, of electromagnetic intensities (E and B , related to forces). The exterior differential system is a topological constraint that in effect defines field intensities in terms of the potentials. Now search for all vector fields that leave the 2-form F an absolute invariant of the flow; that is, search for all vectors that satisfy Cartan’s magic formula, L(V) F = i(V )dF + di(V )F = 0 + di(V )F = 0.

(6.61)

For C2 functions, the term involving dF vanishes, leaving the expression, L(V) F = di(V )F, = d{(E + V × B) · dr − (E · V)dt}, = {curl(E + V × B)}z dyˆdz... + {∂(E + V × B)/∂t + grad(E · V)} · drˆdt, = 0.

(6.62) (6.63) (6.64) (6.65) (6.66)

Setting the first three factors to zero yields, curl(E + V × B) = 0.

(6.67)

From the Maxwell Faraday equations for C2 functions, curlE = −∂B/∂t, and when this expression is substituted into the above equation, the "master" equation given above is the result. Now recall that dF generates the limit points of A, and if F = dA is a flow invariant, then all limit points are flow invariants relative to the Cartan topology. This result implies that the vector fields, V, that satisfy the constraints of the "master equation" are uniformly continuous evolutionary processes, as the limit points, F = dA, of the 1-form A are flow invariants, and the lines of vorticity are "frozen-in" the flow. Non-uniform continuity would imply that the limit points are not invariants of the process, but that the closure of the limit points of the target range includes the limit points of the initial domain. Such processes would correspond to a folding of the "lines" of vorticity, which preserve the limit points, but not their sequential order. A second criteria for limit point invariance is given by the equation, {∂(E + V × B)/∂t + grad(E · V)} = 0.

(6.68)

The formula indicates that limit point invariance can occur in the presence of inputoutput power, E · V 6= 0. The criteria for frozen-in fields is established as a constraint of uniform continuity on the admissible vector fields, Uniform Continuity:

di(V )dA = di(V )F = 0.

(6.69)

Applications of Cartan’s Topological Structure

321

The solution vector fields, V, subject to this constraint can be put into three global categories: 1. Extremal (Hamiltonian) i(V )F = 0. 2. Bernoulli-Casimir (Hamiltonian) i(V )F = dθ. 3. Helmholtz-Symplectic i(V )F = dθ + γ harmonic . The first category can exist only on domains of support of F which are of odd Pfaff dimension, but then the solution vector is unique to within a factor. In the other categories, the solution vector need not be unique. Vector fields that satisfy the equation for uniform continuity are said to be symplectic relative to the 1-form, A. Vector fields that belong to categories 1 and 2 have a Hamiltonian representation. Vector fields that belong to category 1, are said to be "extremal" relative to the 1-form, A. When the concepts are applied to the integrals of the 2-form F , then the criteria for invariance of the flux integral depends on the topology of the integration domain. If the integration area of the 2-form is a boundary or a cycle of a threedimensional domain, the flux integral over the closed boundary or cycle is always a flow invariant. If the integration area is bounded, then by Stokes theorem the flux integral depends only on the boundary conditions; F or i(V )F must vanish, on the boundary, or when integrated over the boundary.

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Cartan’s Topological Structure

Chapter 7 CONTINUOUS TOPOLOGICAL EVOLUTION 7.1

Introduction

In this chapter, the theory of Continuous Topological Evolution∗ will be developed in terms of Cartan’s theory of exterior differential forms. The motivation is based upon the concept that thermodynamic irreversibility implies topological change. The basis for such a postulate follows from the fact that if an evolutionary process is described by a map, Φ, between initial and final states, and if the map is not continuously reversible, then the observable topology of the final state is different from the observable topology of the initial state. Cartan’s methods can be used to extend these concepts to the dynamics of physical systems that admit description in terms of exterior differential forms. It is remarkable that the mathematical development leads to recognizable thermodynamic features which permit the determination of classes of processes which are reversible or irreversible. For example, all Hamiltonian processes are thermodynamically reversible. In fact, all Helmholtz processes are thermodynamically reversible. An essential feature of irreversible processes is that they involve the evolution of what has been defined as Topological Torsion. The appearance of Topological Torsion is a signal that the thermodynamic system is not in equilibrium. The observation of topological change, with the production and destruction of defects and holes, lines of self-intersection and other obstructions, will be the signature of topological irreversible evolution. Topological change can occur discontinuously as in a cutting process, or continuously, as in a pasting process. Such continuous but irreversible processes can be used to study the decay of turbulence, but not its creation. The production of disconnected components will be the signature of those discontinuous processes which are necessary to describe the creation and evolution of chaotic but perhaps reversible evolution, or turbulent, irreversible evolution. In this monograph, emphasis will be place upon those processes which are C2 continuous, but not reversible. The more difficult problem of C1 continuity in producing tangential discontinuities can be applied to the generation of wakes [213], [211]. Remarkably such C1 processes are locally adiabatic. Processes or maps that preserve topology are technically described as homeomorphisms [97]. Homeomorphisms are both continuous and reversible. Homeo∗

The fundamental ideas were initially formulated about 1981.

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Continuous Topological Evolution

morphic reversibility means that the inverse function, Φ−1 , must exist and must be continuous. Topological properties, such as orientability, compactness, connectivity, hole count, lines of self-intersection, pinch points, and Pfaff dimension are invariants of homeomorphisms, but geometrical properties such as size and shape are not necessarily invariants of homeomorphic deformations. In fact an elementary method of recognizing topological properties is to observe those properties that stay the same under continuous deformations that do not preserve size and shape. The theory of Continuous Topological Evolution is developed herein in terms of physical systems that undergo certain thermodynamic processes. The physical system is assumed to be modeled in terms of the topological features inherent in Cartan’s theory of exterior differential systems. The thermodynamic process will be defined in terms of a vector field, V , and its effect on the differential forms that make up the exterior differential system. The action of the process will be defined in terms of the Lie differential with respect to V acting on the differential forms that make up the exterior differential system, and which in turn approximate the physical system. The methods lead to concepts that are coordinate free and are well behaved in any reference system. A precise non-statistical definition of thermodynamic irreversibility will be stated, and a cohomological equivalent of the first law of thermodynamics will be derived and studied relative to the single constraint of continuous but irreversible topological evolution. Remarkably, many intuitive thermodynamic concepts can be stated precisely, without the use of statistics, in terms of the theory of continuous topological evolution based on the Cartan topology. Given a topology on the final state and a map from an initial state to the final state it is always possible to define a topology on the initial state such that the given transformation, or even a given set of transformations, is continuous. However, the topologies of the initial and final states need not be the same; hence the map need not be reversible. Recall that with respect to a discrete topology all maps from the initial to final state are continuous, while relative to the concrete topology, only the constant functions are continuous [21]. A first problem of a theory of topological evolution is to devise a rule for constructing a topology that is physically useful and yet is neither too coarse nor too fine. The rule chosen herein is based on the concept of topological continuity of an evolutionary transformation relative to the different topologies of the initial and final states. In this monograph the topological rules will be made by the specification of an exterior differential system that will model the physical system of interest. Many physical systems appear to be adequately modeled by a 1-form of Action. Physical exhibitions of continuous and discontinuous transformations can be achieved through the deformations of a soap film attached to a wire frame. For example, a soap film attached to a single closed, but double, loop of wire can be deformed from a non-orientable surface into an orientable surface continuously (the topological property of orientability is changed). That is, the soap film can be transformed continuously from a Moebius band into a cylindrical strip. As another example, consider

Continuity

325

an initial state where a soap film is attached to two slightly separated but concentric circular wire loops. The resulting surface is a minimal surface of a single component. As the separation distance of the concentric rings forming the boundary of the soap film is slowly increased, the minimal surface is stretched until a critical separation is reached. Then, without further displacement, the surface spontaneously continues to deform to form a "two sheeted" cone connected at a singular vertex point. The surface separates at the conical singularity, and the two separate sheets of the cone continue to collapse to form a minimal surface of two components. The final state consists of two flat films attached, one each, to each ring. The originally connected minimal surface undergoes a topological (phase) change to where it becomes two disconnected (still minimal) surfaces. An example of this topological transition in the surface of null helicity density has been described in conjunction with the parametric saddle node Hopf bifurcation of a Navier-Stokes flow [203]. In this monograph the fundamental set, X, will be the points {x, y, z, t, ...} that make up an N-dimensional space. Upon this fundamental set will be constructed arbitrary subsets, such as functions, tensor fields and differential forms. Many different topologies may be constructed on the fundamental set in terms of special classes of subsets that obey certain rules of logical closure. In fact the very existence of subsets can be used to define a course topology on X in terms of a topological base. The topological base consists of those subsets whose unions form a special collection of all possible subsets that is closed under logical union and intersection. This special collection of subsets will be defined as the open sets of a topology. The topological base can be used to define a topological structure. A space is said to have a topological structure if it is possible to determine if a transformation on the space is continuous [79]. 7.2

Continuity

The classic definition [141] of a continuous transformation between a set X with topology T 1 to a set Y with a topology T 2 states that the transformation is continuous if and only if the inverse image of open sets of T 2 are open sets of T 1. This definition can be made transparent by use of a simple point set example.

Figure 7.1 A map and its inverse image

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Continuous Topological Evolution

Consider two sets of 4 points, an initial state, {a, b, c, d} and a final state {x, y, z, t}. Define an open set topology on the initial state as T 1 = [X, ∅, a, ab, abc]. Also define an open set topology on the final state T 2 = [X, ∅, x, y, xy, yzt]. The transformation considered is exemplified by the Figure 7.1 above. The open set (y) has a pre-image (a) which is open. The open set (yzt) has a pre-image (abc) which is open. Hence the Map is continuous. (The open sets that involve x are not included as the map does involve x.) However, the Inverse Image mapping is not continuous for the open set (ab) has a preimage as (yz) but (yz) is not an open set of Y . The point set example demonstrates the idea of a continuous but not homeomorphic mapping. The objective herein is to examine such maps in terms of exterior differential systems. There exists another more useful method of defining continuity which does not depend explicitly on being able to define open sets and their inverse images. This second method of defining continuity is based on the concept of closure. The closure of a set can be defined in (at least) two ways: 1. The closure of a set is the union of the interior and the boundary of a set. 2. The closure of a set is the union of the set and its limit points. The first definition of closure is perhaps the most common, and is often exploited in geometric situations, where a metric has been defined and a boundary can be computed easily. The second definition of closure is independent from metric and is the method of choice in this monograph, both for defining continuity and establishing a topological structure. In terms of the concept of closure, a transformation is continuous if and only if for every subset, the image of the closure of the initial subset is included in the closure of the image of that subset [141]. Another way of stating this idea is 3. A map is continuous iff the limit points of every subset in the domain permute into the closure of the subsets in the range. If a method for constructing a closure operator (a Kuratowski closure operator KCl of a subset relative to a topology) can be defined, then a strong version of continuity would imply that the Kuratowski closure operator commutes with those transformations which are continuous. The test for continuity would be to construct the closure of an arbitrary subset on the initial state, and then to propagate the elements of the closure to the final state by means of a transformation. If this result is the same as the result obtained by first propagating the subset to the final state by means of the transformation, and then constructing its closure on the final state, then the map is continuous. Note that such a procedure has defined a topological structure which will be exploited in this monograph, for the subsets of interest will

The evolutionary process

327

be defined as a Cartan system of exterior differential forms, Σ, on X. The topological base defined by this class of sets is too course to be of interest. Hence the Cartan exterior differential will be used to generate additional sets of forms, dΣ, which when adjoined to the initial system of forms defines the Kuratowski closure of the Cartan system as the system of forms, KCl (Σ) = {Σ ∪ dΣ}. The Cartan exterior product may be used as a convenient intersection operator between sets of differential forms. Starting from the system, {Σ}, the Cartan topology is then determined by the construction of the Cartan-Pfaff sequence, which consists of all possible intersections that may be constructed from the subsets of the closure of the differential system, P f aff Sequence : {Σ, dΣ, ΣˆdΣ, dΣˆdΣ, ...}.

(7.1)

The subsets of the Cartan topological space consist of all possible unions of the subsets that make up the Pfaff sequence. The Cartan topology will be constructed from a topological basis which consists of the odd elements of the Pfaff sequence, and their closures, the Cartan topological base : {Σ, KCl (Σ), ΣˆdΣ, KCl (ΣˆdΣ), ...}.

(7.2)

With respect to a topological base constructed from a single 1-form of Action it has been shown in Chapter 6 that the Cartan exterior differential may be viewed as a closure or limit point operator. Given any subset of the Cartan topological space, the exterior differential of that subset generates its limit points, if any. This is a remarkable result, for as will be demonstrated below, all C2 vector fields acting through the concept of the Lie differential on a set of differential forms, with C2 coefficients, generate continuous transformations with respect to the Cartan topology. Moreover, the Cartan topology is disconnected if ΣˆdΣ 6= 0 is not zero. 7.3

The evolutionary process

7.3.1 Cartan’s Magic formula An arbitrary evolutionary process, X ⇒ Y , is defined by a map Φ. The map, Φ, may be viewed as a propagator that takes the initial state, X, into the final state, Y . In this monograph the evolutionary processes to be studied are asserted to be generated by vector fields, V. However, evolutionary vector fields need not be topologically constrained such that they are generators of a single parameter group. In other words, kinematics without fluctuations is not imposed a priori. The local trajectories defined by the vector fields may be viewed as propagators that carry domains into ranges in the manner of a convective fluid flow. The evolutionary propagator of interest to this monograph is the Lie differential with respect to a vector field ,V, acting on differential forms, Σ. The Lie differential has a number of interesting and useful properties.

328

Continuous Topological Evolution

1. The Lie differential does not depend upon a metric or a connection. 2. The Lie differential has a simple action on differential forms producing a resultant form that is decomposed into a transversal and an exact part, (7.3)

L(V) Σ = i(V)dΣ + di(V)Σ. Marsden [148] calls this Cartan’s Magic Formula (see below).

3. The Lie differential may be used to describe both deformations and topological evolution. For vector fields V that are singly parameterized, Cartan’s magic formula has a dynamic interpretation as a propagator down a set of flow trajectories. However, the formula can be used in an algebraic manner such that the vector field need not be singly parameterized, and can admit (topological) fluctuations about the Kinematic Perfection required by a singly parameterized definition of a limit set. 4. If the Lie differential of Σ is zero, then Σ is a (Bernoulli type) invariant along the flow trajectories generated by V. 5. With respect to vector fields and forms constructed over C2 functions, the Lie differential commutes with the Kuratowski closure operator. In fact dL(V) Σ = L(V) dΣ. Hence, the Lie differential generates transformations on differential forms which are continuous with respect to the Cartan topology. Note that the action of the Lie differential on a 0-form (scalar function) is the same as the directional derivative of ordinary calculus, L(V) ϕ = i(V)dϕ + 0 ⇒ V · gradϕ.

(7.4)

7.3.2 The Lie differential and C2 continuity The first four properties of the Lie differential listed above appear in the literature, but the extraordinary property that all C2 vector fields that propagate C2 differential forms in either the dynamic or algebraic manner are continuous, relative to the Cartan topology, requires proof, Proof Given Σ, first construct the closure, Σ ∪ dΣ. Next propagate Σ and dΣ by means of the Lie differential to produce the decremental or residue forms, say Q and Z, L(V) Σ = Q

and L(V) dΣ = Z.

(7.5)

Relative to the Cartan Topology, the first equation propagates the object (Σ), and the second equation propagates the limit points. Now compute the contributions to the closure of the final state as given by Q ∪ dQ. If

The evolutionary process

329

Z = dQ, then the closure of the initial state is propagated into the closure of the final state, and the evolutionary process defined by V is continuous. However, dQ = d(L(V) Σ) = di(V)dΣ + dd(i(V)Σ), (7.6) and Z = L(V) dΣ = (i(V)ddΣ) + d(i(V)dΣ).

(7.7)

The difference becomes, Z − dQ = (i(V)ddΣ) − dd(i(V)Σ).

(7.8)

The concept of continuity requires that Z − dQ ⇒ 0, forming an exterior differential system. For vector fields and differential forms with coefficient functions that are twice differentiable, the continuity condition is always satisfied relative to the Cartan topology (the Poincare lemma states that ddω = 0 where ω is any differential p-form with C2 coefficients). therefore subject to the constraint of C2 differentiability, every vector field, V, generates a continuous evolutionary process relative to the Cartan topology. The set {Σ, dΣ} forms a differential ideal (closure) which is permuted into the differential ideal {Q, dQ} by the action of the Lie differential with respect to V. QED. The Lie differential also can be used to make some sense out of certain classes of discontinuous evolutionary processes (which are not C2). For example, consider a vector field V = ρv where the support function, ρ, is not C2. Then, the action of the Lie differential produces the discontinuity or excess function, Z − dQ = −d(d(i(ρv)Σ)) = d{dρˆ(i(v)Σ) + ρ · d(i(v)Σ} = d(Θ).

(7.9)

If (i(ρv)Σ) ⇒ 0, then the second differential is zero even though the vector field ρv is not C2 continuous. This equation is of use in the study of tangential discontinuities, such as wakes in hydrodynamic systems, and shocks, in physical systems, which unlike the processes, are C2 smooth. 7.3.3 C1 Continuity Note that a special situation arises when (i(v)Σ) = 0, for then {dρˆ(i(v)Σ) + ρ · d(i(v)Σ} ⇒ 0 without the second differentiation. The process is continuous even when ρv is C1. Such special vector fields were defined above to be associated vector fields, and have the properties that the Lie differential has the same abstract form as the covariant derivative. In a thermodynamic system, with Σ ⇒ A, the 1-form of Action that encodes the properties of the physical system, the associated direction field describes a locally adiabatic process. Such adiabatic processes are continuous even though the density function ρ has discontinuities. Also, it can be shown that for even-dimensional symplectic manifolds, there is a unique vector direction field

330

Continuous Topological Evolution

that satisfies i(T4 )Σ = 0 and L(T4 ) Σ = Γ Σ. This direction field, T4 , will generate thermodynamically irreversible evolution, and is continuous if C0. The continuity of T4 is not uniformly continuous. 7.4

Topological Evolution

7.4.1 Evolutionary Invariants If the direction field generated by V acting on a Cartan system of forms satisfies the equations, L(V) Σ = 0 and L(V) dΣ = 0,

(7.10) (7.11)

then, with respect to such evolutionary processes, the forms of the closure are said to be absolute invariants. It follows that each element that makes up the Cartan topological base (see Chapter 6) is also invariant, such that the whole Cartan topology is invariant. As V is continuous, and the topology is preserved, those vector fields, V, that satisfy the equations above must be homeomorphisms, and are reversible. In other words, Q = 0 and dQ = 0 are sufficient conditions that V be reversible. However, for continuous transformations on the elements of the C2 Cartan topology the general equations of topological evolution become (for the form, Σ), L(V) Σ = Q,

(7.12)

L(V) dΣ = dQ,

(7.13)

L(V) (ΣˆdΣ) = QˆdΣ + ΣˆdQ ,

(7.14)

L(V) (dΣˆdΣ) = 2dQˆdΣ.

(7.15)

and (for the limit points, dΣ), from which it follows that and As these equations of continuous topological evolution imply that the elements of the topological base may not be constant, then specific tests must be made to determine what features of the topology are changing, if any. For if it can be determined that the topology is indeed modified by the evolutionary process, then the process generated by this class of vector fields, V, is continuous, but need not be reversible. When dQ 6= 0, the limit points of the Cartan topological structure are not invariants, and it would be natural to expect that the topology is not constant. However, even if Q is closed, such that dQ = 0, it may be true that Q contains harmonic components, such that deRham cohomological classes of Σ are not evolutionary invariants. Even though the topology of the initial state is not the same as the topology of the final state (for the "hole" count of the initial state is not the same as the hole count of the final state) it is not necessarily true that such continuous processes are thermodynamically irreversible.

Topological Evolution

331

7.4.2 Deformation Invariants Recall the topological requirements that define a continuous process. The objective is to find a method of construction of topological properties and then see if they change under the application of a continuous process. Definition A continuous process is defined as a map from an initial state of topology Tinitial into a final state of perhaps different topology Tf inal such that the limit points of the initial state are permuted among the limit points of the final state [141]. If the ordering of the limit points is invariant, the process is uniformly continuous. If the ordering (as in a folding of a boundary) or the number of the limit sets is changed the process is non-uniformly continuous. A simple description of a topological property (invariant of a homeomorphism) is an object that is a deformation invariant. Consider a rubber sheet with three holes. Stretch the rubber sheet. The holes may be deformed but the fact that there are three holes stays the same under small deformations. The concept of three holes is a topological property. If for some reason a fourth holed should appear under continuous deformation, then the topological property has changed. It is remarkable that topologically coherent objects can be constructed in terms of open and closed integrals which are deformation invariants. Definition A topological deformation invariant is defined as an integral of a p-form over a p-dimensional manifold, which may or may not be a cycle, zpd† , such that the Lie differential of the integral of the p-form ω with respect to a singly parameterized vector field, ρV k , vanishes, for any choice of deformation parameter, ρ. Integral Deformation Invariant : L(ρV k )

Z

ω=0

any ρ.

(7.16)

p

Those objects that are integral deformation invariants represent topological, not geometric properties. Absolute Integral Invariants There are two types of invariant integrals, Absolute and Relative integral invariants. If the p-form is exact, the Absolute integral invariant places conditions only on the boundary of the domain of integration. For example, consider physical systems that can be defined by a 1-form of Action, A, the derived 2-form F = dA, is exact. It follows from Stokes theorem that the two-dimensional integral of F is an absolute †

zpd is a shorthand notation for a cycle (z) of (p) (d)imensions.

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Continuous Topological Evolution

integral deformation invariant with respect to all continuous processes that can be defined by a singly parameterized vector field, subject to a boundary condition that the net flux of F across the one-dimensional boundary of M is zero, Z Z Z Z Z Z k L(ρV k ) F = i(ρV )dF + d(i(ρV k )F ) (7.17) M M M Z = 0+ i(ρV k )F. (7.18) boundary M

This concept is at the basis of the Helmholtz theorems of vorticity (or angular momentum per unit mass) conservation in hydrodynamics, and the conservation of flux in classical electromagnetism. Herein, this concept of deformation invariance of a topologically coherent structure will be written in the form of an exterior differential system, F − dA = 0, which is to be recognized as a topological constraint. From Stokes theorem, the (two dimensional) domain of finite support for F can not, in general, be compact without boundary, unless the Euler characteristic vanishes. There are two exceptional cases for two-dimensional domains, the torus and the Klein-Bottle, but these situations require the additional topological constraint that F ˆF ⇒ 0. The fields in these exceptional cases must reside on these exceptional compact surfaces, which form topological coherent structures. Note that an evolutionary process could start with F ˆF 6= 0, and possibly evolve to a state with F ˆF = 0. If such residue states are compact without boundary, then they must be either tori or Klein bottles. The same technique can be applied to non-exact but closed p-forms. Relative Integral Invariants If the integration of the exact 2-form, F , is over a closed two-dimensional closed chain, designated as a two-dimensional cycle, z2d (which may or may not be a twodimensional boundary), then the Integral is invariant for any deformation factor, ρ, Z Z Z Z Z Z k L(ρV k ) F = i(ρV )dF + d(i(ρV k )F ) = 0 + 0. (7.19) z2d

z2d

z2d

Close integrals of exact forms are always relative deformation integral invariants. However, the same technique can be applied to non-exact but closed p-forms. If the conditions of relative integral invariance are applied to an arbitrary 1-form of Action, then the relative integral invariance condition becomes, Z Z Z k A = i(ρV )dA + d(i(ρV k )A) (7.20) L(ρV k ) z1d z1d z1d Z = i(ρV k )F + 0. (7.21) z1d

Simple Systems

333

It follows that i(ρV k )dA must be zero on the cycle z1d for any deformation parameter ρ. Cartan has shown that this is the condition that implies the process ρV k satisfies the constraint i(ρV k )F = 0, and has a "Hamiltonian" representation. Cartan’s Hamiltonian example Consider the flow lines tangent to a given vector direction field, V(x, y, z, t...) that generates a dynamical system, dr − Vdτ = 0. By reparameterization, V ⇒ β(x, y, z, t...)V, the "speed" at which points move down the lines of flow can be changed, but the points that start on a particular flow line, remain upon the same flow line. Next consider a closed curve, Z1, intersecting the flow lines transversely for say τ = 0. The flow lines that intersect Z1 form a "tube of trajectories" As τ increases to some value, say τ = 1, the points of the closed curve appear to flow down the "tube of trajectories". The result of this convective evolution is to produce a new closed curve, Z2. Now choose another parameterization function β‘, which is equal to the original β at τ = 0. The points that make up the closed curve Z1 now flow down the same tube of trajectories, but at τ = 1 form a new closed curve def ormed Z2 that may be considered as a deformation of the closed curve Z2.

Figure 7.2 Cartan’s method for Hamiltonian systems 7.5 7.5.1

Simple Systems The Action 1-form and its Pfaff Sequence

Consider an arbitrary 1-form, A, on an N-dimensional variety of independent functions. The exterior differential of A produces a 2-form of closure points, F = dA, whose components are given by the expression, Fµν dxµ ˆdxν . The combined set {A, F } forms the closure of the set {A}. All possible intersections of the closure, {A, F, AˆF, F ˆF...}, form what is defined herein as the Pfaff sequence for the domain {x, y, z, t}. As defined in the previous chapter (for a four-dimensional variety) these

334

Continuous Topological Evolution

elements are defined as: Topological ACTION A Topological VORTICITY dA Topological TORSION AˆdA Topological PARITY dAˆdA

: = : = : = : =

A Aµ dxµ F = dA Fµν dxµ ˆdxν H = AˆdA Hµνσ dxµ ˆdxν ˆdxσ K = dAˆdA Kµνστ dxµ ˆdxν ˆdxσ ˆdxτ .

(7.22) (7.23) (7.24) (7.25) (7.26) (7.27) (7.28) (7.29)

The union of all elements of the Pfaff sequence and their closures forms the elements of the Cartan topological base, {A, A ∪ F, H, H ∪ K...}.

(7.30)

In order to take into account projective (and certain discontinuous) features, the vector fields of interest often will be scaled by a support function, ρ, such that J = ρV. The fundamental equations of continuous evolution become L(ρV) A L(ρV) F L(ρV) H L(ρV) K

= = = =

(7.31) (7.32) (7.33) (7.34)

Q, dQ, QˆF + AˆQ, 2(dQˆF ) = 2d(QˆF ).

Note that for the even-dimensional elements of the Pfaff sequence, (F and K), the action of the Lie differential is to produce an exact form, dQ, for the Lie differential of F, and 2d(QˆF ) for the Lie differential of K. As integrals of exact forms over closed cycles or boundaries of support vanish, then it is possible to formulate the first theorem. Theorem 53 All even-dimensional Pfaff classes of p-forms, dA = F, dAˆdA = K... are relative integral deformation invariants of continuous evolutionary processes relative to the Cartan topology. The closed integrals of F, K,... are invariants of a continuous process as each integrand is exact, and the integral of an exact form over a closed domain vanishes. Hence if the functions are twice differentiable, Z Z Z F = {i(ρV)dF + di(ρV)F } = dQ ⇒ 0. (7.35) L(ρV) z2

z2

z2

Simple Systems

335

The closed integrals of F, K, ... are invariants of any process generated by ρV for integration domains, z2, that are boundaries or cycles. This theorem is an extension of Poincare’s theorem for even-dimensional pforms which are absolute integral invariants (the integration domain is not necessarily closed) with respect to the restricted set of Hamiltonian processes. It is important to realize that the theorem expresses the existence of (relative) integral deformation invariants (topological properties) with respect to processes that may be thermodynamically reversible or irreversible. It should be noted that the domains of support of the even-dimensional Pfaff classes can not be compact without boundary. 7.5.2 The Action 1-form and Topological Fluctuations For purposes of expose, the Cartan system, Σ, will be limited to a single 1-form of Action, A, and perhaps a single pseudoscalar field, or N-form density, ρ. The 1-form of Action, A, can be written in several equivalent formats known as the Cartan-Hilbert action: A = Aµ dxµ = p · dx − H(x, v, p, t)dt = L(x, v, t)dt + p · (dx − vdt).

(7.36)

The last representation indicates that the Action may be viewed abstractly in terms a Lagrangian function, L(x, v, t), and the kinematic fluctuations in position, ∆x = (dx − vdt).

(7.37)

It is to be noted that the usual assumption for physical systems is to assume that there are zero kinematic fluctuations. In this sense, Kinematic Perfection prevails, ∆x = (dx − vdt) ⇒ 0.

(7.38)

It is rarely appreciated that Kinematic Perfection is equivalent to an exterior differential system which imposes topological restrictions on the variety. For this example, the fluctuations, ∆x, are not presumed to be zero. A simple count of the independent variables that are used to define the CartanHilbert 1-form of Action indicates that the "fluctuation" space is a variety of 3n+1⇒10 dimensions (t, x, v, p). (For simplicity, the "particle" index n has been chosen to be unity.) The coefficients, p, act as Lagrange multipliers for the fluctuations, ∆x. However, it can be determined that the maximum Pfaff dimension of the sequence {A, dA, AˆdA, dAˆdA...) is of dimension 2n+2⇒8 and not dimension 10. Hence the ten-dimensional space is redundant, and an eight-dimensional space is adequate to describe the physical system in terms of a 1-form of Action. The given 1-form of Action therefore generates a non-compact symplectic manifold of dimension 8. If the Lagrange multipliers p of the kinematic fluctuations (dx − vdt) are restricted to be the canonical momenta, as defined by the ubiquitous formula, p =∂L/∂v, the maximum Pfaff dimension is 7, forming a contact manifold historically defined as state space. If the Lagrange function L(x, v, t) is homogeneous of degree 1 in v, then

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Continuous Topological Evolution

the maximal Pfaff dimension is 6, forming a symplectic Finsler manifold of dimension 6, the phase space of classical mechanics. This manifold cannot be compact without boundary. If the contact manifold of dimension 7 is constrained by the equations of kinematic closure, d(∆x) = d(dx − vdt) ⇒ 0, (7.39) then the space of interest becomes the configuration space of four dimensions, a submanifold of the original symplectic structure of eight dimensions. The constraints of kinematic closure imply that the velocity field is expressible as functions of a single variable, t; v ⇒ v(t). Note that the more severe constraint of Kinematic Perfection, ∆x = (dx − vdt) ⇒ 0, implies that the maximal Pfaff dimension is 2, as in this case AˆdA =L(x, v, t)dtˆdL(x, v, t)ˆdt = 0. The Action defines a completely integrable two-dimensional submanifold that, in this circumstance, is not compact without boundary. These concepts will be exploited in other examples given below‡ . 7.6

Continuous Processes

7.6.1

Uniform and Non-Uniform Continuity

The continuous processes are naturally divided into two main categories. Those categories, for which dQ = 0, defined as uniformly continuous flows (in the sense that the limit points of A are invariant) and those categories, for which dQ 6= 0, which are not uniformly continuous flows. Therefore, relative to the Cartan Topology, Uniform Continuity : L(V) dA = dQ = 0,

(7.40)

defines a uniformly continuous closed process, while, non-Uniform Continuity : L(V) dA = dQ 6= 0,

(7.41)

defines a non-uniformly continuous process. The vector field on a four-dimensional domain that is in the direction of the current AˆF is a continuous, but a not uniformly, continuous process. Non-uniform continuity, dQ 6= 0, is necessary for thermodynamic irreversibility. Uniform continuity implies that the limit sets are invariant. Continuity only requires that the limit points permute amongst themselves. For example a fold into pleats which are then pasted together is a processes that rearranges the limit points and is not therefore uniformly continuous. Hence uniform continuity is a more constrained situation. When dQ = 0, it is possible to formulate immediately the following theorem (Poincare) for closed flows: Theorem 54 All even-dimensional Pfaff classes of p-forms, dA = F, dAˆdA = K, ... are invariants of evolutionary processes that satisfy L(V) (dA) = dQ = 0 relative to ‡

For more detail about topological fluctuations, see Chapter 12 in [276]

Continuous Processes

337

the Cartan topology. The forms F, K, ... form a set of absolute integral invariants with respect to uniformly continuous processes. The difference between Theorem 53 and Theorem 54 is that in Theorem 54, the integration chains need not be closed. Moreover, the process is uniformly continuous in Theorem 54, while the continuity is non-uniformly continuous in Theorem 53. The proof of the theorem follows immediately by application of the Leibniz rule, using the constraint, dQ = 0, as, L(V) (dAˆdAˆdA..ˆdA) = integer ×{L(V) (dA)}ˆdAˆdA..ˆdA) = 0.

(7.42)

The integrands of the selected integrals are local invariants and so are their convected integrals. The first application of theorem gives, L(V) (dA) = L(V) F = 0,

(7.43)

which is the equivalent of Helmholtz’ theorem [125]. The theorem often is interpreted as the local conservation of angular momentum per unit moment of inertia, or the conservation of Topological Vorticity. The second application of theorem gives, L(V) (dAˆdA) = L(V) (F ˆF ) = L(V) K = 0,

(7.44)

which leads to the local conservation of Topological Parity, with respect to uniformly continuous flows. In general, L(V) (dAˆdAˆ...dA) = 0, (7.45) which expresses the invariance of a 2N-dimensional area with respect to uniformly continuous flows. 7.6.2 Continuous Hydrodynamic Processes Consider the domain of four independent variables of space-time, {x, y, z, t}, and the 3-form of topological torsion, H = AˆdA = AˆF = i(T4 )dxˆdyˆdzˆdt.

(7.46)

The continuous evolution of this 3-form is determined relative to an arbitrary process, V4 = [V, 1], by the equation, L(βV4 ) H = L(βV4 ) (AˆdA) = i(βV4 )dH + di(βV4 )H = QˆF + AˆdQ.

(7.47)

For local invariance of the 3-form with respect to arbitrary parametrizations, the evolutionary vector βV4 must be collinear with the topological torsion vector (T4 )

338

Continuous Topological Evolution

such that the term i(βV4 )H ⇒ 0. This constraint implies that the 3-form H then must be of the format, H = AˆF ≈ ρ(x, y, z, t)(dx − V x dt)ˆ(dy − V y dt)ˆ(dz − V z dt) = ρ i(V4 )dxˆdyˆdzˆdt.

(7.48) (7.49)

The invariance of the 3-form H then requires that a function ρ(x, y, z, t) exist such that dH ⇒ 0. But this constraint becomes the equivalent of the famous hydrodynamic equation of continuity, dH = {div3 ρV + ∂ρ/∂t}dxˆdyˆdzˆdt ⇒ 0,

(7.50)

which is interpreted physically as the conservation of mass. The implication is that those vector fields, βV4 , that define a continuous hydrodynamic current, need not satisfy necessarily the formulas of topological kinematic constraint, dx − Vdt = 0, but instead must be collinear with the topological torsion vector, J4 = λ(x, y, z, t)T4 , if it exists. The important idea is that local deformable conservation of mass is to be associated with the conservation of the 3-form of Topological torsion as an absolute evolutionary invariant. These results are to be compared with the even-dimensional Poincare absolute integral invariants [290] for the more restrictive case of Hamiltonian (extremal) evolution of a Hamiltonian action, A = Aµ dxµ = p · dx − H(x, p, t)dt,

(7.51)

on a (2N+1)-dimensional state space. It is the result (7.43) which is interpreted in statistical mechanics as the invariant "area" of phase space with respect to extremal, or Hamiltonian, evolution. The fact of the matter is that uniform continuity alone produces a set of absolute integral invariants for any action, in Hamiltonian format or not. Hamiltonian extremal flows satisfy the equation, dQ = 0, and are therefore uniformly continuous, but they are not the only flows that satisfy this constraint. The invariance of "phase space area" is a consequence of uniform continuity alone, and does not require the additional constraints of constant homogeneity that limit the set of continuous flows to that subset of continuous vector fields which are extremal, and Hamiltonian. 7.6.3 DeRham categories of Continuous Vector Fields DeRham’s cohomology theory [182] may be used to classify p-forms, and such ideas may be applied to the 1-form W defined by W = i(ρV)F . Correspondingly, the vector fields that are used to construct the 1-forms W of virtual work permit processes to be put into the following categories, depending on whether the virtual work, W, is null, exact, closed, or not closed with respect to exterior differentiation. These categories are either uniformly continuous (reversible), or non-uniformly continuous (necessary for irreversibility, but not sufficient):

Continuous Processes

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

339

Uniform Continuity Hamiltonian_extremal Bernoulli_Eulerian Helmholtz_Symplectic Non-Uniform Continuity Navier_Stokes_T orsion

W = i(ρV)F 0 −dB −dB + γ

Q dU d(U − B) d(U − B) + γ

dW 0 0 0

dQ 0 0 0

arbitrary

arbitrary

6= 0 6= 0

⎤

⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(7.52) The Bernoulli-Casimir functions, B, must be first integrals as in general, i(V)W = −i(V)dB = 0.

(7.53)

For uniformly continuous processes the first law insures that the 1-form W is closed, dW = dQ = 0, but W need not be exact and may contain harmonic components. That is, the 1-form W is not necessarily representable over the variety x, y, z, t in terms of the gradient of a single scalar function. The classic example of a non-exact 1-form is given by the expression, γ = σ z (ydx − xdy)/(x2 + y 2 )

(7.54)

γ ⇒ Γ = σ z {ydx − xdy + (r × V)z dt}/(x2 + y 2 ),

(7.55)

R for which dγ = 0, but z1 γ = 2πσ z . The coefficient σ z is assumed to be a constant. Such 1-forms, γ, generate period integrals and the deRham cohomology classes. The number of independent forms of the type given by (7.54) determine the Betti numbers of a variety for which the singular point (at the origin in the example) has been excised. The Betti numbers can be interpreted as a method for counting the number of holes or handles in the variety. It is these contributions to the general differential form that carry topological information about the domain of support. The duals to these forms are also closed, leading to the definition, harmonic forms. From the first law the harmonic contributions to W are equal to the harmonic contributions to Q. If the harmonic contributions to Q are not zero, then the number of "holes and handles" in the Cartan topology of the final state is different from the number of holes and handles in the Cartan topology of the initial state, and the evolutionary process is continuous but not reversible. In order to make the 1-form transversal, use the Cartan trick of substituting i dx − V i dt for each dxi . The transversal harmonic form becomes,

which demonstrates the close relationship to transversal harmonic forms and angular momentum. The format may be extended to a spin vector of components, σ = [σ 1 , σ 2 , σ 3 ] = [σ x /(y 2 + z 2 ), σ y /(z 2 + x2 ), σ z /(x2 + y 2 )],

(7.56)

340

Continuous Topological Evolution

such that the harmonic form becomes, Γ = σ 1 (zdy − ydz) + σ 2 (xdz − zdx) + σ 3 (ydx − xdy) + (σ ◦ r × V)dt.

(7.57)

The last term is recognized as a "spin orbit" coupling term. The idea of harmonic contributions to a 1-form is closely related to the concept of a complex number or ordered pair representation; i.e., the form cannot be represented by a map to a space of one dimension. Other formats for harmonic 1-forms are given by the expressions, Γ = {φdχ − χdφ}/(aφp + bχp )2/p , (7.58) where φ and χ are arbitrary functions on the base space, and for the complex function, ψ, Γ = {ψdψ ∗ − ψ∗ dψ}/(ψ∗ ψ). (7.59) The last representation of a harmonic form is in the format of the "probability current" of quantum mechanics, and gives a clue as how to adapt the formalism of this monograph to quantum systems. Such a development is deferred to a later article. For uniformly continuous processes on space-time, the fundamental equations of evolution are given by the expressions for the 2-form dA = F and the 4-form d(AˆF ) = K = dF ˆdF . The even forms are invariant. The two fundamental equations of uniformly continuous evolution are, L(ρV) d(A) = L(ρV) F = dQ = 0, and L(ρV) d(AˆF ) = L(ρV) K = L(ρV) F ˆF = dQˆdA = 0.

(7.60) (7.61)

It should be noted that the first equation defines the uniformly continuous evolution of the limit points of the Action 1-form, while the second equation defines the uniformly continuous evolution of the limit points of the topological torsion 3-form. If dQ 6= 0 , then the ordering of the limit points of the 1-form of Action is perturbed, as the continuity is not uniform. If dQˆdA 6= 0, then the ordering of the limit points of the 3-form of topological torsion is perturbed, as the continuity is not uniform. It is conjectured that a change of ordering of the limit sets of the 1-form of Action is related to an "entropy" of translation (folding process), while the change of ordering of the limit sets of the 3-form of topological torsion is related to an "entropy" of rotation. It should be remarked that if the 1-form of Action, A, is completely integrable in the sense of Frobenius, then the 3-form AˆF is evanescent, and the evolutionary equation for H = AˆF has no applicability. Such evolutionary processes (H = 0) are the equivalent to laminar flows in fluid dynamics and completely integrable, nonchaotic, Hamiltonian processes. It is known that if a Lagrangian system is not chaotic, then the action, A, is reducible to two variables (or less), and the 3-form H is necessarily zero. However when there exists a sense of helicity in the evolutionary process,

Continuous Processes

341

or chaos is present, then the formula for H describes the appropriate topological evolution. The first expression (7.60) may be put into correspondence with the evolution of energy, while the second fundamental equation (7.61) may be described as the evolution of complexity, or perhaps better as the evolution of defects, links, knots, or in abstract terms, the evolution of an entropic concept. If the Heat 1-form Q is zero, then the evolutionary process is adiabatic, and topology is preserved. However, as the Cartan topology is not connected when H 6= 0, then continuous evolution of H can be accomplished only between connected subsets. The transition from a connected topology with H = 0 to a disconnected topology with H 6= 0 can only take place via a discontinuous transformation. The idea is that the continuous rate of change of H is definite (and arbitrarily taken to be positive). This feature is one of the key properties of entropy. Entropy can never change its sign. The creation of topological torsion, H, is a discontinuous process from a state of zero topological torsion, but once created, the growth (or decay) of H can be described by a continuous process (relative to the Cartan topology). These entropic features of the topological torsion 3-form will be useful in the description of the transition to turbulence. 7.6.4 The Hamiltonian Extremal Sub-Category It should be remarked, that Cartan has proved, on a domain of dimension 2n+1, that if i(ρV)F = W = 0, Q = dU, (7.62) for any reparameterization, ρ, then V generates a Hamiltonian system, and visa versa [43]. This remarkable result indicates that Hamiltonian flows are not only continuous, but preserve many topological properties. Note that if F is a 2-form of even dimensions and maximal rank, Extremal fields do not exist. There are no eigenvectors of zero eigenvalue for any even-dimensional square antisymmetric matrix such that the determinant of the matrix is not zero. All such eigendirection fields are complex spinors, with complex eigenvalues. The 1-form Q must be exact for Hamiltonian flows. Hence the observable holes and handles are topological invariants of Hamiltonian flows, as the ρ terms vanish. However, the fact that Q is exact for Hamiltonian flows does not completely establish a proof that Hamiltonian systems preserve all topological properties of the Cartan topology. In the calculus of variations, vector fields that satisfy i(ρV)dA = 0 [113] are defined as extremal vector fields; associated vector fields satisfy i(V )A = 0. Characteristic vector fields are both extremal and associative. They satisfy the equations, L(V) A = 0 and L(V) F = 0.

(7.63) (7.64)

In other words, continuous characteristics preserve the Cartan topology (Q = 0 and dQ = 0). Characteristic Hamiltonian vector fields generate waves in systems that

342

Continuous Topological Evolution

can be endowed with the additional structure of a metric. Note that extremal and characteristic vector fields do not exist for 1-forms of maximal rank in even Pfaff dimensions. 7.6.5

The Bernoulli-Euler Sub-Category

The Bernoulli-Euler category is not quite Hamiltonian. W is not zero, but must be a perfect differential, W = −dB. However, this perfect differential must be a first integral in order to satisfy the transversality condition, i(ρV)W = 0. The 1form Q is not necessarily so constrained. The abstract flows of this category are to be compared with the equations of motion of a compressible Eulerian fluid in which there may be stratification. If the pressure, P , is a function of the density, ρ, alone, then the Eulerian flow can be reduced to a Hamiltonian system [7]. If there exists some anisotropy due to stratification, then the Hamiltonian reduction is not perfect. Note that the first integral, B, acts as a Bernoulli constant along a given streamline, but the constant can vary from streamline to streamline because the function is transversal. 7.6.6

The Stokes Sub-Category

The Stokes category admits topological evolution in the sense that the harmonic contributions to W are not null, and therefore the "hole and handle" count of the Cartan topology is changing in an evolutionary manner. Such closed flows are not reversible. Note that all closed flows preserve topological vorticity and topological parity, and so if the flow is without vorticity in the initial state, then the flow is without vorticity in the final state. The Pfaff topological dimension remains less than 2. However, if the initial state has vorticity, that vorticity will be preserved, but the Topological Torsion 3-form can change. In fact the Topological Torsion 3-form could be non-zero in the initial state, and zero in the final state, for the decay rate of topological torsion is proportional to QˆF . Both the 1-form of Action and its hole count, and the 3-form of Topological Torsion, and its twisted handle count, are not necessarily invariants of a Stokes flow. A method of distinguishing between "holes and twisted handles" is of some interest. Note that physically a handle can be constructed by deforming the rims of two holes in a surface into tubes and pasting the tubular ends together. If the rims are twisted by half integer or integer multiples of pi before the ends are glued together, then the handles have torsion. Note that a handle cannot be constructed in the plane, so it is an intrinsically three-dimensional thing. If the 3-form H vanishes, then there are no handles in the initial state, and as the Hamiltonian evolution produces no more new holes, there can be no more new handles in a Hamiltonian flow. However, existing handles may become twisted or knotted, because QˆF 6= 0, even for Hamiltonian flows. These facts correspond to the physical result that Hamiltonian systems are not dissipative and preserve energy, but that does not mean that entropy must be conserved. It should be noted that for all uniformly continuous flows, dW = 0. It follows that the transversality condition, i(ρV)W = 0, implies that the 1-form of virtual

Global Conservation Laws

343

work W is an absolute invariant of the flow, Uniformly continuous Flows : 7.7

L(ρV) W = 0.

(7.65)

Global Conservation Laws

7.7.1 First Variation Extremal (or Hamiltonian) flows and Eulerian flows induce a set of global conservation laws in the sense that the closed integrals of all odd-dimensional Pfaff classes of the fundamental forms are relative integral invariants of uniformly continuous evolution. The result follows from the fact that the evolutionary rates, Q and QˆF , with respect to such flows, are zero. Integrals of exact forms evaluated over closed cycles, whether the cycle (z1d or z3d) or not, vanish. Hence all closed integrals of oddR is a boundary R dimensional sets, z1d A and z3d H, are evolutionary invariants of Hamiltonian and Eulerian flows. For the closed flows of the Stokes category, the evolutionary rates of all odd Pfaff classes are closed, but not necessarily exact. That is, the equations, dQ = 0, and d(QˆF ) = 0,

(7.66)

imply closure, but Q and QˆF are not exact. The deRham classes are not empty and are not flow invariants. Topology changes during such evolutionary processes. Hence a global set of conservation laws in terms of closed integrals of A and H can be devised only for those closed chains that satisfy Stokes theorem, and those chains must be boundaries (of support). Arbitrary closed integralsR are not evolutionary invariants. This lack of relative integral invariance [185] for z3d H corresponds to the production or destruction of three-dimensional defects, and these new defects are indications of changing topology and changing inhomogeneity. Formally, a closed integral over a closed form is a period integral whose value, by Brouwer’s theorem [193], is an integer multiple of some smallest value. A variation of a period integral signals a change in a Betti number and hence a change in topology. Such flows can produce three-dimensional defects. These results point out the limitations of Moffatt’s and Gaffet’s claims [78] that the volume integral of helicity density, v•curlv, is an evolutionary invariant. Helicity is NOT necessarily an invariant of a continuous flow. Moreover, open or closed integrals of Helicity are not necessarily integral invariants of continuous evolution. In particular, the closed volume integral of helicity density, the fourth component of the Helicity 4D current, is not an invariant of continuous flows for which there is a torsion current. A theorem depending on only the first variation can be stated for the continuous evolution of flows restricted to Hamiltonian or Eulerian flows: Theorem 55 The (uniformly) continuous evolution of all odd-dimensional Pfaff classes of the Cartan base with respect to Hamiltonian or Eulerian flows (dQ = 0, Q exact)

344

Continuous Topological Evolution

are exact. Hence, the closed integrals of A and H = AˆdA over closed cycles or boundaries are relative integral invariants with respect to Hamiltonian or Eulerian flows. The proof of the theorem is as follows: Proof: L(V) A = i(V)dA + d(i(V))A) = d[P + i(V))A] = Q, which demonstrates that Q is exact. Therefore, Z Z Z L(V) A = Q= d[P + i(V))A] = 0 z1 z1 z1 Z ⊃ invariance of A.

(7.67)

(7.68) (7.69)

z1

Similarly,

L(V) H = L(V) (AˆF ) = (L(V) A)ˆF = QˆF = d[P + i(V))A]ˆF, (7.70) which demonstrates that H is exact. Therefore Z Z H = d[P + i(V))A]ˆF = 0 L(V) z3 z3 Z ⊃ invariance of H.

(7.71) (7.72)

z3

In the hydrodynamic case of a compressible Eulerian fluid, this theorem is the generalization of the "invariance of Helicity theorem" often stated for a barotropic domain or isentropic constraints. Closed flows therefore exhibit global conservation laws based on relative integral invariants of A and H, as well as absolute integral invariants of F and K. As will be demonstrated below, the integral of the 3-form of topological torsion, not the helicity density, over a boundary is an invariant of all flows that satisfy the Navier-Stokes equations and for which the vorticity vector field satisfies the Frobenius complete integrability conditions. This result is independent from the magnitude of the viscosity coefficient. On the other hand, the continuous destruction of three-dimensional defects can be associated with closed flows of the Stokes category. Helicity is NOT necessarily a relative integral invariant of Stokes flows. Remarkably, such flows also admit a set of relative integral invariants, but these are determined only in terms of a second variational process.

Global Conservation Laws

345

7.7.2 Second Variation It should be noted that the second Lie differential of the odd-dimensional Pfaff classes (represented by A and H) does produce a set of global conservation laws for uniformly continuous processes. The result follows from the fact that the second Lie differential of the Action with respect to closed flows is exact, where the first Lie differential is closed! The fundamental theorem is then: Theorem 56 The (uniformly) continuous evolution of all odd-dimensional Pfaff classes of the Cartan base with respect to closed flows (dQ = 0) are closed, but not R R necessarily exact. The second Lie differential is always exact so that z1 Q and z3 QˆF are relative integral invariants of (uniformly) continuous (dQ = 0) evolution. The proof of the fundamental theorem is as follows: Proof L(ρV) A = i(ρV)dA + d(i(ρV))A) = Q. L(ρV) L(ρV) A = L(ρV) Q = R = i(ρV)d(i(ρV)dA) +di(ρV)di(ρV)A = i(ρV)d(Q) + di(ρV)di(ρV)A = 0 + d(Λ),

(7.73) (7.74) (7.75) (7.76) (7.77)

which demonstrates that R is exact. Similarly, L(ρV) L(ρV) AˆdA = L(ρV) QˆF = d(Λ F ), which is exact. It follows that, Z Z L(ρV) QˆF = d(Λ F ) = 0, z3

such that

R

z3

(7.78)

(7.79)

z3

QˆF is a relative integral invariant. Q.E.D.

Uniform continuity requires that d(L(ρV) A) = L(ρV) dA = dQ = 0, which insures that Q and QˆF are closed. Hence closed integrals of the odd-dimensional p-forms of Q and QˆF (and not necessarily A andRH) are relative integral invariants R of uniformly continuous evolution. The integrals z1 Q and z3 QˆF generate global conservation laws for uniformly continuous processes in which dQ = 0. In elementary terms, on a space-time variety, the fundamental theorem of uniformly continuous

346

Continuous Topological Evolution

evolution states that the Lorentz force has zero curl, and the torsion defect production rate has zero divergence (K = 0), whether the system is dissipative or not. The successive Lie derivations with respect to a uniformly continuous vector field J = ρV produces an exact sequence, starting from the concept of action-angular momentum, A, evolving to a closed set, Q, which under continued Lie derivation evolves to an exact kernel of radiation-power, R [193]. A similar exact sequence can be constructed for all odd-dimensional Pfaff classes, A, AˆdA, AˆdAˆdA, .... 7.7.3 Continuity and the Integers A most remarkable feature of the fundamental theorem of uniformly continuous evolution is that the integral of any radiation 1-form, R, through a container which is a maximal cycle is in relation to the integers. This concept is another application of the Brouwer degree of a map theorem, that says that all period integrals are integer multiples of some smallest value. The maximal cycle is a closed set that is not a boundary but can contain a system with internal defects, hence the name, the "container". As a simple example consider a disc with several internal holes; the maximal cycle is the cycle which would be the boundary if the disc had no holes. The global conservation laws stated above imply that radiation through the maximal cycle must be compensated by a change in the cohomology class, or the production of a defect of inhomogeneity in the interior. Radiation defects ("holes and torsion handles") are quantized, for it is impossible to create half a hole. It would appear from the above argument that Planck’s hypothesis of quantized radiation oscillators may be considered to be a consequence of the fundamental theorem, and Uniformly Continuous evolution, as defined by equation (7.65). 7.8

Pfaff’s Problem, Characteristics, and the Torsion Current

Closely related to the concept of topological torsion is the Pfaff problem that asks about the solubility of the system of differential equations defined by setting each element of the Cartan closure to zero. The problem is equivalent to finding characteristic vector fields which, if continuous, generate an evolutionary flow that preserves the Cartan topology. The key idea of Pfaff’s problem is to find maps from spaces of q dimensions into the variety, X, such that when these maps and their differentials are substituted into the system of forms that make up the Cartan closure, then the new forms are equal to zero. In this sense, the pullback of the forms of the Cartan closure to the spaces of dimension q are zero. In the case of usual interest to physics, the maps are of a single parameter which almost always is associated with the concept of time. However, there may exist higher-dimensional solutions of say two parameters or more. The question arises as to the largest dimension of such a "solution" and is determined in terms of the "characters" and "genus" of the Pfaff system [242]. It is the objective of this section to demonstrate that the genus of the Pfaff system built from a single 1-form of Action is 3 if the Torsion current, T, vanishes, and can be 2

Pfaff’s Problem, Characteristics, and the Torsion Current

347

only if T 6= 0. The genus is an arithmetic invariant and a topological property. A change of genus implies topological evolution. However for the special Pfaff system described, the characters are such that only 1-parameter solutions are possible, when T = 0, and a unique 2-parameter solution is admissible only when T 6= 0. In other words the Pfaff problem admits a "string" solution (a 2-parameter solution) only when the Torsion current is not zero. Consider an electromagnetic format. For the electromagnetic case, the Cartan 1-form may be defined in terms of the vector and scalar potentials, A = A • dr − ϕdt.

(7.80)

Using the classical notation of Sommerfeld, define the E and B field intensities as, B = curlA,

E = −∂A/∂t − gradϕ.

(7.81)

Then the components of the Darboux-Cartan-Maxwell field, Fµν , may be written as an antisymmetric matrix (or as a Sommerfeld 6-vector) of components, F12 = Bz , F13 = −By , F23 = Bx , F14 = Ex , F24 = Ey , F34 = Ez ,

(7.82)

such that the components of dA = F = Fµν dxµ ˆdxν . The Topological torsion, H, becomes, H = AˆdA = −i{E × A + ϕB, A • B}dxˆdyˆdzˆdt,

(7.83)

with the torsion current defined as, T = −[E × A + ϕB],

(7.84)

h = −A • B.

(7.85)

and the helicity density, The Topological Parity 4-form becomes the global top Pfaffian on the four-dimensional space-time variety, and is equal to, K = dAˆdA = −2E • Bdxˆdyˆdzˆdt.

(7.86)

divT + ∂h/∂t = 2E • B.

(7.87)

Note that, The 3-form of axial current, H, is NOT conserved when K 6= 0. This result has been observed by Berger [18]. Following Chern, the Euler index on a compact manifold would be the integral Z χ=

z4

2E • Bdxˆdyˆdzˆdt.

(7.88)

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Continuous Topological Evolution

The Pfaff problem is determined by the equations, A = 0, F = 0.

(7.89)

Following Slebodzinsky [242], as there is only one 1-form in the Pfaff system, the first character, s0, of the Pfaff system is equal to 1. Multiply F by ϕ, and use A = 0 to eliminate ϕdt in the equation, F = 0. The result is given by the equation, {E × A + ϕB}µν dxµ ˆdxν = {T}µν dxµ ˆdxν = 0,

(7.90)

which is an expression that does not contain dt. The polar system of these resultant equations determines the genus of the Pfaff system. In particular, if T, the torsion current vanishes, then (7.90) vanishes, the second character, s1 is zero and the genus of the Pfaff system is 3. All higher characters vanish, so the Pfaff system is special. Only 1-parameter homeomorphic evolutionary solutions are possible for the Pfaff system in four dimensions, when T = 0. On the other hand, for any arbitrary vector field, V, such that the two 1-forms, {T × V}µ dxµ and A, are linearly independent, then the second character, s1, equals 1, and the genus is 2. There then exists a 2-parameter characteristic evolutionary system (a string). In other words, the presence of the torsion current is necessary for the existence of a 2-parameter solution to the Pfaff problem. There are no 3parameter solutions to this Pfaff problem in four dimensions. This extraordinary connection between the concept of the Torsion current and the solubility of Pfaff’s problem serves to further emphasize the content of the often neglected quantity of topological torsion. 7.8.1 The Euler index The coefficients of the Action 1-form globally define a covariant vector field on the variety. This vector field need not be a section without singularities. Arnold has shown how the singular points (zeros) of the Action 1-form, A, can be used to define the Euler index of the topology induced on the variety. Another method for evaluating this key topological property has been devised by Chern [90], [53]. Following Chern, the Euler index becomes the integral, Z Z χ= K= 2E • Bdxˆdyˆdzˆdt. (7.91) z4

z4

In Lagrangian field theories, a non-zero value for K implies that the second Chern class is not empty and signals the demise of time reversal and parity symmetry [36] (hence, the name Topological Parity 4-form). It should be remarked that K is the exterior differential of the 3-form of topological torsion, H, and that this 3-form, H, can be put into correspondence with Pfaff’s problem and the Pfaff topological dimension, or class, of the 1-form A. The class of a 1-form and its relationship with Lagrangian field theory dates back to Forsyth [77], Vanderkulk [238], and Post [175]. In effect the evolutionary law for the 3-form of Topological Torsion given

Pfaff’s Problem, Characteristics, and the Torsion Current

349

by (7.83) is a Lagrangian field theory built on Pfaff’s problem. Much later than these Pfaff problem methods, Chern and Simons developed an action principle based upon the connection matrix of 1-forms, and its exterior differential of this matrix. The concepts became known as Chern Simons theory, but is different from the Pfaff problem and topological torsion, which does not require a connection. When the electric field is orthogonal to the magnetic field, then the Euler index is zero. The idea that this Poincare 4-form might have deeper meaning led Eddington [67] to state: "It is somewhat curious that the scalar-product of the electric and magnetic forces is of so little importance in classical theory, for..(7.91).. would seem to be the most fundamental invariant of the field. Apart from the fact that it vanishes for electromagnetic waves propagated in the absence of any bound electric field (i.e., remote from electrons), this invariant seems to have no significant properties. Perhaps it may turn out to have greater importance when the study of electron-structure is more advanced." A non-zero value of the Topological Parity 4-form, K, implies that the divergence of T is not zero. therefore, torsion lines can stop or start within the variety even though the evolution is C2 continuous. The torsion current is not necessarily conserved and three-dimensional defects can be produced internally. String theorists describe this effect as an anomaly of the axial (Torsion) current. In the same sense that the closed but not exact 1-form leads to a complex representation involving ordered pairs of variables, a closed but not exact 3-form leads to a quaternionic representation. The concept of a domain of non-null Euler index (K 6= 0) now appears to be useful to the theory of magnetic reconnection in the electromagnetic case [166] and to vortex reconnection [152] in the hydrodynamic case. The correspondence between the bridging and rib structures produced in numerical simulations of turbulent fluid flows and the 4-string interaction of superstring theory is remarkable [106]. The concept (K 6= 0) appears to be applicable to the understanding of the stretching of lines and surfaces in turbulent flows where time-reversal symmetry is violated [63]. The appearance of large scale structures in certain flows has been associated with the lack of parity invariance [254]. The concepts of macroscopic violations of P and T symmetries appear to have application to the theory of the quantum Hall effect [287]. With regards to hydrodynamic systems, the evolution of a flow from a laminar flow to a turbulent flow involves topological evolution. For the Navier-Stokes system, the Euler index depends upon the viscosity and the lack of Frobenius integrability of the vorticity field [209]. Such a term yields a local source for the creation of Torsion currents. The lack of reversibility of such flows, and the irreducible time dependent, three-dimensional features of such flows, implies that K can not be zero for the turbulent state. It is conjectured that the Euler index of the flow (the integral of K over the domain) is not zero during the transition to turbulence. That is, K is not a last multiplier of the spatial volume element, dxˆdyˆdz for the flow describing the continuous (relative to the Cartan C2 topology) transition to turbulence. If dQˆF = 0

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Continuous Topological Evolution

then the function K defines an integrating actor in the sense of a mass density such that, div(KV) + ∂K/∂t = 0. (7.92) If K were a mass density, this equation is often called the "equation of continuity", but it is more accurately described as the "conservation of mass". Relative to the Cartan topology all C2 vector fields are continuous. The transition to the turbulent state, however, must be discontinuous, for the Cartan topology in the turbulent state is disconnected. 7.8.2 Evolution of Topological Torsion In summary, a topology has been constructed on a variety in terms of the elements of closure of a Cartan system of C2 differential forms and their intersections. The associated topological structure indicates that all processes generated by the Lie convective derivative (relative to a C2 vector field, V) are continuous relative to the Cartan topology. However, the processes so generated are not necessarily homeomorphisms for they need not be reversible; i.e., the topology of the initial state can evolve continuously into a different topology on the final state. The method for constructing the Cartan topology is the same on both the initial and the final state, but, for example, the "hole and handle" count on the initial state can be different from the "hole and handle" count in the final state. In terms of a single 1-form of Action, A, a Cartan topological base was constructed in terms of a set of distinct elements, defined as a Pfaff sequence, and their closures. The fundamental laws of evolution of each of the elements of the topological base was formulated relative to an arbitrary vector field. It was determined that there are two categories of continuous flows, those which are "uniformly continuous" and those which are " non-uniformly continuous". A special sub-category of closed flows describe a Hamiltonian evolution, an evolutionary process which preserves the number of "holes and handles". Relative to the uniformly continuous category of continuous processes, all even dimension elements of the Cartan topological base are evolutionary invariants. For uniformly continuous flows, topological evolution takes place only in terms of the odd elements of the topological base. The first odd element of the topological base is the Action, and its law of evolution is equivalent to the evolution of energy. The next odd element (and the only other odd element on space-time) of the Cartan topological base is formulated as the novel 3-form of Topological Torsion. The evolution of this 3-form is studied, for although it does not necessarily satisfy a local conservation law, the anomalous source term, defined as topological parity, can be computed. It is a source of system evolutionary defects. However, it is still possible to establish a set of global conservation laws for the category of non-uniformly continuous and irreversible evolutionary flows. Although the evolution of topological torsion may be described by a continuous process, the creation of topological torsion from a state without topological torsion is not described by a continuous process. As the Cartan topology is

Pfaff’s Problem, Characteristics, and the Torsion Current

351

not connected, the creation of topological torsion must involve discontinuous processes or shocks. The decay of topological torsion can be described by a continuous process. The fundamental equation of topological evolution, L(ρV) A = Q, is equivalent to cohomological format of the first law of thermodynamics, W + dU = Q. The Heat 1-form, Q, may be used to form a Pfaff sequence whose Pfaff dimension may be used to further classify evolutionary flows. For example, if the Pfaff dimension of Q is 2 or less, then Q can be written in the equilibrium format, Q = T dS. An example of a non-uniformly continuous system of flows (defined as dQ 6= 0) is presented in Chapter 4 in terms of the Navier-Stokes equations, for which the anomalous source term, can be computed. In effect it was demonstrated that C2 irreversible flows are among the solution set to the Navier-Stokes system. An abstract example was also given for an electromagnetic Action in Chapter 2, in which the concept of time reversal and parity symmetry breaking was associated with a non-null Euler characteristic of the Cartan topology. 7.8.3

Thermodynamic processes

In thermodynamics, a reversible process is defined as a process for which the 1-form of heat, Q, admits and integrating factor, and an irreversible process is a process for which the 1-form of heat does not admit an integrating factor (of reciprocal temperature) [160]. This definition may be made precise in terms of Cartan’s magic formula and the Frobenius theorem, for if the 1-form of heat, Q, does not admit an integrating factor then the 3-form, QˆdQ, does not vanish. However, for a given physical system defined in terms of a 1-form of Action, A, and its Pfaff sequence, those processes, ρV, that satisfy the equation L(ρV) A ˆ L(ρV) dA = 0 are reversible. Definition : of a reversible process V L(ρV) AˆL(ρV) dA = QˆdQ = 0.

(7.93)

This precise definition of thermodynamic reversibility will be subsumed, and the cohomological equivalent of the first law of thermodynamics will be studied relative to the constraint of continuous reversible or continuous irreversible topological evolution. Many intuitive thermodynamic concepts can be stated precisely in terms of the theory of continuous topological evolution based on the Cartan topology. For example, those processes, ρVLA , for which i(ρVLA )L(ρV) A = i(ρVLA )Q = 0 are locally adiabatic. Local adiabatic process, VLA :

L(ρV) i(ρVLA )A = i(ρVLA )Q = 0.

(7.94)

As must be the case in thermodynamics, there is a fundamental difference between the 1-form W and the 1-form Q. From the definition W = i(ρV)dA, it follows that, i(ρV)W = i(ρV)i(ρV)dA ⇒ 0 (transversality). (7.95)

352

Continuous Topological Evolution

This fact implies that the 1-form W must be constructed from first integrals, φ, of the flow V , or from transversal fluctuations in the kinematics, W = dφ + f ◦ (dx − vdt).

(7.96)

Although W can be included in the concept of Q, there are parts of Q that are not transformable into W . A precise difference between the 1-form of (virtual) work and the 1-form of heat can be established; the 1-form of work is necessarily transversal to the process, while the 1-form of heat is not. This issue is at the heart of the second law of thermodynamics. The argument is pleasing for it gives formal substance to the intuitive differences between the thermodynamic concepts of heat and work. 7.8.4 The Kinematic Topological Base For continuous evolution in space-time, the key idea is that the exterior differential system consists of a Pfaff sequence constructed from a single 1-form of Action A, plus (perhaps) some additional constraints defining a domain of support and its boundary. The work of Arnold (and others) [8] has established that the singular points (zero’s) of a global 1-form carry topological information. This idea is to be extended to the singular points of all elements of the Pfaff sequence, or topological base. In Chapter 9, the idea of how a global 1-form of Action, A, existing on a space of dimension N+1 can be put into correspondence with a line bundle on a variety of dimension N is worked out in detail. The key features are that the Jacobian matrix of the projectivized 1-form of Action carries most of the information about the subspace. The trace and determinant of the Jacobian matrix determine the mean and Gaussian curvature of the subspace. The antisymmetric components of the Jacobian are the functions that make up the 2-form, F = dA. The polynomial powers of F form the Chern classes for the line bundle [55]. 7.8.5 Comments 2006. The perspective of this chapter (which was developed over the period 1981-2004) is somewhat modified by the recognition that processes that are not extremal, and which satisfy the equation i(ρV )dA = W 6= 0, must contain components associated with complex Spinor direction fields with zero quadratic form. All non-equilibrium systems and topological fluctuations are due to Spinor eigen direction fields of the 2-form, dA. These newer developments will be studied in [277].

Chapter 8 PERIODS ON MANIFOLDS, QUANTIZATION, HOMOGENEOUS P-FORMS, AND FRACTALS 8.1 8.1.1

Introduction Historical

More than 25 years ago (1977), I published an article with a title similar to the heading of this article. The article was entitled, "Periods on Manifolds, Quantization and Gauge" [193]. At that time, it had become apparent that at least some of the quantum mechanical features of measurables with rational ratios (the quantum numbers) could be interpreted in terms of topological period integrals. Further motivation for the original publication was based on the idea that the Sommerfeld integrals (which could be interpreted as one-dimensional period integrals) might be used to explain the details of that Copenhagen mystery whereby the quantum jump, or radiative transition from one quantum state (initial state period integral value) to another quantum state (final state period integral value), is described as a "miracle". What was, and still is, needed was a method of describing the dynamics of topological evolution. A period integral is a topological invariant of a homeomorphism, and to describe the change of a period integral would require a process that is not a homeomorphism. It was apparent from Cartan’s work [43] that all Hamiltonian processes preserve the Sommerfeld integrals (closed 1-forms of mechanical action), and could not describe the dynamics of topological evolution, much less the dynamics of a radiative transition. Prior work had indicated that a modification of the Hamiltonian method using Cartan techniques might be used to explain topological evolution [185]. Part of the presentation in this chapter will be the demonstration of Cartan techniques that can be used to describe continuous topological evolution and thermodynamic irreversibility. The major part of this chapter, however, is to give examples and methods of construction of closed p-forms, which may serve as the integrand of period integrals with non-zero values along closed integration chains which are not boundaries. The basic idea stems from the recognition that the integrands of topological period integrals can be encoded in terms of homogeneous p-forms of degree zero. Homogeneous p-forms of degree not zero are always exact [140], hence would yield zero values for their period integrals along closed integration chains. Homogeneous p-forms of de-

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

gree zero are independent from "scale changes", not only at a point, but globally over the homogeneous domain, even though the scale factor is not a global constant. The most common of such objects is to be found in projective geometry, where the fractional linear, or Moebius transformation, is used to deduce the important projective invariants, which are universally homogeneous functions of cross ratios. It is remarkable that the transition probability of quantum mechanics, according to Fermi’s golden rule, is such a cross ratio invariant. The concept of "gauge invariance", as introduced by Weyl, was an attempt to answer the question: Can parallel displacement change the length, or scale, of a vector. Before Weyl, it was recognized that parallel displacement in Riemannian geometry around closed circuits could change the orientation of a vector. The orientation at the start of the process of parallel transport around a closed path need not be the same as the orientation of the vector when it returned to the starting point. Orientation changes in the tangent plane of the starting point were known to be related to curvature, and orientation changes orthogonal to the tangent plane had been related to the concepts of torsion. Apparently, before Weyl, the idea of a length change had been ignored in framing the conditions of what was meant by a parallel displacement in a Riemannian geometry. Could the geometric concepts of metric and connection be reformulated beyond the constraints of Riemannian geometry to produce scale change and path dependence relative to parallel transport? The details of such reformulations in terms of metrics and connections appear most cogently in the last article of Eddington [67], and in the concept of Finsler spaces [5]. In the language of differential forms, without recourse to geometric assumptions of metric or connections, the concept of displacement inducing a change of scale is encoded in terms of the Lie differential with respect to a direction field, X = [xk ], acting on p-forms, ω(xk , dxk ), to create the same form magnified by a scale factor, D. A homogeneous differential form satisfies an equation of the type, [140] L(X) ω = i(X)dω + d(i(X)ω) = Dω.

(8.1)

Differential forms that satisfy such a formula are said to be homogeneous of degree D. The formula is exactly equivalent to Euler’s formula for homogeneous scalar functions, Θ(X) of homogeneity degree D. The homogeneity index need not be an integer, and the components xk of the process X need not be functions of the same dimension, ¯ ¯ ® ­ L(X) Θ(xk ) = i(xk )dΘ = ∂Θ/∂xk ¯ ◦ ¯xk = D · Θ(xk ), (8.2) Θ(X) =

a zero form.

(8.3)

The homogeneity formula applied to any p-form, ω, does not insure that the p-form is closed. If the p-form is closed, such that dω = 0, then the homogeneous closed p-form must be exact when D is a non-zero constant. Remark 57 When D = D(xk ),the homogeneous formula demonstrates that continuous topological evolution can describe self similarities due to changes of scale, without

Introduction

recourse to specific geometrical constraints. Cartan topology is invariant.

355

When D = 0, for closed 1-forms, the

The homogeneous formula can be extended to processes, X, of arbitrary direction fields, acting on both p-forms and p-form densities. Herein, the concept of relative of functional form is related to homogeneous functions of degree D, and absolute (projective) invariance to homogeneous functions of degree D equal to zero. One of the principle results of the first cited article [193] was the presentation and utilization of three period integrals, of dimension 1, 2, and 3, which have dominant physical significance. A period integral is defined as a closed p-form, ω, with dω = 0, integrated over a (closed) cycle of dimension p, zp , which is not a boundary. In this chapter, another three-dimensional period integral (originally presented in 1977 [190], [202]) is added to the list. The format chosen will emphasize, for purposes of more rapid comprehension, an electromagnetic application, but the basic ideas apply to many other areas of physical speciality, such as hydrodynamics and thermodynamics. The four topological period integrals presented in electromagnetic format are: R 1. The Flux quantum = z1 A. The integrand A is a pair 1-form, and the cycle is a one-dimensional closed integration chain, z1 , where dA = 0. In electromagnetic format the physical unit of the flux quantum period integral is h/e. RRR 2. The Topological Torsion or Polarization quantum = AˆF. The intez3 grand AˆF is a pair 3-form, and the closed cycle is three dimensional, z3 , in a domain where d(AˆF ) = 0. In electromagnetic format the physical unit of the Topological Torsion quantum period integral is (h/e)2 . Note that this is equal to the spin quantum times the Hall coefficient, h · h/e2 = ~ZHall . Also recall that the non-zero value of AˆF , indicates that the Cartan topology is a disconnected topology (see Chapter 6). RR 3. The Charge quantum = G. The integrand G is an impair 2-form density, z2 and the closed cycle is two dimensional, z2 in domains where dG = 0. In electromagnetic format the physical unit of the charge quantum period integral is e. The fact that the charge quantum is impair implies that charge is a pseudo-scalar, a fact not in agreement with the current mainstream convention. RRR 4. The Topological Spin quantum = AˆG. The integrand AˆG is an imz3 pair 3-form density, and the cycle is three dimensional, z3 in domains where d(AˆG) = 0. In electromagnetic format the physical unit of the Topological Spin quantum period integral is h. The fact that the spin quantum is impair implies that spin is a pseudo-scalar. As the integration cycles are in domains where the exterior differentials of the integrands vanish, then the values of the integrals have rational ratios [182], a fact which leads to the idea of topological "quantization" and the term "period

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

integral". The integrands for the Flux quantum and Topological Torsion quantum behave as scalars with respect to transformations of the independent variables in their arguments. Such scalars are, in the language of invariant theory, called "absolute" invariants. The Charge quantum and the Topological Spin quantum, are tensordensities derived from integration of "impair" exterior differential form densities. Impair exterior differential forms are sensitive to odd permutations, or in the case of diffeomorphic maps, they are sensitive to the determinant of the diffeomorphism. Sometimes this sensitivity is defined as sensitivity to orientation. The other species of tensor densities are related to the integration of "pair" exterior differential form densities. These pair tensor-densities, by the dogma of tensor analysis, depend only upon the magnitude of the Jacobian determinant, and are not sensitive to odd permutations or the sign of the determinant of the diffeomorphic transformation. Such objects are, in the language of invariant theory, called "relative" invariants [268]. Those densities that are sensitive to the sign of the Jacobian determinant are called pseudo-scalars in the tensor literature [176]. Exterior differential forms are either scalars, or "scalar-densities" (of both the pair and impair type), with respect to tensor diffeomorphisms. However, differential forms are functionally well behaved with respect to functional substitution of C1 transformations that are not invertible (the pullback). Such transformations are not diffeomorphisms. Indeed, such transformations admit topological change. The Flux quantum and the Charge quantum were more or less well known in 1977, but the concept that these period integrals were independent from any metrical constraints was not so well known. Even now (2006) the fact of metrical independence is not fully appreciated. The fact that these objects depend upon the determinant of differentiable transformations is almost completely ignored. In 1977, the third period integral, the Topological Spin quantum, was somewhat novel, having been discovered just a few years before in a somewhat different context [184]. About the same time [190], the second three-dimensional period integral of Topological Torsion was created to study the topological transition from turbulence to the streamline state in a fluid. It took some 10 to 20 years before it was appreciated that the nonzero closure of the pair 3-form, AˆF, defined domains that could be put into correspondence with thermodynamic irreversibility. In addition, it is only very recently that it has been appreciated that the 3-form of Topological Torsion has a direction field that is composed of Spinors, not classical diffeomorphic vectors. Hence in a hydrodynamic context, as a turbulent flow must be irreducibly four dimensional, d(AˆF ) 6= 0, then the cause of turbulence ultimately must be traced back to the Spinor content generated by the eigendirection fields of the 2-form, dA. The 3-form AˆF is of utmost importance to (and is nonzero in) the thermodynamic theory of non-equilibrium systems. Although each of these period integrals appear to have application to the microphysical world, they also should have applicability to the macrophysical world. After all, period integrals are topological objects independent from metric constraints

Introduction

357

of size and shape. When written in terms of a homogeneous degree zero format, all components of a homogeneous p-form can be multiplied by a factor, β, and the pform, like a cross-ratio in projective geometry does not change. Size and shape are not important to these continuous deformation invariants. This fact initially posed an ontological conflict, for experience (or prejudice) seems to indicate that "quantum" features are artifacts of the microphysical world, alone. Recently it has become apparent that the concept of Spinors is another topological idea based upon the eigendirection fields of infinitesimal rotations, and does not depend upon scales. Spinors are not just things to be associated with microscopic elementary particles such as electrons. Spinors and their importance on macroscopic physical systems has long been ignored. Remark 58 As E. Cartan [44] has demonstrated, Spinors are not vectors (tensors) with respect to infinitesimal rotations. E. J. Post became interested in this predicament, and now champions the idea that Quantum Mechanics of the microworld should be developed in terms of metric free ideas [174]. On the other hand, the physics of gravity, constitutive relations, and the synergetic aggregates of the macrophysical world appear to have geometric, metric-dependent, features. Indeed, many of these geometric features are topological properties, especially when they are elements of a diffeomorphic equivalence class. In order to examine metric-based topological features, Post recommends the use of general diffeomorphic invariance principle be used to determine metrical based topological features. That is, the diffeomorphic maps should not be restricted to some particular geometrical group, such as is presumed in gauge theories. The problem with the use of diffeomorphic maps is that they miss the discrete symmetry breaking features of handedness of polarization and to-fro evolution. Diffeomorphic maps imply covariance with respect to both translations and rotations. Spinors are diffeomorphic covariants with respect to translations, but they are not diffeomorphic covariants with respect to rotations. Perhaps a better method to discover metric independent features is to choose a metric arbitrarily, and then show (as did Hodge) that certain topological invariants arise which do not depend upon the choice of metric. Such invariants include those invariants which are "gauge" invariant, in the sense that they are independent from metric based scales. At what physical level a metric-based topology evaporates into a non-metric based topology is still unknown. Conversely, at what level a non-metric based topology condenses or "emerges" into a metric based topology is intuitively at the level of forming coherent quantum macro states, such as those that appear in superconductivity, or as non-dissipative solitons in macro structures. It is conjectured that such a process occurs when the closed, but not exact, homogeneous differential forms used to construct period integrals become harmonic. Another suggestive concept that requires investigation is related to how and if a given metric can undergo topological evolution and change. In particular,

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

Remark 59 The signature of a metric may be a process dependent topological feature. As mentioned in [273] and in more detail in Volume 2, [274], the experimental observations of the features of the non-equilibrium Falaco Solitons appear to be best represented by a 3D Minkowski metric of signature {+,+,-}, yet the initial state of the fluid, and the ultimate (equilibrium) state, appear to be Euclidean with a signature {+,+,+}. If the observations are correct, the Falaco Solitons [274] yield some of the first experimental results that physics recognizes situations where 3 spatial dimensions will support a signature which is negative, and non-Euclidean. Any change in signature is a topological change, and one which appears in the experiment to occur in finite time. 8.1.2 Tensors Tensors are ordered sets of functions of base variables that obey certain (linear) transformation laws with respect to Jacobian differentials of diffeomorphisms amongst the base variables. Diffeomorphisms are special homeomorphisms (which preserve topology) represented by a functional non-linear map, φ, of base variables from domain xm to range y k = φk (xm ). To be a diffeomorphism requires that the map be differk entiable, dy k = (∂φk /∂xm )dxm = Jm dxm , and that both the non-linear map, φ(x), and the linear (Jacobian) map, dφ, have inverses, φ−1 and dφ−1 . The dimension m of the domain is the same as the dimension k of the range. There are two species of "tensors" constructed as ordered arrays of functions of base variables, and epitomized by what are called contravariant vector arrays and covariant vector arrays. A contravariant vector is constrained to transform linearly with respect to the Jacobian map of the base variables, dφ. A covariant vector is constrained to transform linearly with respect to the Jacobian of the inverse map of the base variables, dφ−1 . Figure 8.1 displays the transformational properties of fields that are constrained to be "tensor" fields. For diffeomorphisms, all of the horizontal arrows can be reversed.

Figure 8.1

Collineations with Tensor constraints.

Introduction

359

It is important to realize that the differential of a tensor is not necessarily a tensor, for the maps α and β need not be linear. To devise a differential process that creates a tensor from a tensor leads to the concept of the "covariant" differential, and the theory of connections. 8.1.3 Exterior differential p-forms are not p-tensors. The field structures used in the theory of exterior differential forms need not be so constrained. Contravariant fields are defined as maps, α (collineations if linear), from a set of base variables to a range of ordered functions. Covariant fields are defined as maps, β (correlations if linear), from a set of base variables to a range of ordered functions. The map of base variables, φ, is presumed to be differentiable such that dφ exists. However, it is not assumed that φ−1 and/or dφ−1 exist. Such transformations are not homeomorphisms, hence can be use to describe topological change. What is most remarkable is that exterior differential forms defined on the range permit retrodiction [192] to exterior differential forms on the domain, by means of functional substitution which is defined as the pullback. However, the converse is not true. Exterior differential forms on the domain are NOT uniquely predictable to differential forms on the range. It is often claimed that differential p-forms are p-tensors, but that is not true. As shown below, exterior differential forms are (with respect to diffeomorphisms) either scalars, or densities. Recall that a scalar field is defined such that under functional substitution, Θ(x) ⇐ Θ(y(x)).

(8.4)

A scalar exterior differential form satisfies the equation, Σ(x, dx) ⇐ Σ(y(x), dy(x)).

(8.5)

The operation of scalar equivalence is defined as the pullback, where: 1. Coefficient functions of variables on the final state are replaced by functional substitution, as defined by the map, φ, from base variables, xm , of the initial state to base variables, y k , on the final state, given by the formula, φ : y k = φk (xm ).

(8.6)

¯ ® 2. Differentials, ¯dy k , on the final state are replaced in terms of the differentials, |dxm i, on the initial state using the differential mapping, ¯ ® (8.7) dφ : ¯dy k = [∂φk /∂xm ] ◦ |dxm i .

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

The C1 mapping, φ, is not necessarily a homeomorphism, as both the inverse map, φ , and the inverse differential map, dφ−1 , need not exist. Such a case is ˙ > N. typical of projections of base variables of dimension M to dimension N, with M −1 −1 If φ, φ , dφ, and dφ exist, then the map is called a diffeomorphism. 8.1.4 Exterior differential p-forms are pullback Scalars or pullback Densities Exterior differential p-forms are of two classes: −1

1. The Scalar class: The scalar class is fully described as an exterior differential p-form scalar, with an example given by the 1-form, A = Ak (y)dy k . ¯ ® hAk (y)| ◦ [∂φk (x)/∂xm ] ◦ |dxm i ⇐ hAk (y)| ◦ ¯dy k Scalar 1-form on the final state:

k

m

m

hAk (y(x))| ◦ [∂φ (x)/∂x ] ◦ |dx i ⇐ hAm (x))| ◦ |dxm i ⇐ ¯ ® Therefore, hAm (x))| ◦ |dxm i = hAk (y)| ◦ ¯dy k .

(8.8) (8.9) (8.10) (8.11) (8.12)

2. The Density class: The density class is fully described as an exterior differential p-form density, with an example given the (N-1)-form density (to within an arbitrary factor): Density N-1 form on the final state J = i(J m (y))dy 1 ˆ...ˆdy N = i(J m (y))dV ol(y), m ..ˆdy N , d = J m dy 1 ˆ..dy = J 1 dy 2 ˆ...ˆdy N − J 2 dy 1 ˆ...ˆdy N ... ± J N dy 1 ˆ...ˆdy N−1 , = i(J m (y))Ω(y).

(8.13) (8.14)

The (N-1)-form : Density pullback m i(J (x))Ω(x) ⇐ i(J m (y))Ω(y), |J m (x)i ⇐ [∂φk (x)/∂xm ]Adjoint ◦ |J m (y(x))i .

(8.18) (8.19) (8.20)

(8.15) (8.16) (8.17)

It is of interest to note that the differential volume element, Ω(y), is a monomial N form on the final state, but not necessarily a monomial on the initial state. If the mapping is such that dφ−1 exists (by limiting the domain), then the weighted differential volume in the initial state becomes equal to, ∆(x) · ρ(x)Ω(x) ⇐ ρ(y)Ω(y),

∆(x) = det[∂φk (x)/∂xm ].

(8.21) (8.22)

The determinant of the Jacobian matrix, ∆, acts as a density multiplying factor of the weighted differential volume element on the initial state.

Introduction

361

Historically it has been recognized that exterior differential p-form densities (depending on the arbitrary weighting factor) are of two types: The Pair Type: Which are p-form densities that are insensitive to orientation, or odd permutations of the final state base variables, and The Impair Type: Which are p-form densities that are sensitive to orientation, or odd permutations of the final state base variables, and lead to pseudoscalar integrals. It is possible to construct a diagram representing the natural covariance of exterior differential p-forms. Follow the arrows in Figure 8.2 to demonstrate the retrodictive properties of the pullback. The field maps, α and β, on the initial state are functionally well defined in terms of the field maps, α and β, on the final state, relative to the (evolutionary) base variable maps given by φ and dφ. α = (φ, dφadjoint ) ◦ α ◦ (φ, dφ), β = (φ, dφtranspose ) ◦ β ◦ (φ, dφ),

(8.23) (8.24)

Figure 8.2 Differential form Scalars and Differential form Densities Note that, in practice, the dimension, k, of the (thermodynamic) range is large enough to accommodate the largest Pfaff topological dimension of the 1-forms under consideration; k is smaller than or equal to the (geometric) dimension, m, of the domain. For such cases with m > k, the contravariant and covariant objects that are pulled back to the domain are (geometric) subspaces. If k is 2 or less then the geometric subspace is connected, but need not be simply connected. If k is greater

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

than 2, the geometric subspace is disconnected. These subspaces form topological defects in the domain. Integrals of impair p-form densities generate pseudoscalars. A physical example of an Impair pseudoscalar density yielding a pseudoscalar integral is given by the closed integral of the electromagnetic excitation 2-form density, G [276], [176]. Historically, charge, like mass, has been presumed to be a scalar, and therefore should not be orientation dependent. However, Post, through his studies of magnetic effects in crystals, demonstrated long ago that if charge was a scalar, magnetic permeability would vanish in crystals that had a center of symmetry, counter to experimental observation [174]. Post, over the years has championed this fact that charge is a pseudoscalar, but only recently has the physical community started to take note of this important experimentally confirmed result [95]. Most of the physics community still hangs on to the dogma that charge is a scalar. If the arbitrary density, ρ(X), has an odd power of the determinant, ∆(x), then the density coefficient, ρ(x), is NOT sensitive to the sign of the determinant, ∆(x). The density coefficient is still sensitive to the magnitude of the determinant, ∆(x), so it is still not a true scalar. The p-form densities which have this behavior are defined as Pair p-form densities. Historically it has been presumed that mass density is a physical example of a Pair scalar density. Lagrangian Pair and Impair N-forms It is usual in a field theory to start with a Lagrange "density", L, as the coefficient of a volume integral, and then apply the calculus of variations to the integrand (an N-form density) to find "stationary" solutions that define the "field equations". (N1)-forms, or "currents" can be deduced that satisfy (or generate) conservation laws. From an alternate point of view, "equations of motion" can be obtained by considering a Lagrange scalar, L, and the Cartan-Hilbert 1-form of Action as the starting point for theoretical analysis. The theory of continuous topological evolution generates field equations and equations of motion representing a dynamical process V m by evaluating the effect of applying the Lie exterior differential (with respect to V m ) to exterior differential systems. Although this latter method is favored in this book, it is of some interest to consider density p-forms as well as scalar p-forms. For rapid comprehension, it is best to start off with the concept of a diffeomorphic map (φ, φ−1 , dφ, and dφ−1 exist). The value of the determinant, ∆(xm ), can be greater or less than zero. If the determinant is less that zero, the effect is a change of orientation of the base functions. The negative determinant class is usually associated with a reflection. The positive determinant class is usually associated with an evolutionary motion. Consider a (N)-form with a density coefficient, L(X), of the type, L(X)dX 1 ˆdX 2 ..ˆdX N = L(X) · Ω(X).

(8.25)

Introduction

363

Now use the given mapping formulas to express the volume element Ω(X) in terms of the volume element Ω(x): α(x) · Ω(x) = L(X)·∆(xm ) · Ω(x) ⇐ L(X) · Ω(X).

(8.26)

α(x) = L(X(x))·∆(xm ).

(8.27)

H(X) = |∆| /∆) · L(X),

(8.28)

β(x) = L(X(x))· |∆| (xm ).

(8.29)

It follows that the pullback density obeys the formula,

The pullback density is proportional to both the magnitude and the sign of the transformation Jacobian determinant, ∆(xm ), times the original density function, L(X). Such density coefficients are called pseudoscalars. The differential N-form, α(x) · Ω(x), is called an Impair N-form. If the original density function was multiplied by the ratio, (|∆| /∆), defining a new density function of the form,

then the transformation law for the density would be of the form,

The density function depends only on the magnitude of the determinant of the transformation. The differential N-form, β(x) · Ω(x), is called a Pair N-form. Remark 60 The integrals of pair p-form densities do not depend upon the choice of orientation of the integration chain. The integrals of impair p-form densities depend upon an orientation of the integration chain. If the volume element is to be an evolutionary invariant relative to the contravariant vector V k it follows that the exterior differential of the (N-1)-form must vanish, d(i(V k )Ω) ⇒ 0. This statement is equivalent to saying the divergence of the "current density" V k must be zero. The integral over a boundary of the volume element is an absolute invariant relative to βV k independent from any factor β. 8.1.5 Closed differential forms Period integrals (which lead to topological quantization of macrostructures) are constructed by integrating exterior differential p-forms over integration chains in regions where the exterior differential of the p-form vanishes, but the form itself is not zero. When the exterior differential of a p-form is zero, the p-form is said to be closed. Closed p-forms can be of two types, Closed and Exact, and Closed but not Exact. 1. Closed and Exact p-forms, dω p−1 , can be expressed in terms of the exterior differential of a global C2 (p-1)-form, such that the Poincare lemma is valid, ddω p−1 = 0 globally. Integrals of exact forms over a boundary or a closed cycle vanish.

364

Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

2. Closed but not exact p-forms, γ, have singular points, or regions where γ is singular, and dγ is not defined uniquely. If the singular regions are excised, then in the remaining region the p-form γ is not singular, and dγ = 0. Such p-forms are closed, almost everywhere, but they are not globally exact. The singular sets define topological defects in the "global" domain. Integrals of closed but not exact forms are not zero when the integration chain is a cycle that "encircles" the excised defect singularity. Such integrals over cycles of closed but not exact forms are defined as period integrals. The integration chain (a cycle) may "surround" several singular region(s) and will yield (according to deRham theory) values which are related to the integers (times some constant). This result is the basis of Topological Quantization. From the formula (8.1) for homogeneous p-forms, note that if the p-form is closed and homogeneous of degree D greater than zero, then it must be exact. All related period integrals are zero, and therefore are invariants of those processes that produce a non zero homogeneity index, D 6= 0. Note that it is conceivable that D is a function of time and space such that continuous topological evolution would cause the homogeneity index to decay to zero, thereby describing the emergence of a period integral and a macroscopic quantum state. Examples are given in [274]. In that which follows, algorithms will be used to generate closed, but not exact, p-forms. Closed but not exact p-forms can be constructed from p-forms that can be mapped to p-forms that are homogeneous of degree zero, and cannot be represented (globally) by the exterior differential of a unique (p-1)-form. 8.1.6 Pullbacks, Immersions and Submersions Exterior differential forms are well behaved as scalars or densities with respect to diffeomorphisms. More importantly, exterior differential forms are functionally well behaved with respect to C1 mappings that are without inverse. Immersions Consider a domain of N variables, {xm }, and their differentials mapped into a range of N+1 variables, {ξ k }, and their differentials: φ : xm ⇒ ξ k = ξ k (xm ), m

k

k

(8.30) m

n

n

dφ : dx ⇒ dξ = {∂ξ (x )/∂x }dx .

(8.31)

The Jacobian matrix, £ ¤ [J] = ∂ξ k (xm )/∂xn ,

(8.32)

is without inverse as there are N+1 rows and N columns. However, an augmented N+1 x N+1 square matrix can be fabricated by adding a last, N+1, column composed of the vector elements, ξ k , ¤ £ (8.33) [J; R] = ∂ξ k /∂xm ; ξ k .

Introduction

365

Construct the N+1 differential volume element, ΩN+1 = dξ 1 ˆ...ˆdξ N+1 ,

(8.34)

J = i([ξ 1 ...ξ N+1 )ΩN+1 .

(8.35)

and the N-form Current

This N-form on the (N+1)-dimensional range, {ξ k }, has a pullback as a Volume element on the domain, {xm }, with a "density" coefficient ρ(xm ). The pullback operation consists of inserting the well defined expressions (in terms of the immersion) for ξ k and dξ k . The pullback density coefficient consists of the determinant of [J; R] : ρ(x)Ω(dx)N = det [J; R] dx1 ˆ...ˆdxN ⇐ i([ξ 1 ...ξ N +1 )Ω(dξ)N+1 .

(8.36)

As a simple example, consider the immersion, and its augmented Jacobian: φ : xm ⇒ ξ k = xk ; ξ N+1 = Θ(xm ), ⎡ ⎤ 1 0 .. x1 ⎢ 0 1 .. x2 ⎥ ⎥. [J; R] = ⎢ ⎣ .. .. .. .. ⎦ ∂S/∂x1 ∂S/∂x2 .. Θ

(8.37)

ρ(xk ) = {Θ − xm ∂Θ/∂xm }.

(8.39)

(8.38)

The determinant of this matrix is

Note that the determinant, det [J; R] , vanishes if the function Θ is homogeneous of degree 1 in the variables, xm . When Θ(xm ) is homogeneous of degree 1 in the variables, xm , then the variables {∂Θ/∂xm } are thermodynamically conjugate and homogeneous of degree zero in this example. Note that there are other augmentations of the non-square Jacobian matrix associated with the immersion that are useful to differential geometry. Submersions Consider a domain of N variables, {xm }, and their differentials mapped onto a range of M ≤ N variables, {V k }, and their differentials: φ : xm ⇒ V k = V k (xm )

dφ : dxm ⇒ dV k = {∂V k (xm )/∂xn }dxn . The Jacobian matrix,

(8.40) (8.41)

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

£ ¤ [J] = ∂V k (xm )/∂xn ,

(8.42)

is without inverse as there are M rows and N columns, so the map φ is not a diffeomorphism. Construct the differential volume element (an M-form, ΩM ) on the range, ΩM = dV 1 ˆ...ˆdV M , (8.43) and the (M-1)-form Current, J0 , J0 = i([V 1 ...V M )ΩM .

(8.44)

Divide the (M-1)-form current by a function, λ, (an inverse integrating factor) and ask what must the form of the integrating divisor be such that the (M-1)-form on the range is closed (but not exact): J = J0 /λ = (1/λ)i([V 1 ...V M )ΩM .

(8.45)

Evaluate the exterior differential of J : dJ = d(1/λ)ˆJ0 + (1/λ)d(J0 ) = −(1/λ2 )dλˆJ0 + (1/λ)(M · ΩM ) = (1/λ2 ){−V k ∂λ/∂V k + M · λ}ΩM .

(8.46) (8.47) (8.48)

If the (M-1)-form current, J , is to be closed, then the integrating denominator must be homogeneous of degree M, making the reduced current, J = J0 /λ, homogeneous of degree zero. There are two important algebraic ways to construct homogeneous functions. The first is based on the Holder Norm, and the second is based on the Buckingham Pi theorem. Application of the Buckingham Pi theorem will be found in Volume 5 of this series [277], while some examples using the Holder Norm appear below. The Holder Norm A suitable function for the integrating denominator is given by the Holder norm, λ:

Holder Norm : λ(V k ) = {a1 (V 1 )σ + a2 (V 2 )σ ... + an (V n )σ }M/σ , λ(βV k ) = β M λ(V k ), which leads to m m J(x , dx ) ⇐ (J0 /λ)(V k , dV k ), dJ(xm , dxm ) = 0, if M = the number of vector components.

(8.49) (8.50) (8.51) (8.52) (8.53)

Introduction

367

This (M-1)-form current can be pulled back to the domain space, producing a closed, but not exact, (M-1)-form, which is a candidate for a period integral. The method can be used to construct p-form candidates for period integrals of any degree p. The ak are usually treated as constants that determine the "isotropy" and signature of the Norm. Note that the length of a vector defined as the square root of a quadratic (Euclidean) inner product is a homogeneous function of degree D = 1, isotropic in its components, and of signature zero (no minus signs). In terms of the vector components, Vk , λ(V k ) = hV| ◦ |Vi = {(V 1 )2 + (V 2 )2 + (V 3 )2 }1/2 , σ = 2, D = 1, a1 = a2 = ... = ak = 1.

(8.54) (8.55)

The homogeneous Holder norm will be of importance to pair differential forms and the quantization of line integrals (flux) and 3-forms of topological torsion. Closure Examples M=2 dV 1 ˆdV 2 , + V 1 ˆdV 2 − V 2 dV 1 , {a1 (V 1 )m + a2 (V 2 )m }(2/m) , {(V 1 )2 + (V 2 )2 } a common format, 0.

(8.56) (8.57) (8.58) (8.59) (8.60)

dV 1 ˆdV 2 ˆdV 3 , V 1 ˆdV 2 ˆdV 3 − V 2 dV 1 ˆdV 3 + V 3 dV 1 ˆdV 2 , {a1 (V 1 )m + a2 (V 2 )m + a3 (V 3 )m }(3/m) , {(V 1 )2 + (V 2 )2 + (V 3 )2 }(3/2) a common format, 0.

(8.61) (8.62) (8.63) (8.64) (8.65)

Ω2 J0 λ λ d(J0 /λ)

= = = = =

Closure Examples M=3 Ω3 G0 λ λ d(G0 /λ)

= = = = =

Closure Examples M=4 Ω4 = dV 1 ˆdV 2 ˆdV 3 ˆdV 4 , H0 = V 1 ˆdV 2 ˆdV 3 ˆdV 4 − V 2 dV 1 ˆdV 3 ˆdV 4 +V 3 dV 1 ˆdV 2 ˆdV 4 − V 4 dV 1 ˆdV 2 ˆdV 3 , λ = {a1 (V 1 )m + a2 (V 2 )m + a3 (V 3 )m + a4 (V 4 )m }(4/m) , λ = {(V 1 )2 + (V 2 )2 + (V 3 )2 + (V 4 )2 }2 a common format, d(H0 /λ) = 0.

(8.66) (8.67) (8.68) (8.69) (8.70)

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

The Buckingham Pi Product The second useful homogeneous function, π(ξ k ), is given by the product Buckingham Pi Product, π(ξ k ) = {(ξ 1 )a (ξ 2 )b (ξ 3 )c ....}, π(βξ k ) = β (a+b+c...) π(ξ k ).

(8.71) (8.72)

This homogeneous construction enjoys some degree of fame in the engineering world of hydrodynamics and chemistry, [35], where it can be used to make statements about the behavior of physical systems, without having a complete formal theory. The methods have been employed also by economists [48] and biologists. The homogeneous Buckingham Pi product function will be of importance for volume elements, impair differential forms, and differences between orientable and non-orientable domains. The concepts will have importance with regard to the quantization of charge and spin. 8.2

Homogeneous Functions

The concept of a homogeneous function, relative to the variables xm , is encoded by the equation, Θ(xm ; sσ ) ⇒ Θ(βxm ; sσ ) = g(β)Θ(xm ; sσ ) = β D Θ(xm ; sσ ).

(8.73)

Ψ(β, xm ; sσ ) = {Θ(βxm ; sσ ) − β D Θ(xm ; sσ )} ⇒ 0, and ∂Ψ(β, xm ; sσ )/∂β = 0.

(8.74) (8.75)

Every variable, xm , in the argument of the function is multiplied by the same factor, β. The homogeneous function is self similar upon change of scale, β. The variables, xm , need not be "dimensionally" equivalent. The scale, or magnitude, of β is arbitrary. This requirement of scale independence requires that the function,

The constraint, ∂Ψ(β, xm )/∂β = 0, leads to Euler’s theorem for homogeneous functions, {xm ∂Θ(xk ; sσ )/∂xm − D · Θ(xm ; sσ )} = 0.

(8.76)

The exponent D is said to determine the homogeneity degree. The interest herein will be focused on functions that are homogeneous of degree D = 0. Remark 61 Note that neither the exponent D, nor the scale factor β, need be a constant, and D need not be an integer, a notion popularized by the concept of fractals. D = D(xm ; sσ ), β = β(xm ; sσ ).

(8.77) (8.78)

Homogeneous Functions

369

For functions, the Lie differential, with respect to a position vector of base variables, X = [xm ] can be written as, L(X) Θ(xk ) = i(X)dΘ(xk ) = xm ∂Θ(xk )/∂xm ,

(8.79)

so that the Euler formula can be expressed in a metric independent manner using the Lie differential, L([xm ]) Θ(xk ) = xm ∂Θ(xk )/∂xm = D(xk ) · {Θ(xk )}.

(8.80)

Θ(xm ; sσ ) = γ ∆ Θ(xm ; sσ ),

(8.81)

The homogeneity concept can be extended to functions that contain two sets of variables,

where the function is homogeneous in the second set, sσ , of degree ∆. The function can be homogeneous in both sets of variables with different homogeneity degrees. This situation occurs in thermodynamics where (with a change to a more suggestive notation): Θ(Tm , βS m ) = β 1 Θ(Tm , S m ), Θ(γTm , S m ) = γ 0 Θ(Tm , S m ).

(8.82) (8.83)

The thermodynamic function, Θ(Tm , S m ), is homogeneous of degree 1 in the additive thermodynamic variables (quantities), S m = {S, V, U..} and is homogeneous of degree zero in the intensive thermodynamic variables Tm = {T, P, ...} In the defining formula for homogeneity, 1. The factor β (or γ) can be interpreted as a scaling parameter. It need not be a domain constant!. 2. The factor D (or ∆) can be interpreted as the homogeneity degree, or the fractal dimension, and it does not need to be a domain constant! 3. The factor β D (or γ ∆ ) could be interpreted as the number of copies, N (of Θ(xm ; sσ )). These interpretations lead to the formula, N = βD,

(8.84)

D = ln N/ ln β,

(8.85)

and the "Hausdorf" dimension,

where the equivalent notations in terms of γ and ∆ can be used.

370

8.3

Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

Homogeneous differential forms

In general, a homogeneous exterior differential p-form, ω(Am (ξ k ), dξ m ), will satisfy the equation, [140] L(V a ) ω = k ω. (8.86) Consider those situations where there exists a map from a set of a parameters σ a to the n independent (coordinate) variables, ξ k , and therefore another map (obtained by functional substitution) from the σ a to the coefficient functions, Ak (ξ k ) : σ m ⇒ ξ k (σ m ), bk (σ m ) = Ak (ξ k (σ a )). σm ⇒ A

(8.88)

σ a ∂ξ k /∂σ a = D(ξ) ξ k .

(8.89)

(8.87)

Assume that the ξ k are homogeneous of degree D, relative to the parameters, σ a ,

Note that the Lie differential, with respect to σa , of the elements dξ k is, L(σa ) (dξ k ) = d(i(σa )dξ k ) + i(σ a )ddξ k k

a

= = = = k k where (ξ ∂D/∂ξ ) =

a

(8.90)

a

(8.91) (8.92) (8.93) (8.94) (8.95)

d{i(σ )((∂ξ /∂σ )dσ )} + 0 d(σ a ∂ξ k /∂σ a ) = d(D(ξ) ξ k ) D(ξ) dξ k + (ξ k ∂D/∂ξ k )dξ k D(ξ)(1 + B(ξ))dξ k B(ξ)D(ξ).

In the theory of Finsler spaces it is assumed that the differentials dξ k are homogeneous of degree 1. When D(ξ)(1 + B(ξ)) = 1, Chern has defined the system as being "projectivized" [53]. A simple case is where D(ξ) = 1, and B(ξ) = 0. Euler’s equation for homogeneous functions can be replicated for exterior differential forms in terms of Cartan’s magic formula for the Lie differential. Consider the Lie differential of a set of functions homogeneous functions, Ak (ξ k ), homogeneous of degree H relative to the variables, ξ k . Hence the assumption implies that, ξ m ∂Ak (ξ k )/∂ξ m = H(ξ) Ak (ξ m ).

(8.96)

Note that the Lie differential of the function set Ak (ξ k ) becomes: L(σa ) Ak (ξ k ) = d(i(σ a )Ak (ξ k )) + i(σ a )dAk (ξ k ) = = ⇒ =

a

k

m

m

(8.97) a

0 + i(σ )(∂Ak (ξ )/∂ξ )(∂ξ /∂σ )dσ (∂Ak (ξ k )/∂ξ m )(σ a ∂ξ m /∂σ a ) D(ξ) (∂Ak (ξ k )/∂ξ m )(ξ m ) D(ξ) · H(ξ) Ak (ξ k ).

a

(8.98) (8.99) (8.100) (8.101)

Homogeneous differential forms

371

Combining these results for a 1-form yields the equation, L(σa ) (Ak (ξ k )dxk ) = {L(ηa ) Ak (ξ k )}dξ k + Ak (ξ k )L(σa ) (dξ k ) k

k

= D(ξ){H(ξ) + (1 + B(ξ))}Ak (ξ )dξ = (H(ξ) + 1 + B(ξ)) D(ξ) Ak (ξ k )dξ k .

(8.102) (8.103) (8.104)

This formula generalizes, for p-form, to read,

where and

L(σa ) ωp ξ m ∂Ak /∂ξ m σ a ∂ξ k /∂σ a ξ k ∂D/∂ξ k )

= = = =

(H(ξ) + p(1 + B(ξ)))D(ξ) ω p = k ωp , H(ξ) Ak , D(ξ) ξ k , B(ξ) D(ξ).

(8.105) (8.106) (8.107) (8.108)

A special class of p-forms are those which are closed; i.e., the exterior differential of the p-form vanishes. From the definition of a homogeneous form, if the homogeneity index is not zero, then the closed homogeneous p-form is exact, L(σa ) ωp = d(i(σ a )ω p ) + i(σ a )dω p , for closed homogeneous p-forms = d(i(σ a )ω p ) + 0 = k ω p .

(8.109) (8.110)

Special interest is placed upon those closed p-forms which are not exact, for they lead to topological period integrals. If these closed homogeneous p-forms are not exact, they must be homogeneous of degree zero. Using this theorem, an algorithm will be constructed below to generate closed, but not exact p-forms, in terms of homogeneous of degree zero p-forms. There are two ways to generate homogeneous of degree zero p-forms, either D(ξ) = 0, or H(ξ) = −p. The choice D(ξ) = 0 is not acceptable, for that would imply zero displacements∗ . The choice H(ξ) = −p appears to work well for the construction of period integrals. Fractal differential forms and evolutionary processes From the arguments presented above, it would appear that if the coefficients of a projectivized differential form are of homogeneity index H(ξ) not an integer, then the exterior differential form is a fractal exterior differential form. The evolution of a fractal differential form can be studied in terms of Cartan’s Magic formula. Of particular interest (especially to an expanding universe) is the homogeneity index of a volume element. The projective mapping can be used, by functional substitution, to pullback the closed (M-1)-form C(V m , dV m ) on M to a closed (M-1)-form, C(xk , dxk ), on N, as an integer. However, since the popularization of fractals, it is to be recognized that D does not need to be an integer. ∗

In fact, a popular choice is that D = 1, a choice that is called the projectivized p-form, and which is the choice utilized to define Finsler geometries.

372

8.4

Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

Some applications of closed but not exact p-forms

Herein, the discussion will be based upon use of the procedure derived above to produce closed but not exact p-forms on any space of N independent base variables. The method employs projective maps from the base space to vector spaces of dimension M ≤ N, and the Holder Norm. The closed p-forms generated by the algorithm will be of degree p = M − 1. Special attention will be paid to closed 1-forms, and closed (N-1)-forms, and to all closed forms on base spaces of dimension 4. Closed 2-forms in 3D and closed 3-forms in 4D will be related to Linking and Braid integrals. 8.4.1 Closed but not exact 1-forms The classic example is given by the closed, but not exact, 1-form, γ, γ = Γ(ydx − xdy)/(x2 + y 2 ), dγ = 0.

(8.111) (8.112)

Such objects are of use in hydrodynamics where they are the source of circulation (which makes a wing fly), in the theory of the Bohm-Aharanov effect, and in the theory of super conductivity, and perhaps surprisingly in the theory of Schroedinger quantum mechanics, for example. Note that the 1-form γ(x, y, dx, dy) is homogeneous of degree zero for constant Γ and constant values of the scale change β: γ(βx, βy, d(βx), d(βy)) = = = =

Γ{βyd(βx) − βxd(βy}/((βx)2 + (βy)2 ) β 2 Γ{yd(x) − xd(y}/β 2 ((x)2 + (y)2 ) γ(x, y, d(x), d(y)) γ(x, y, d(x), d(y)).

(8.113) (8.114) (8.115) (8.116)

The 1-form γ is well defined everywhere except at the origin, which is a "topological obstruction". A closed integration chain that encircles the origin can not be "shrunk to zero". Integration over any closed chain that does not encircle the origin yields zero, but integration of γ over a closed one-dimensional cycle, z1, that encircles the origin {x = 0, y = 0) once will yield the value Γ·(2π). If the integration chain encircles the origin n times, then the value of the integral is n · Γ · 2π. These statements can be demonstrated by considering the map from {r, θ} to {x, y}, {r, θ} ⇒ {x, y} = {r cos θ, r sin θ}, (8.117) {dr, dθ} ⇒ {dx, dy} = {dr cos θ − r sin θdθ, dr sin θ + r cos θ dθ}. (8.118) Substitution for {x, y, dx, dy} in terms of {r, θ, dr, dθ} yields, Z γ = Γ(ydx − xdy)/(x2 + y 2 ), z1 Z = Γdθ = Γ · (2π), z1

(8.119) (8.120)

Some applications of closed but not exact p-forms

373

where the cycle z1 ≈ x2 +y 2 = r2 = 1 (a constant) encloses the topological obstruction r ⇒ 0. This metric free topological counting mechanism is valid for domains where a metric can be defined, and for domains where a metric is not defined. It should be noted that the closed but not exact integrand in terms of non-homogeneous variables, (x, y), is homogeneous of degree zero, and therefore independent of scales, γ = Γ(ydx − xdy)/(x2 + y 2 ), = Γ((ηy)d(ηx) − (ηx)d(ηy))/((ηx)2 + (ηy)2 ), η = constant.

(8.121) (8.122) (8.123)

The 1-form, ”dθ” = (ydx − xdy)/(x2 + y 2 ), appears to be an exact differential, but the statement is not valid globally. Note that dθ is mapped to a closed 1-form that is homogeneous of degree zero. The divisor, λ = (x2 + y 2 ),in the formula for γ looks like a quadratic inner product of a vector [x, y] with itself. However, the fact that dγ = 0 is valid if λ = (± ax2 ± by 2 ), where a and b are arbitrary constants. The closure condition, as an inner product, is satisfied for any signature or isotropy constraint. The key issue is that the 1-form is homogeneous of degree zero. The integration chains which are used to evaluate the period integrals are limited to closed cycles. Closed cycles can be of two types, cycles which are boundaries and cycles which are not. Boundaries are composed of cycles. As an example, consider the punctured disc. It has an inner cycle, and an outer cycle. The boundary of the interior of the punctured disc is composed of the two cycles, the inner one (say clockwise) and the outer one (say anti-clockwise). The integration of closed form, γ, whose exterior differential vanishes on the interior of the disc, over the inner cycle is a finite number. The value of the integral over the outer cycle is also a finite number, and is equal and opposite in sign to the first period integral. The integral over the boundary, being the sum of the two cycles, vanishes. Note that the Stokes theorem cannot be applied directly to the evaluation of a period integral when the closed integration chain is not a boundary. It is not appreciated (usually) that the classic example given above is in effect a canonical representation for many 1-D period integrals. The concept of differential closure can be satisfied by any pair of functions, α(xm ) and β(xm ) on a space of 1 ≤ m ≤ N independent variables. For example the 1-form defined below in terms of 2 scalar functions, α(xm ) and β(xm ), is closed: Assume A = Γ · (α(xm )dβ(xm ) − β(xm )dα(xm ))/λ, (8.124) m 2 m 2 (8.125) λ = (α(x ) + β(x ) ), then dA = 0 (the vector of coefficients has zero "curl" in 3D). (8.126) What is even more remarkable is that the 1-form denominator can be any form of

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the Holder norm [241]. If A = Γ · (α(xm )dβ(xm ) − β(xm )dα(xm ))/λ, and λ = {a · α(xm )p + b · β(xm )p }n/p a Holder norm, then dA = 0 Any constant a, any constant b, any p, n = 2.

(8.127) (8.128) (8.129)

The Holder norm has the exponent n = 2, which makes the 1-form, A, homogeneous of degree zero in terms of the two functions, α(xm ) and β(xm ). As another ubiquitous example, note that if α(xm ) is a complex function, Ψ = u(xm ) + iv(xm ) and β(xm ) is its complex conjugate, Ψ∗ = u(xm ) − iv(xm ), then the period integral becomes related to the "probability current" of Copenhagen quantum mechanics, Probability current Period Integral Z Z P = J= {Ψ∗ dΨ − ΨdΨ∗ }/(Ψ∗ Ψ), z1

dJ = 0.

(8.130)

z1

(8.131)

In this article, these results will be generalized to produce examples of closed p-forms for any p. In physics, a closed 1-form has zero "curl". A closed (N-1)-form has zero "divergence" and is called a "current". Physical theories recognize that vector direction fields with zero divergence are the basis of many conservation laws. Hence, the formulation of period integrals based upon p-forms where p = N − 1 are of particular interest. For the physical arena of space-time, where N = 4, integrals of closed 3-forms will represent topological properties invariant with respect to continuous differentiable processes. Simply said, when the divergence of a vector is zero, that which goes in equals that which comes out. There is no accumulation in the interior. In both hydrodynamics and electromagnetism there is interest in the possibility that such divergence-free direction field lines (often interpreted as frozen in lines of vorticity or magnetic field) are linked or knotted. The evolution, or the creation, of such a topological state is an unsolved problem. In general, the concept of a contravariant direction field in the Cartan calculus is represented by a (N-1)-form on a space of N dimensions. If the (N-1)-form is closed (which is related to the idea that the direction field is divergence free) then the closed integrals of such (N-1)-forms are deformation invariants (hence represent topological properties) of all evolutionary processes that can be represented by a single parameter semi-group. If the (N-1)-form is closed, then either the lines that represent the direction field begin and terminate on boundary points of the domain, or are cyclic and close upon themselves. A closed (divergence free) direction field never stops or starts in the topological interior. The lines which stop and start on a boundary are of two types, those that stop and start on the same boundary component, and those the stop and start on a different

Some applications of closed but not exact p-forms

375

boundary components. The fundamental idea starts with the concept of divergence free vector fields. However, the arguments of deformation of closed integrals extends to p-forms which are closed. 8.4.2 Closed 1-forms in 3D and 4D In this section, the method will be specialized to show examples for domains of 3 and 4 base variables. The 1-form current will be generated from the formulas given by the set (8.56), (8.66), J0 = V 1 ˆdV 2 − V 2 dV 1 .

(8.132)

Consider the projective map φ from (x, y, z, s) to [U, V ] = [V 1 , V 2 ], p x2 + y 2 − s2 , U(x, y, z, s) = V (x, y, z, s) = z,

(8.133) (8.134)

p dU(x, y, z, s) = {xdx + ydy + sds)/ x2 + y 2 − s2 , dV (x, y, z, s) = dz.

(8.135) (8.136)

with differentials dφ :

Assume (one of many) closure factor(s), λ = (U 2 + V 2 )2/2 = x2 + y 2 + z 2 − s2 ,

(8.137) (8.138)

to yield the pullback 1-form on x,y,z,s: p φ∗ Γ = γ = ((x2 + y 2 − s2 )dz − zxdx − zydy − zsds)/{ x2 + y 2 − s2 λ}.

(8.139)

This 1-form current, J0 /λ = γ is closed, implying that a covector field on {x, y, z, s} with components, p (8.140) A = [−zx, −zy, (x2 + y 2 − s2 ), −zs]/{ x2 + y 2 − s2 λ},

has zero exterior differential. From electromagnetic theory, in 4D, the exterior differential of a 1-form of electromagnetic potentials has 6 components, three of which are B-like and three of which are E-like. As the 1-form generated above is closed, the 1-form of potentials defines a region where both the B field and the E field are excluded. Yet there is a finite period integral, which can be interpreted as the flux quantum. In 3D, the variable s could be considered to be a constant. The resulting 1-form has coefficients that could represent a stationary fluid flow, without vorticity, but with finite Kelvin circulation.

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8.4.3 Closed 2-forms in 3D and 4D The 2-form current will be generated from the formulas given by the set (8.61), G0 = V 1 ˆdV 2 ˆdV 3 − V 2 dV 1 ˆdV 3 + V 3 dV 1 ˆdV 2 .

(8.141)

Consider the projective map φ from (x, y, z, t) to [U, V, W ] = [V 1 , V 2 , V 3 ], p x2 + y 2 = ρ, (8.142) U (x, y, z, t) = V (x, y, z, t) = z, (8.143) W (x, y, z, t) = t, (8.144) with, p dU (x, y, z, t) = {xdx + ydy)/ x2 + y 2 , dV (x, y, z, t) = dz, dW = dt.

(8.145) (8.146) (8.147)

Construct the Holder norm, λ = (aU p + bV p + eW p )n/p , specializing to, λ = (x2 + y 2 + z 2 + et2 )3/2 .

(8.148)

The pullback of the form, G = G0 /λ, becomes the closed 2-form on {x, y, z, t}, {ρ2 dzˆdt − z xdxˆdt + txdxˆdz + tydyˆdz − zydyˆdt} ρλ ∗ dφ G = 0. φ∗ G =

(8.149) (8.150)

8.4.4 Closed 3-forms in 4D The 3-form current will be generated from the formulas given by the set (8.66),

H0 =

V 1 ˆdV 2 ˆdV 3 ˆdV 4 − V 2 dV 1 ˆdV 3 ˆdV 4 +V 3 dV 1 ˆdV 2 ˆdV 4 − V 4 dV 1 ˆdV 2 ˆdV 3 .

(8.151)

The extension to four dimensions follows from the assumption that the four components of the vector field are functions of {x, y, z, t}. From a given array of N = four functions, {U(x, y, z, t), V (x, y, z, t), W (x, y, z, t), S(x, y, z, t)}, construct the (N-1=3)-form, H = (UdV ˆdW ˆdS − V dU ˆdW ˆdS +W dU ˆdV ˆdS − SdUˆdV ˆdW )/λ, where λ = {aU p + bV p + cW p + eS p }n/p .

(8.152) (8.153)

Some applications of closed but not exact p-forms

377

Then for any choice of the constants a,b,c,e,p, dH = 0 when n = 4.

(8.154) (8.155)

By choosing the closure factor in the form of a Holder norm and with proper selection of the homogeneity index, the methods above permit the construction of globally closed 3-forms given an arbitrary direction field in 4D. In particular, consider the direction field given by the map, {x, y, z, t} ⇒ [U, V, W, S] = [U, V, W, (x2 + y 2 + z 2 − c2 t2 )2 ].

(8.156) (8.157)

For any Holder norm of the form, (e 6= 0), λ = {aU p + bV p + cW p + e(x2 + y 2 + z 2 − c2 t2 )2p }n/p ,

(8.158)

it may be demonstrated, by direct substitution, that the 3-form, H = (UdV ˆdW ˆdS − V dU ˆdW ˆdS +W dU ˆdV ˆdS − SdUˆdV ˆdW )/λ,

(8.159)

has an exterior differential such that dH = m(x, y, z, t)(x2 + y 2 + z 2 − c2 t2 )(n − 4)dxˆdyˆdzˆdt.

(8.160)

The factor m(x, y, z, t) depends upon the functions that compose the direction field. The curious result is that the 3-form T is closed globally when n=4, but is also closed on the submanifold (x2 + y 2 + z 2 − c2 t2 ) = 0 (the lightcone) for any n. This result has applicability to electromagnetism. For example, when the constants are a = b = c = 1, e = ±r0−2 and p = 2, the closure renormalization denominator is λ = {U 2 + V 2 + W 2 ± r0−2 (x2 + y 2 + z 2 − c2 t2 )4 }n/2 .

(8.161)

The closure factor looks like a Euclidean norm on the lightcone. Note that the scale factor r0 is arbitrary. (Could it be equal to the Hubble radius?) Closure of the Topological Torsion 3-form, A^dA Of particular importance is the closure of the 3-form of topological Torsion, AˆdA = i(T4 )(dxˆdyˆdzˆdt). By defining the Holder norm in terms of the components of T4 leads to the closure factor, λAˆdA = {a(T(x) )p + b(T(y) )p + c(T(z) )p + e(T(t) )p }4/p ,

(8.162)

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such that division by the Holder norm leads to a closed 3-form, d{AˆdA/(λAˆdA )} = 0.

(8.163)

In Electromagnetic theory, the result is equivalent to the fact that the second Poincare invariant is zero. In thermodynamics, the result implies that associated processes are NOT thermodynamically irreversible. The closure factor does not depend upon metric signature as the coefficients can be of any sign. The closure factor does not depend upon the exponent p. Ubiquitous choices are p = 2, a = b = c = −1, e = +1.

8.4.5 Direction Fields with Zero Divergence The algorithms presented above for construction of closed p-forms can also be viewed as the projective construction of an (M-1)-form current on an M-dimensional vector space (range) pulled back to the N-dimensional domain of base variables {xk }. When M is less than N, the Jacobian matrix of the projective mapping is not square, and will have an obvious zero determinant. When the range has the same dimension as the domain, then the zero determinant property for the projective mapping is not so obvious. However, given a N-dimensional vector field, X, if the components of the vector field are divided by a Holder norm with homogeneity index n = 1, Renormalized t = X/λ (n = 1),

(8.164)

then the Jacobian matrix, [J(t)], of the rescaled (or in physics - renormalized) vector field, t, will have a zero determinant globally, det[J(t)] = 0.

(8.165)

If the rescaled vector field is multiplied by the adjoint of the said Jacobian matrix, then the resulting vector field, j, has zero divergence, if j = [J(t)]adjoint ◦ |ti , div j = 0.

(8.166) (8.167)

By contraction with the N-dimensional volume element, the construction produces a closed (N-1)-form, or current: j = i(j)dx1 ˆdx2 ..dxN = i(j)ΩN , d(j) = 0.

(8.168) (8.169)

The closed (N-1)-form computed by this method agrees to within a factor N with the closed (N-1)-form computed from algorithmic equation (8.19). These results have obvious consequence to the Cartan-Frenet theory of the moving Frame. For then, the Holder norm is explicitly given as Euclidean norm, and

The Gauss Integrals (2-forms)

379

implies that the identity matrix is a metric for the space: λ = {aU p + bV p + cW p + eS p }n/p with a = b = ... = 1, p = 2, n = 1, X p ⊃ λF renet = ( (X k )2 )1/2 = hX ◦ Xi, p t = X/λF renet = X/ hX ◦ Xi = "unit tangent vector", t ◦ t = 1.

(8.170) (8.171) (8.172) (8.173) (8.174)

The differential of t with respect to the arclength s, is equivalent to, dt/ds = [J(t)] ◦ |[ti = κ |ni .

(8.175)

As det[J(t)] = 0, it follows that t is orthogonal to n with respect to the Euclidean version of the Holder norm, ht| ◦ |dt/dsi = ht| ◦ [J(t)] ◦ |[ti = κ ht| ◦ |ni = 0. (8.176) £ ¤ The concept can be extended to a constant anisotropic metric η jk with arbitrary signature, q­ £ ¤ ® λF renet = Xk ◦ η jk ◦ Xk p = 2, n = 1 (8.177) metric η jk = 0, j 6= k, η kk = ak , t = X/λF renet £ ¤ ® ­ t ◦ t = Xk ◦ η jk ◦ Xk /λ2F renet = 1.

(8.178) (8.179) (8.180)

If the renormalized vector is used as the coefficients of a 1-form, then a correspondence can be made between Frenet - Serret theory of a space curve and the evolution of a hypersurface. 8.5

The Gauss Integrals (2-forms)

The basic issue is that not all divergence free fields (differential forms) are exact. It is true that all divergence free fields in a three-dimensional Euclidean topology are exact, but that is precisely where the topological features enter into the picture. A Euclidean topology is simply connected and without obstructions. Even in three dimensions (and with Euclidean dogma) there are still two species of 3-component fields. Every one learns from Gibbs vector analysis that the 3-vector of angular momentum L = r × p is never to be added to the 3-vector of momentum, p. The angular momentum and the linear momentum are "two different species" of vectors (direction fields). However, with regard to non-Euclidean topological domains there is also another concept, defined as a vector density. There is a topological difference, even in 3D, between a covariant tensor and a contravariant

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tensor density, but not detectable if volume deforming processes are excluded (the typical non-dissipative case). To demonstrate these ideas, first consider ordinary three-dimensional vector fields and a C1 map of a 3-space onto a Euclidean domain of three dimensions, φ : {x, y, z} ⇒ {U, V, W } = φj (xk ).

(8.181)

This map defines a vector field, Z = [U(x, y, z), V (x, y, z), W (x, y, z)]. The (3 × 3) Jacobian matrix is defined by the equations, ¤¯ ® £ dφ : {dx, dy, dz} ⇒ {dU, dV, dW } = [J] ◦ |dri = ∂φj /∂xk ¯dxk .

(8.182)

(8.183)

Next construct the volume element generated by these three functions, Ωz = dUˆdV ˆdW = det[J]dxˆdyˆdz,

(8.184)

and the (N-1=2)-form density,† G = i(Z)dUˆdV ˆdW = UdV ˆdW − V dUˆdW + W dUˆdV = Dx dyˆdz − Dy dxˆdz + Dz dxˆdy.

(8.185) (8.186)

The Vector Z induces a preimage, D, on {x, y, z}. Formally, the vector, D, is defined in terms of the adjoint mapping, by the matrix equation, ¯ ® £ ® ¤adjoint ¯ Contra−variant tensor density ¯Dk (xm ) pb = ∂φj /∂xk ◦ ¯Z(xj ) . (8.187)

The functional substitution and pullback (pb) construction works even though the Jacobian map does not have an inverse. In this respect the retrodictive process [192] resembles the pullback of a 1-form, where the covariant tensor field is functionally well behaved with respect to the transpose mapping, ¤transpose ¯ £ ® Co − variant tensor |Ak (xm )ipb = ∂φj /∂xk ◦ ¯Z(xj ) . (8.188) Now, the extraordinary result is that if Z is rescaled by the divisor, λ(U, V, W ) = {aU p + bV p + cW p }3/p ,

(8.189)

then the 2-form density,

†

b = i(Z/λ)dUˆdV ˆdW, G = i(D/λ)dxˆdyˆdz,

(8.190) (8.191)

The 2-form G as used above is not the same as the field intensity 2-form F of electromagnetism.

The Gauss Integrals (2-forms)

381

b = 0. This result implies that the rescaled vector field, D= b D/λ, has is closed, dG zero divergence. The notation above is deliberate, for in four dimensions it distinguishes the electromagnetic Intensities, E, B, (as components of a covariant tensor deduced from Ak ) from the electromagnetic Quantities (or excitations) D, H (as components of a contravariant tensor density). The assumption of a Euclidean domain masks these topological features. The topological closure of |Di is the concept of zero divergence; the topological closure of |Ai is a zero curl concept. In 3D, for C2 differentiable fields where |Bi = curl |Ai , it follows that the closure of the 2-form generated from the components of |Bi is always empty, in a global manner! However, the closure of |Di need not be globally empty! On a space of three dimensions there are 2-forms of three components that are exact, and there are 2-forms of three components that are not exact. Although the 2-form with covariant components |Bi constructed from the curl of a vector potential A is closed and exact, the 2-form with tensor density components |D/λi is closed, but not necessarily exact. The fundamental idea is that for a non-bounding closed cycle (nbcc) (such as formed by a closed twisted ribbon), ZZ ZZ B ◦ d(Area) = 0, but D ◦ d(Area) 6= 0, (8.192) nbcc

nbcc

where for a boundary (such as toroidal surface), ZZ ZZ B ◦ d(Area) = 0, and boundary

boundary

D ◦ d(Area) = 0.

(8.193)

If the integration chain is closed in the sense of cycle, and is not a boundary, then there must exist points of the integration domain which must be excluded. These points form the topological defects (the point charges in EM theory or "topological holes") or the topological obstructions that are of interest to the theory of "Links and Braids". In particular, the theory of links depends upon such obstructions and is represented by integrals of the form, RR RR z (D dyˆdz − Dy dxˆdz + Dz dxˆdy)/λ = G 6= 0, dG = 0, (8.194) Lk = nbcc

nbcc

and should have nothing to do with magnetic flux, RR RR (Bx dyˆdz − By dxˆdz + Bz dxˆdy) = F = 0, dF = 0, Φm = nbcc

(8.195)

nbcc

which has no obstructions, as the integrand is globally exact. If F was to have obstructions, the pre-images global postulate of potentials, F − dA = 0, must fail, and the conservation of flux would not be true. Such a failure implies the existence of magnetic monopoles (the obstructions to F being globally exact). My personal

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

personal view (along with E.J.Post and many others) subsumes that the failure to detect magnetic monopoles is proof that classical electromagnetism is defined by the postulate of potentials; i.e., F − dA = 0, globally. On the other hand, the 2-form of field excitations, G, is not exact. 8.5.1 Example 1. The Gauss Link Integral The first application is to the divergence free vector field on three dimensions which is not exact, but is closed, and requires three functions for its description. The generic form for the integral of interest is given by the expression, ZZ ZZ Lk = i(Z)Ωz = (UdV ˆdW − V dUˆdW + W dUˆdV )/λ. (8.196) closed

As an example of the Gauss integral, Lk, consider the case where the displacement vector is the difference of two position vectors to two separate space curves. Define, Z = (R2 − R1 ), R2 = [x2 , y2 , z2 ] , R1 = [x1 , y1 , z1 ] , λ = (a(x2 − x1 )p + b(y2 − y1 )p − +c(z2 − z1 )p )3/p ,

(8.197) (8.198)

where R1 defines the position vector to one field of space curves, and R2 defines the position vector to a second field of space curves. Space curves from different families can have different parametrizations. Hence, the vector Z represents the vector difference of points on two different space curves which cannot be synchronized parametrically. Next assume that the displacements of interest are constrained by two parametric curves given by the exterior differential system, dR1 − V1 dt = 0 and

dR2 − V2 dt0 = 0,

(8.199)

where the parameters dt and dt0 are not functionally related, such that: dtˆdt0 6= 0, but dtˆdt = 0 and dt0 ˆdt0 = 0.

(8.200)

The vector D can be interpreted‡ as the displacement vector between points on the space curve C1 parametrized by t, and the points on another space curve C2 parametrized by t0. The integral to be evaluated is, ZZ ZZ Lk = Γ= i(Z/λ)d(x2 − x1 )ˆd(y2 − y1 )ˆd(z2 − z1 ) closed closed ZZ = (1/λ)(V21 − V11 )ˆ(V22 dt0 − V12 dt)ˆ(V23 dt0 − V13 dt) + ... Z Zclosed = (1/λ)(V21 − V11 )ˆ(V22 V13 − V12 V23 )dtˆdt0 + ... , (8.201) closed

‡

D is the pullack on x, y, z space of Z on U, V, W space.

The Gauss Integrals (2-forms)

383

using dtˆdt0 6= 0, but dtˆdt = 0 and dt0ˆdt0 = 0. Rewriting the formula using the isotropic Gauss format, a=b=c=1, p=2, leads to the classic Gauss Linkage formula, RR HH Lk = G= {(R2 − R1 )◦V1 × V2 }dtˆdt0 /λ, (8.202) closed

t t0

λ = (R1 ◦ R1 − 2R1 ◦ R2 + R2 ◦ R2 )3/2 .

(8.203)

However, the zero divergence formula works for the anisotropic case, for any a,b,c and for any exponent p. From Stokes theorem, if the closed two-dimensional integration domain is a boundary of a three-dimensional domain, then the Link integral vanishes. However, if a particular integration chain is a closed cycle (not a boundary of a three-dimensional domain) then the linking integral has values with rational ratios. These closed integrals are deRham period integrals in two dimensions. Points where D vanishes are excluded. When the two curves are distinct, the integration is over the two bounding cycles of a closed ribbon. The ribbon surface is closed but it is not a boundary of any volume. Then the two non-intersecting cycles (that form the boundary of the ribbon area) are defined by the two distinct parameters, dt, and dt0 . When integrations are computed along these closed curves whose tangent vectors are V1 and V2 , then the integer values of the closed integral may be interpreted as how many times the two curves are linked. The interpretation of the closed surface integral as a orientable ribbon works if the triple product divided by lambda does not change sign as t and t0 are varied. If the integrand changes sign, then the ribbon in non-orientable. The constraint that dtˆdt0 6= 0 implies that the "motion" along the curve generated by R1 is independent of the "motion" along the curve generated by R2 . If the curve generated by R1 is a conic in the x-y plane and the curve generated by R2 is a conic in the x-z plane, then the surface swept out by the vector D is a Dupin cyclide. Such surfaces have application to the propagation of waves in electromagnetic systems. 8.5.2 Example 2: Flat tangential developables From another point of view, consider the ruled surface defined by the vector field of two parameters, {t, µ} (isotropic, a=b=c=1, p=2). The ruled surface will be defined by the position vector R(t) to a space curve and a ruling parameter µ times the tangent Velocity vector to the space curve, V(t). Use the general methods above to create the doubly parametrized divergence free vector field, Z(µ, t) = {R(t) ± µV(t) }, λ(µ, t) = (R(t) ◦ R(t) ± 2µR(t) ◦ V(t) + µV(t) ◦ µV(t))3/2 .

(8.204) (8.205)

Vector fields of this type are primitive examples of "strings" for fixed values of the parameter, t, and string parameter, µ. Direct substitution of the physical constraints,

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

dR−Vdt = 0, and dV−Adt = 0, such that dZ = d{R(t)±µV(t) } into the definition of the linking integral, ZZ i(Z/λ)dZ 1 ˆdZ 2 ˆdZ 3 , (8.206) closed on N

leads to yet another realization and interpretation of the Gauss formula, RR RR G= {R◦µV × A}dtˆdµ/λ Q = closed

=

RR

closed

(8.207)

closed on µt

{A ◦ R×µV}dtˆdµ/(R ◦ R ± 2µR ◦ V + µV ◦ µV)3/2 .

It is apparent that the interaction of the "angular" momentum, L = R × µV, and the acceleration, A, produces a topological invariant whose values are "quantized" (in the sense that the ratios of the closed integrals are rational). Note that the triple vector product of the integrand numerator is proportional to the Frenet torsion of the orbit. For an orbit that is planar the Frenet torsion is zero everywhere, and the Gauss integral vanishes. Recall that if the space curve is an edge of regression, then the ruled surfaces associated with the forward and backward motions (the ± signs in the formula) are not the same to the second degree. Such a result demonstrates an obvious distinction between forward and backward motion that breaks time reversal symmetry. Linear rulings in one direction are on 1 sheet of the ruled surface, and rulings in the opposite direction are on the other surface. The two surfaces meet at an edge of regression. Similar time reversal symmetry breaking effects have been observed macroscopically in dual polarized ring lasers. 8.5.3 Example 3. Scrolls The two parameter surface described above is closely related to the ruled surface known as the tangential developable. Such ruled surfaces (parametrized by arclength s rather than time, t, and with the directrix of the ruling in the direction of the unit tangent vector, and multiplied by µ) have zero Gauss curvature. Though bent, such surfaces can be rolled out flat. By constructing the ruled surfaces in terms of the normal and/or binormal to a space curve, other forms of ruled surfaces yield negative values for the Gauss curvature of the surface, and are not "flat". They are defined as Scrolls. Of particular interest to physics are those ruled surfaces of negative Gauss curvature, which are also minimal surfaces. They have application in describing hydrodynamic wakes. These surfaces can be viewed as double edged ribbons for given values of µ. The equations for the ruled surface of a scroll, with a directrix in the direction of the binormal, b(s), are, D(µ, s) = {R(s) ± µb(s)}/λ, λ = (R(s) ◦ R(s) ± 2µR(s) ◦ b(s) + µb(s) ◦ µb(s))3/2 .

(8.208) (8.209)

Braids, Spin and Torsion-Helicity (3-forms)

385

When the parameter µ takes on the constant values µ = κ/τ 3 , (with κ = the Frenet curvature, and τ = the torsion of the space curve) then the ruled surface is a minimal surface, and the binormal field twists about the space curve generated by R(s). Another interesting scroll is that generated by the Darboux vector, D(θ, s) = {R(s) ± (n(s) cos(θ) + b(s) sin(θ)}/λ, λ = (D ◦ D)3/2 ,

(8.210) (8.211)

which seems to be of interest to Longcope [135]. 8.6

Braids, Spin and Torsion-Helicity (3-forms)

8.6.1 Chaos and the Unknot Much interest of late has been shown in knot theory and its application to an understanding of the trajectories of dynamical systems. The conjecture is that somehow an understanding of knot theory will give a better understanding of chaos. Counter intuitively is the idea that chaos is to be related to the unknot. Of particular interest will be those cases where lines of vorticity have an oscillatory Frenet torsion with a period equal to 2/3 of the fundamental period of closure. The topological Gauss integral will average to zero for such systems; but these systems can be created by continuous deformations of folding and twisting a closed loop of vorticity, producing a period 3 system which is known to be related to chaos [295]. In the non-deformed circular state, tubular neighborhoods guided by the vortex lines can continuously evolve into domains without stagnation points or tangential singularities, or knots, or twists. However, when the closed vortex line is in the deformed period 3 configuration, tangential (hyperbolic) singularities are created by the flow lines of the velocity field, and the evolution becomes highly convoluted and chaotic (see Figure 8.1). These topological features may be demonstrated visually by taking a long strip of paper and wrapping the strip three times around your fingers. Close the strip by going under one strand and over the next before pasting together. The strip is of obvious period three. Now slide the closed strip from the fingers and note that it can be deformed continuously into a cylindrical strip without twists or knots (Spin-0). If the same procedure is used, except that a double over or a double under crossing is used before pasting the strip ends together, the resulting closed loop will have a continuously irreducible 4π twist (Spin-2). Both the Spin-2 and the Spin-0 strips have a zero Euler characteristic. However, the Spin-2 strip can be continuously deformed into a Klein bottle, or a double lapped Mobius band, and is not homeomorphic to the Spin zero strip [187]. If a model of the Spin-0 and Spin-2 systems (deformed to their period 3 configurations) is made from a copper tube, and if flexible bands are created to link any pair of neighboring tubular strands, then it is readily observed that the paired domain twists and folds as it is propagated unidirectionally along the vortex lines.

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

For the Spin-2 system the flexible bands will return to their original state in 3 revolutions. However, the paired domain continues to twist and fold, becoming ever more complicated as it follows the evolution around the Spin-0 configuration. The folded Spin-0 system has chaotic neighborhoods. This result indicates that the source of chaos in dynamical systems may be due to the unknot, and not the knot! The Cartan theory thereby predicts that the source of chaos in turbulent systems does not require a discontinuous cut and connect process, but may be induced by vortex lines that continuously evolve by twisting and folding into a closed, Spin-0, period three configuration. 8.6.2

The Torsion 3-form and the Braid integral

For n = 4 the same procedures used above can be used to produce a period integral over a closed three-dimensional domain. In fact, the same vector field that is used to define the Cartan 1-form of Action may be used to construct a dual (N-1)-form that is closed. The algorithm is to substitute for the functions of the vector field, V, the functions that make up the covariant 1-form of Action, A. This construction is equivalent to constructing the Jacobian matrix of the original vector field on the N-dimensional velocity space, computing its cofactor matrix, multiplying the original vector by the cofactor matrix, and then dividing by the quadratic form, λ. When these operations are completed, functional substitution will lead to an conserved axial vector current density on (x,y,z,t). Another form of the topological integral invariant is constructed in the following way. First, for the classic Cartan action, A = Pk dxk − Edt/c, construct the N-volume, Ω = −dPx ˆdPy ˆdPz ˆdE/c. Next contract Ω with the vector, (P x, P y, P z, −E/c), and then divide by λ = {±P ◦ P ± (E/c)2 }2 . For sake of simplicity, assume that E/c is a constant such the dE = 0. Then the closed 3-form or current becomes equivalent to, J = (E/c)dPx ˆdPy ˆdPz /λ

with

dJ = 0.

(8.212)

Now invoke the same Cartan trick of individual parametrization as used above. Consider a total momentum vector composed of three individual vector components, P = p1 +p2 +p3 . Assume that the Cartan topology is constrained in such a way that for each vector component a Newtonian kinematic law of parametrization is maintained such that, dp1 −f 1 dt = 0, dp2 −f 2 dt0 = 0,

dp3 −f 3 dt00 = 0.

(8.213)

Also note that dtˆdt0 ˆdt00 6= 0; that is, the parameters used in the Newtonian kinematic descriptions are not sychronizable. If they were functionally related the value of J must be zero. Substitute these expressions into the equation for the closed current J and integrate over a closed three-dimensional chain to yield a triple Braid

Braids, Spin and Torsion-Helicity (3-forms)

387

integral, Braid =

H

H3

H J = (E/c)dPx ˆdPy ˆdPz /λ 3

= (E/c){ f1 ◦ (f2 × f3 )} dtˆdt0 ˆdt00 / {±P ◦ P ± (E/c)2 }2 .

(8.214)

3

The integrations are now over three closed curves whose tangents are the Newtonian forces, f, on three "particles". Where in the two-dimensional Gauss integral, of the previous section, the evaluation was along the closed curves of two particles that formed the ends of a string, in this case the integrations are along the closed trajectories of three "particles" which form the vertices of a triangle. In every case, the trajectories are the trajectories of a system of limit points. The idea that three "lines" are used to form the integral (whose values form rational ratios) is the reason that this topological integral in the format given above is defined as the braid integral. Of course the 3-form of topological torsion is a variant of the braid integral, but applies to those topologies where the system is not reducible to three factors dt, dt0 and dt” (such systems are said to have torsion cycles). An example of a period 3 braid with Braid integral zero (chaotic) and Braid integral 2 (non-chaotic) is given in Figure 8.2 below.

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

Figure 8.2 Chaotic and non-chaotic braids The equivalent to Figure 8.2, and the fact that there are two distinct period 3 configurations, one chaotic and one non-chaotic, was brought to my attention during a stimulating lecture given by J. Los at the August, 1991, Pedagogical Workshop on Topological Fluid Mechanics held at the Institute for Theoretical Physics, Santa Barbara UCSB. It is to be noted that the 3-form of topological torsion is related to the braid integral, a three-dimensional thing in four dimensions, and not the Gauss linkage integral, which is a two-dimensional thing in three dimensions. The literature of helicity is sometimes confused on this point; the helicity integral is related, incorrectly, to the linkage integral. 8.6.3

Braids

In slightly different notation, for n = 4, the same general procedures described above may be used to produce a period integral over a closed three-dimensional domain. The technique is to define a four-dimensional vector field, Z =[Z1 , Z2 , Z3 , Z4 ]. Then use the general Holder renormalization function, λ = {αZ1p + βZ2p + γZ3p + Z4p }n/p ,

(8.215)

Braids, Spin and Torsion-Helicity (3-forms)

389

and set n = 4, for zero 4-divergence of the renormalized current, Z/λ. The 3-form so constructed is closed, J = i(Z/λ)dZ1 ˆdZ2 ˆdZ3 ˆdZ4 , dJ = 0.

(8.216) (8.217)

Assume the 4-component vector has a realization as, (8.218)

Z = P1 +P2 +P3 ,

where the three independent fields P represent three space-time curves that obey the kinematic constraints, dP1 −f 1 ds = 0, dP2 −f 2 ds0 = 0,

dP3 −f 3 ds00 = 0.

(8.219)

Substitute for each of the differentials in J (and further assume that the domain {x,y,z,t} of interest is further constrained such that dt = 0) to yield the 3-form, Br = { f1 ◦ (f2 × f3 )}dsˆds0 ˆds00 /λ, λ = {αZ1p + βZ2p + γZ3p + Z4p }4/p . The spatial braid integral becomes equal to, HHH f1 ◦ (f2 × f3 )}dsˆds0 ˆds00 /λ. Braid Integral =

(8.220) (8.221)

(8.222)

s s0 s00

The integrations are now over three closed curves whose tangents are the "Newtonian forces", f, on three "particles". Where in the two-dimensional Gauss integral, of the previous section, the evaluation was along the closed curves of a ribbon, in this case the integrations are along the closed trajectories of three "particles" which form the vertices of a triangle. The idea that three "lines" are used to form the integral (whose values form rational ratios) is the reason that this topological integral in the format given above is defined as the braid integral. Of course the 3-form of topological torsion is a variant of the braid integral, but applies to those topologies where the system is not reducible to three factors ds, ds0 and ds00 . An example of a period-3 braid with Braid integral B0 (chaotic) and Braid integral B2 (non-chaotic) is given in the Figure 8.1. Another interesting feature can be generated by assuming that the 3 different braid segments become connected into 1 closed chain but of period-3. It is illuminating to construct the two braids by wrapping a long flat ribbon of paper smoothly around the palm of your hand. Close the ribbon surface by pasting the ends together. Then make another example, where this time thread the loose end underneath the middle wrap, rather than over the middle wrap, before gluing the ends together. Take the two examples from your hand and note that one is continuously deformable

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

into a closed cylinder (Tw = zero) while the other has a 4π twist (Tw = 2). What is surprising is that it is the Tw = 0 configuration that has a chaotic neighborhood, while the Tw = 2 structure is not chaotic. To test for chaos construct the equivalent of the closed braid from copper tubes. Then link any pair of tubes with a large loop of elastic or thread. Push the looping thread around the period-3 copper tube, and note that for a Tw = 2 configuration, the looping thread becomes untangled after 6π revolutions about the central axis. For the Tw = 0 configuration, the looping thread never unwinds, but becomes more and more twisted and complex. It is to be noted that the 3-form of topological torsion is related to the braid integral, a three-dimensional thing in four dimensions, and not the Gauss linkage integral, which is a two-dimensional thing in three dimensions. 8.7

Electrodynamic Applications

8.7.1 Phase-Polarization and Orientability in 4D The physical experiments yielding states of matter which are superconducting, and states of matter that emit laser coherent radiation indicate that these properties are deformation invariants and therefore should have a topological basis. In the superconductor, it has been suggested that the electrons form a Boson pair with Spin zero, such that the orientability feature of Fermion spin-up or spin-down enantiomorphism has been lost (or averaged to zero). In the laser, the radiating electrons form a phase-polarization coherent state, such that the concept of phase-polarization differences (between electrons) is "averaged to zero". What (different?) topological features are to be associated with orientability and phase? Topological Spin and the Spin quantum - a closed 3-form density A topological basis for Electromagnetism [216] is based upon the differential system, F − dA = 0 and

J − dG = 0.

(8.223)

The system admits 3 fundamental 3-forms, The charge-current density 3-form, J = dG, the Topological Torsion Helicity 3-form, H = AˆF, the Topological Spin 3-form, S = AˆG.

(8.224) (8.225) (8.226)

The first 3-form is closed globally (by construction), but the second and third 3forms are not closed, necessarily. However, in a space of four dimensions, the latter two 3-forms admit integrating factors that will cause the 3-forms to be closed and homogeneous of degree zero. On space-time of four dimensions, starting from an electromagnetic Action 1-form, there are two important 3-forms, the 3-form of Topological Torsion and

Electrodynamic Applications

391

the 3-form density of Topological Spin. Each of these 3-forms are represented by a 4D direction field, and involve the 1-form of Action as a factor in the sense of an exterior product. To relate the above definition of a 3-form of Helicity with the six-dimensional formulation involving the Biot-Savart substitution (to me) is an extraordinary constraint on the topology of the domain. The substitution effectively mixes a tensor and a tensor density, where definition 1 above mixes a tensor with a tensor, AˆF . There is however, another well defined electromagnetic 3-form, AˆG , which mixes a tensor (A) and a tensor density (G) . I call AˆG —with physical dimensions of angular-momentum— the Spin 3-form. (The 3-form, AˆF, has physical dimensions of Angular momentum divided by Ohms.) In electrical engineering notation, Spin : S4 = [A × H + φD, A ◦ D], T orsion_Helicity : T4 = −[E × A + φB, A ◦ B].

(8.227) (8.228)

The closure of AˆG defines a measure known as the first Poincare 4-form, P 1 := (B ◦ H − D ◦ E) − (A ◦ J − ρφ) ≡ d(AˆG),

(8.229)

while the closure of AˆF yields the second Poincare 4-form, P 2 := 2(E ◦ B) ≡ d(AˆF ).

(8.230)

When these measures vanish (the divergences of the 4-vectors vanish), then there exist separate topological conservation laws (of Spin and Helicity). The question arises, can the 1-form of Action be multiplied by a closure factor, not for the 1-form A, but for the 3-form AˆF, or the 3-form density, AˆG. The answer is obviously yes, and the appropriate closure factor will have the functional form of the appropriate Holder norm. For example, the Topological Torsion direction field is generated by the exterior product A0 ^F0 = A0 ^dA0 . If the 1-form A0 is divided by the renormalization factor, β, then the Topological Torsion direction field A^dA is proportional to the direction field A0 ^dA0 , AˆdA = (A0 /β)ˆd(A0 /β) = (1/β 2 )A0 ˆdA0 .

(8.231)

As A0 ˆdA0 = T x dyˆdzˆdt − T y dxˆdzˆdt + T y dxˆdyˆdt − T t dxˆdyˆdz,

(8.232)

the 3-form, A0 ˆdA0 , can be rewritten as, A0 ˆdA0 = Z x dZ y ˆdZ z ˆdZ t − Z y dZ x ˆdZ z ˆdZ t +Z z dZ x ˆdZ y ˆdZ t − Z t dZ x dZ y ˆdZ z ,

(8.233)

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

where the components Z k may be computed as, |Z n (x, y, z, t)i = [∂Z m /∂xn ]−1 ◦ |T m (x, y, z, t)i .

(8.234)

The renormalization factor then can be computed from the algorithms given above, such that, (8.235) λ = β 2 = {a(Z x )p + b(Z y )p + c(Z z )p + e(Z t )p }4/p . Hence, β = ±{a(Z x )p + b(Z y )p + c(Z z )p + e(Z t )p }2/p ,

(8.236)

A0 = ydx − xdy + zdt − tdz, dA0 = 2dyˆdx + 2dzˆdt, A0 ˆdA0 = −{+xdyˆdzˆdt − ydxˆdzˆdt +zdxˆdyˆdt − tdxˆdyˆdz}, dA0 ˆdA0 = −4dxˆdyˆdzˆdt.

(8.237) (8.238)

is a closure factor for the 3-form, A0 ˆdA0 , such that, d(1/β 2 )A0 ˆdA0 = 0. It is important to note that β need not be a closure factor for the 1-form, A0 ; that is, d(A0 /β) 6= 0. Example: Consider the (Hopf) 1-form and its Pfaff sequence:

(8.239) (8.240)

The Topological Torsion vector is, T = −[x, y, z, t],

(8.241)

Z = −[x, y, z, t],

(8.242)

which leads to, such that the closure factor, β, becomes β = ±{a(x)p + b(y)p + c(z)p + e(t)p }2/p .

(8.243)

The renormalized Action which produces a closed topological Torsion 3-form is of the format A0 = {ydx − xdy + zdt − tdz}/ ± {a(x)p + b(y)p + c(z)p + e(t)p }2/p .

(8.244)

As d(A0 /β)ˆd(A0 /β) = 0, d(A0 /β) 6= 0, the 3-form of topological torsion is closed globally; but the 2-form F is not zero, F = d(A0 /β) 6= 0. The implication is that there are domains for which the 3-form can be used as a period integral, but the domain supports finite E and B fields. The components of the Topological Torsion vector are proportional to the position vector from the origin.

Electrodynamic Applications

393

The closure of these two 3-forms is not necessarily zero, for, d(AˆF ) = F ˆF = 2(E ◦ B)dxˆdyˆdzˆdt,

(8.245)

and d(AˆG) = F ˆG − AˆJ = {(B ◦ H − D ◦ E) − (A ◦ J − ρφ)}dxˆdyˆdzˆdt. (8.246) Each of these equations can be interpreted in terms of a 4-component vector field, Z4 , and each 3-form admits an integrating factor such that the closure ("divergence") is zero. In such situations, the 3-form is homogenous of degree zero in terms of the component functions of Z4 . 8.7.2 Topological Torsion and the Polarization quantum - a closed 3-form Perhaps it is more important that the theorem of Topological deformation invariance of the closed integrals of the 3-form is also valid in any dimension for which F ˆF is zero. In higher dimensions the integral must be evaluated over 3D-submanifolds that need not be space-like. For four dimensions the 3-form AˆF has 4 components and can be constructed as, Topological Torsion-Helicity: AˆF = i(T4 )dxˆdyˆdzˆdt,

(8.247)

where, T4 = −[E × A + φB, A ◦ B].

From this formulation it must be remembered that the components of this "vector" transform as a third rank covariant tensor field; A ◦ B is merely the fourth component. In the four-dimensional literature such objects are often described as pseudo vectors. They are not pseudo vector densities. If the 3-form, AˆF, is to be an evolutionary invariant, then RRR RRR L(βV) AˆF = di(βV)AˆF + i(βV)d(AˆF ), (8.248) closed closed RRR i(βV)F ˆF, (8.249) = closed RRR = 2 · (E ◦ B)i(βV)dxˆdyˆdzˆdt. (8.250) closed

RRR If the closed integral of the Topological Torsion, AˆF , is to be an closed evolutionary deformation (hence topological) invariant for ANY process, V, then E ◦ B = 0. This requirement, which is equivalent to F ˆF = 0, implies that the Topological Parity must vanish. The condition is sufficient, but not necessary, for topological invariance of the Torsion-Helicity integral, even when the fields are explicitly time dependent. The relative integral invariant in 4D is, RRR H = AˆF, (8.251) closed RRR = {Tx dyˆdzˆdt − Ty dxˆdzˆdt + Tz dxˆdyˆdt closed

−A ◦ Bdxˆdyˆdz}.

(8.252)

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

In all cases the 3-divergence of B vanishes, and the 2-form F is closed, for it is exact, dF = 0. For isochronous domains, dt = 0, and the integral reduces to the standard spatial format of plasma physics. However, there are example electromagnetic fields for which A ◦ B = 0, and yet T4 is not zero. Example: Define A = [0, 0, (x2 + y 2 )/2], φ = z.

(8.253)

Then, B E A◦B T4

= = = =

[−y, x, 0], [0, 0, −1], 0, E ◦ B = 0, −[zB, 0].

(8.254) (8.255) (8.256) (8.257)

Note that in four dimensions (that admit time dependent fields), the frozen-in lines are the lines of the Torsion vector. The Torsion vector is dominated by the B field, but also can have a component due to E × A. The lines of T4 can continuously evolve, with most regions in a plasma dominated by φB 6= 0, E × A → 0. Then as B lines approach one another, induction causes E × A 6= 0, φB → 0. The "B lines" terminate on a null (a boundary point), break apart, and then possibly reconnect different segments, after which φB 6= 0, E × A → 0. (This mechanism appears to be at the foundation of the numeric technique investigated by Horning [99].) Special situations become evident when the integration domain is compact with a boundary, for then Stokes law may be applied, and deformation invariance requires that i(βV)AˆF =0 on the boundary. These are interesting but special cases which are invariants of only a special choice of boundary conditions. For example, if β(x, y, z) = 0 defines the boundary, then for deformation invariance of the integral it must be true that the function β also must be an evolutionary invariant, such that L(V) dβ = i(V)dβ = 0. Classically the function β which is used to define the boundary, is not arbitrary, but must be a first integral of the evolutionary vector field. For such special cases, the field on the boundary need not be tangential! There are other special situations as well. For integration domains which are open, the criteria for absolute integral invariance is much more severe, and requires that d(i(βV)AˆF ) = 0. This constraint is to be recognized as the criteria that the evolutionary vector field βV be an element of the symplectic group. I have demonstrated that all such evolutionary processes are thermodynamically reversible. 8.7.3 The Flux or Circulation Integral 1-form For the Cartan topology constructed from a fundamental 1-form of Action and a fundamental (N-1)-form of Current, several period integrals of closed forms integrated over closed chains appear in a natural manner. In particular on an (N=4)-dimensional

Electrodynamic Applications

395

domain, the four period integrals of most interest are the period integrals of flux (circulation), charge, spin and torsion. The fundamental period integral over a closed 1-form will be defined as the "Circulation" or "flux" integral. When the Pfaff Topological dimension is 2, there exists a submersive map to two dimensions, and the vector fields on this domain will have two irreducible components, say [Φ(x, y, z, t), Ψ(x, y, z, t)]. Following the procedure of the preceding section, construct the two-dimensional volume element defined as Ω = ρdΦˆdΨ. Then as n − 1 = 2 − 1 = 1, the 1-form, A, becomes, A = (ΦdΨ − ΨdΦ)/{±aΦp ± bΨp }2/p . The exterior differential of such a 1-form is exactly zero for all point sets that exclude the null set of the denominator. The classic choice is for p = 2, and a = 1, b = 1, (+,+) signature. The closed integrals of these closed 1-forms then can be expressed as, H H Circulation Γ = A = (ΦdΨ − ΨdΦ)/{Φ2 + Ψ2 }. (8.258) 1

1

By substituting the functional forms in terms of (x,y,z,t), the circulation integral can be written in terms of functions on (x,y,z,t) and their differentials, {dx, dy, dz, dt...}.

As an example, suppose that the domain is three dimensional, N=3. Then the zero sets of Φ(x, y, z) = 0 and Ψ(x, y, z) = 0, represent 2 two-dimensional surfaces which may or may not have one or more lines of intersection. If the surfaces intersect, then, Intersection = dΦˆdΨ 6= 0. (8.259) If the closed integration paths cannot be contracted to a point, because they encircle these lines of intersection, the values of the integrals have rational ratios depending on how many lines are encircled and how many times the integration path encircles a line. The lines of intersection must have zero divergence (and therefore must stop or start on boundary points, or are closed on themselves). Otherwise the integration chains can be deformed and then contracted to a point. The classic example is given by the 1-form, A = (ydx − xdy)/(+x2 + y 2 ) in three dimensions. For integration H contours that encircle the z-axis, the value of Γ = 1 A = 2π. In hydrodynamics, this vector field is called a potential (or Abrikosov) "vortex", even though the vorticity ω =curlv = 0. Stokes theorem does not apply as the closed integration chain is a cycle that is not a boundary. 8.7.4 The Charge Integral 2-form Many different options exist for construction of these invariant topological structures from closed p-forms. The idea is to find a formulation for a closed form on a domain, and then to specify a closed and compatible integration chain. The integration chain need not be a boundary, but only a closed cycle. For example, from the components of the specified vector, Aµ , the Jacobian matrix, [∂Aµ /∂xν ] can be constructed. The rows or columns of the matrix of cofactors of the Jacobian (the adjoint matrix) forms a set of vector fields that have zero divergence [268], and therefore these vectors could

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

be used to construct relative integral invariants. In every case there exists an algebraic construction which produces a vector that is divergence free and whose line of action is uniquely related to original vector that was used to construct the Cartan topology. That vector may be constructed by multiplying the original vector Aµ by the matrix of cofactors and then dividing by the function λ defined above. The construction replicates the previous procedure. As an application for n = 3, p=2, consider the vector that represents the difference between two space curves, z = R2 −R1 . Then compute the 2-form G(z) from the "volume" element Ω = dz 1 ˆdz 2 ˆdz 3 /λ, to give, Gn=3 = {z 1 dz 2 ˆdz 3 − z 2 dz 3 ˆdz 1 + z 3 dz 1 ˆdz 2 }/λ,

(8.260)

λ = (±(z 1 )2 ± (z 2 )2 ± (z 3 )2 )3/2 .

(8.261)

where Next assert that the displacements of interest are constrained by two parametric curves given by dR1 = V1 dt

and

dR2 = V2 dt0 ,

(8.262)

where the parameters dt and dt0 are not functionally related (which would imply that dtˆdt0 = 0). It is important to realize that kinematic constraints are topological constraints that refine the Cartan topology, a topology based solely upon the specified 1-form of Action, A. From a physical point of view, these constraints can be interpreted as constraints of null fluctuations and in certain circumstances can be associated physically with the limit of zero temperature. To demonstrate the utility of such constraints, substitute these differential expressions into the expression for the 2-form G of "current" in N=three dimensions, and carry out the exterior products, using dtˆdt0 6= 0, but dtˆdt = 0 and dt0ˆdt0 = 0. The result is the vector triple product representation for the Gauss integral, H H Q = G = {z ◦ V1 × V2 }dtˆdt0 /(R1 ◦ R1 − 2R1 ◦ R21 + R2 ◦ R2 )3/2 . (8.263) 2

2

The integration domain is the closed "two-dimensional area" formed by the displacements along the non-intersecting curves defined by the two distinct parameters, dt, and dt0 . This double integral is to be recognized as the Gauss linking integral of Knot Theory [81]. (Without the kinematic substitutions, it may also be interpreted as the charge integral of electromagnetic theory.) When integrations are computed along closed curves whose tangent vectors are V1 and V2 , then the integer values of the closed integral may be interpreted as how many times the two curves are linked. Note that the same integer result is obtained when the vector z is interpreted as the sum of the two vectors, z = R2 + R1 , although the values of the integrals have different scales.

Electrodynamic Applications

397

The constraint that dtˆdt0 6= 0 implies that the "motion" along the curve generated by R1 is independent of the "motion" along the curve generated by R2 . If the curve generated by R1 is a conic in the x-y plane and the curve generated by R2 is a conic in the x-z plane, then the surface swept out by the vector z is a Dupin cyclide. Such surfaces have application to the propagation of waves in electromagnetic systems. From another point of view, consider the ruled surface [251] defined by the vector field of two parameters, z(µ, t) = R(t) ± µV(t).

(8.264)

Vector fields of this type are primitive types of "strings" for fixed values of the parameter, t, and string parameter, µ. Direct substitution of the physical constraints, dR − Vdt = 0, and d(V) − Adt = 0 leads to the topological Gauss integral, H H Q = G = {R◦µV × A}/λ 2 2 H = {A ◦ R×µV}dtˆdµ/(R ◦ R ± 2µR ◦ V + µV ◦ µV)3/2 . (8.265) 2

It is apparent that the interaction of the "angular" momentum, L = R × µV, and the acceleration, A, produces a topological invariant whose values are "quantized" (in the sense that the ratios of the integrals are rational). Note that for the classical central field problem where the force (acceleration) and the angular momentum are orthogonal, the orbits are in a plane and the Gauss—linking number is zero. Further note that the triple vector product of the integrand is proportional to the Frenet torsion of the orbit. An orbit that is planar has Frenet torsion zero everywhere. The Gauss linking integral is a special case of the Gauss two-dimensional period integral of electromagnetic theory when the integration domains can be factored into independent products, dtˆdt0 6= 0. 8.7.5 Hedgehog fields, Rotating plasmas, Accretion discs Using Maple (see Volume 6, "Maple programs for Non-Equilibrium systems"), it is possible to find a modification of a closed 1-form solution to Maxwell’s equations that makes the magnetic field lines appear like the spines of a Hedgehog. It is also possible to demonstrate how such modifications of closed 1-forms make the z=0 plane of a rotating plasma a chiral attractor. Consider, A = Γ(x, y, z, t)[−y, x, 0]/(x2 + y 2 ) , p with Γ = −z m/ (x2 + y 2 + z 2 ), and φ = 0.

(8.266) (8.267) (8.268)

These potentials induce the field intensities, E = [0, 0, 0], B = m [x, y, z]/(x2 + y 2 + z 2 )3/2 .

(8.269) (8.270)

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Periods on Manifolds, Quantization, Homogeneous p-forms, and Fractals

The B field is of the format of the famous Dirac Hedgehog field often associated with "magnetic monopoles". However, the radial B field has zero divergence everywhere except at the origin, which herein is interpreted as a topological obstruction. The factor is to be interpreted as an oblateness factor associated with rotation of a plasma, and is a number between zero and 1. It is apparent that the helicity density and the second Poincare 4-form are zero, E◦B=0

and

A◦ B = 0.

(8.271)

In fact, the 3-form of topological torsion vanishes identically (as φ = 0), T4 = [0, 0, 0, 0].

(8.272)

In this example, there is a non-zero value for the Amperian current density, even though the potentials are static. The Current Density 3-form has components, J4 = (3m/2µ) (1 − ) z [−y, x, 0, 0]/(x2 + y 2 + z 2 )5/2 ,

(8.273)

which do not vanish if the system is "oblate" (0 < < 1). This current density has a sense of "circulation" about the z-axis, and is proportional to the vector potential reminiscent of a London current, J = λA. The "order" parameter is (3/2µ) (1 − )/(x2 + y 2 + z 2 )2 . The Lorentz force can be computed as, J × B =(3m2 /4µ) (1 − )[xz 2 , yz 2 , −z]/(x2 + y 2 + z 2 )2 .

(8.274)

The formula demonstrates that the Lorentz force on the plasma, for the given system of circulating currents, is directed radially away (centrifugally) from the rotational axis, and yet is such that the plasma is attracted to the z = 0, x-y plane. The Lorentz force is divergent in the radial plane and convergent in the direction of the z-axis, towards the z=0 plane. This electromagnetic field, therefore, would have the tendency to form an accretion disk of the plasma in the presence of a central gravitational field. Although the 3-form of Topological Torsion vanishes identically, the 3form of Spin is not zero. The spatial components of the Spin are opposite to the direction field of the Lorentz force (in the sense of a radiation reaction), S4 = (m2 /4µ)[xz 2 , yz 2 , −z, 0]/(x2 + y 2 + z 2 )2 .

(8.275)

The components of the Spin 3-form are in fact proportional to the components of the virtual Work 1-form, W „ with the ratio −3(1− ) depending on the oblateness factor. It is also true that the divergence of the 3-form of spin is not zero, for the first Poincare 4-form is, d(AˆG) ⇒ P 1 = (m2 /4µ)(x2 + y 2 + 4(1 − ) z 2 )/(x2 + y 2 + z 2 )3 .

(8.276)

For a more detailed discussion, see Volume 4 "Plasmas and non-equilibrium Electrodynamics" [276].

Chapter 9 TOPOLOGY AND THE CARTAN CALCULUS 9.1 9.1.1

Why differential forms? Pair and Impair exterior differential forms

Exterior differential p-forms should not be confused with p-tensors. All exterior differential p-forms (with respect to diffeomorphic transformations) are either invariant scalars, or densities to within a factor. The exterior p-form densities are of two types, Pair and Impair. Impair p-form densities are sensitive to orientation and magnitude of the determinant of any diffeomorphic mapping of independent variables. Pair p-form densities are not sensitive to orientation changes but are sensitive to the magnitude of the determinant of the diffeomorphism. Scalar p-forms are not sensitive to orientation changes. The 2-form, F = dA, of electromagnetic theory is an example of a scalar p-form. The 2-form density, G, such that J = dG, is an example of an Impair 2-form density. Densities are defined to within a "factor", where the factor is dependent upon some power of the determinant ∆±k (or absolute value of the the determinant, |∆|±k ), of admissible diffeomorphic changes of coordinates. If the p-form density is sensitive to orientation, then its integrals are called pseudoscalars. Exterior differential forms are well behaved with respect to diffeomorphic mappings of the independent variables. This means that the functional form of the coefficients can be pushed forward from given initial data to predicted final data, or pulled back from given final data to retrodicted initial data. However, exterior differential forms are also well behaved with respect to a wider class of differentiable mappings. The larger class of functionally well behaved differentiable mappings includes those mappings that are not reversible, as well as diffeomorphisms - which are reversible. The mapping functions for such transformations are C1-differentiable, and produce a Jacobian matrix of partial derivatives, [J] , but the Jacobian need not be invertible. Such non-invertible differentiable transformations will be called "C1/diffeo" (C1 differentiable mod diffeomorphisms) transformations in order to distinguish them from invertible C1 "diffeomorphisms". The "C1/diffeo" transformations can be used to describe topological evolution, while the subset of diffeomorphisms cannot.

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Non-diffeomorphic C1 maps • Exterior differential p-form scalars are functionally well defined in terms of the PullBack operation. The PullBack process implies that the p-form scalars are defined on the final state with arguments in terms of final state independent variables. The mapping from initial to final state is presumed to be C1, but it is not necessarily invertible. A key feature of the pullback is the correspondence with the transpose of the Jacobian matrix. When applied to the theory of electromagnetism, it is evident that the Scalar forms are related to concepts of Forces and field intensities, E and B, and work. • Pair and Impair exterior differential p-form densities as defined on the final state of a C1 mapping, are functionally well defined in the sense of a pullback operation. A key feature of the pullback is the correspondence with the adjoint of the Jacobian matrix. When applied to the theory of electromagnetism, it is evident that the Impair forms are related to concepts of Sources, and field quantities, D and H. • Note that relative to the constraint of special or proper unimodular groups of transformations, where ∆−1 = +1, the two species of differential forms are not distinguishable. It is also remarkable that, relative to C1/diffeo transformations, the pullback is well defined, but the pushforward is not. Unique prediction fails, but retrodiction is deterministic. An "arrow of time" is built into the logic of differential forms. Diffeomorphic C1 maps (the Constraints of Tensor Analysis) • In tensor analysis, there exist two important categories of tensors, known as covariant tensors and contravariant tensors. These sets are distinguished by their transformation properties with respect to invertible diffeomorphisms. • The functional components of Covariant vectors are pushed forward from the initial state to the final state by means of the linear transformations induced by the Inverse of the Jacobian matrix of the diffeomorphism between independent variables. It is usually not emphasized that scalar exterior differential 1-forms with Covariant coefficients are pulled back by means of maps induced by the transpose of the Jacobian matrix. The pullback does not require that the Jacobian matrix have and inverse. • The functional components of Contravariant vectors are pushed forward from the initial state to the final state by means of the linear transformations induced by the Jacobian matrix of the diffeomorphism between independent variables. It is usually not emphasized that density exterior differential (N-1)-forms with

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401

Contravariant coefficients are pulled back by means of maps induced by the Adjoint of the Jacobian matrix. The pullback does not require that the Jacobian matrix have and inverse. A preferred notation for the two species of exterior differential forms (using ordered or collective indices) is demonstrated by a Scalar 1-form, A, an Impair (N1)-form density, J, a Scalar 2-form F and an impair 2-form density, G: A = a Scalar 1-form, A = Ak dxk . J = an "Impair" differential (N-1)-form density, J F F G

= = = =

ck ...dxN }. J k (x){dx1 ˆdx2 ...dx a Scalar differential 2-form, Fjk (x)dxj ˆdxk . an "Impair" differential (N-2)-form density,

cj ...dx ck ...dxN }. G = Gjk (x){dx1 ˆdx2 ...dx

(9.1) (9.2) (9.3) (9.4)

ck means that that term dxk is left out of the factors The hatted symbol dx that form the volume element. Note that the coefficients of the Impair forms is defined with an upper index which is a collective compliment of "left out" factors in the volume element. 9.1.2 Functional Substitution and the pullback Cartan’s theory of exterior differential forms is NOT just another notational system of fancy. The important fact, often ignored, is that a differential form is not necessarily a tensor. In short, a differential form is a mathematical object that is well behaved with respect to differentiable maps, y k ⇒ xµ = φµ (y k ), from an initial state {y k } of independent variables to a final state {xµ } of independent variables. Such maps need not be homeomorphisms. Hence the topology of the final state need not be the same as the topology of the initial state. A tensor is an object which is well behaved if and only if the differentiable map φµ from initial to final state has a differentiable inverse. Such maps are defined to be diffeomorphisms, a subset of homeomorphisms, and as such the topology of the final state and the initial state must be the same. Differential forms can be used to understand and study topological evolution, while tensors cannot. Contravariant and covariant tensors A tensor is a restricted class of mathematical objects in a multilinear £ µthat transform ¤ µ k manner relative to the Jacobian matrix [Jk ] = ∂φ (y)/∂y of the differentiable map, and its inverse. The motivation for the definitions is most transparent when

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¯ ® one considers the matrix inner product of two vectors, hAk (y)| ◦ ¯V k (y) , on the initial state. Suppose that the Identity matrix can be written in terms of two factors, [I] = [J]−1 ◦ [J] . Then the equation, ¯ ® ¯ ® ­ ¯ µ hAk | ◦ ¯V k ⇒ hAk | ◦ [Jµk ]−1 ◦ [Jµk ] ◦ ¯V k = Aµ (y)¯ ◦ |V (y)i, (9.5) can be interpreted in terms of a matrix transformation, where, ¯ ® µ |V (y)i = [Jµk (y)] ◦ ¯V k (y) ,

(9.6)

and

¯ ­ hAk (y)| ⇒ Aµ (y)¯ = hAk | ◦ [Jµk (y)]−1 ,

(9.7)

in such a way that the value of the matrix row-column vector product before the transformation and after the transformation has the same value. Note that the vector with components indexed by µ on the final state has arguments in terms of the independent variables on the initial state. In order to produce a vector array of functions on the final state, with arguments in terms of the variables, x, on the final state, requires knowledge of the inverse mapping that expresses y in terms of x. The column vector |V µ (y)i is the epitome of a contravariant vector. The row vector hAµ | is the epitome of a covariant vector. The matrix [Jµk (y)] is the Jacobian matrix of the transformation from initial to final state. The matrix transformation rules above give primitive meaning to the concepts of covariance and contravariance. These rules in tensor analysis are extended to multi-linear objects. The classical definitions consider a tensor to be a mathematical object whose functional form given on the initial state is well defined on the final state. Note that the tensor operations are a "pushforward" from the initial to the final state. The transformation rules are of two types are defined as: T he Contravariant Rule : (tensor pushforward) ¯ k ® £ ® ¤¯ ¯V (y) ⇒ |V µ (y)i = ∂xµ (y)/∂y k ¯V k (y) ¯ µ ® ⇒ ¯V (x) = |V µ (y(x))i . T he Covariant Rule : (tensor pushf orward) £ ¤ Ak (y) ⇒ Aµ (y) = Ak (y) ∂y k (x)/∂xµ .

(9.8) (9.9)

(9.10)

In both cases an inverse mapping (x = φ−1 (y) ) is required if the arguments of the functions created on the final state are to be expressed in terms of the independent variables, x, of the final state. When the tensor rule for covariant transformations is multiplied on both sides by the Jacobian matrix, and using the constraint inherent in the tensor definitions that an inverse Jacobian exists, a more general formula for covariant transformations

Why differential forms?

403

can be obtained. If the functions that make up the covariant object are given on the final state, then the functions on the initial state are well defined in terms of the independent variables on the initial state. This more general situation works when the map is differentiable, but no inverse map exists. The process is defined as the covariant pullback. For more detail, see the subsection on the pullback examples, below. T he Covariant Rule : (pullback) £ ¤ |Ak (y)i ⇐ Aµ (x(y)) ∂xµ (y)/∂y k ⇐ Aµ (x).

(9.11) (9.12)

Retrodiction versus Prediction

It has been shown [192] that with respect to maps without differentiable inverses, the classic rules of tensor transformations do not permit the unique prediction of the functional form on the final state of either contravariant or covariant tensors, given the functional form of these tensors on the initial state. Numeric values on the final state sometimes can be predicted, but the functional form (defining neighborhoods) on the final state cannot be uniquely predicted. It also has been shown relative to maps without differentiable inverses that a given the functional form of a contratensor on the final state does not permit a unique retrodiction of the functional form on the initial state. However, there are two situations where unique well defined retrodiction is possible in a functional sense. This retrodiction process is defined as the "pullback" The first situation involves covariant tensors on the final state. The second situation involves contravariant antisymmetric tensor densities on the final state. Given a mapping and its Jacobian, [Jkµ (y)] , the two forms of pullback may be written as, P ullback Covariant Scalar Rule : Ak (y) ⇐ Aµ (x(y)) [Jkµ (y)] ,

(9.13)

£ ¤ k P ullback Contravariant density Rule : C (y) ⇐ AdJµk (y) C µ (x(y)). (9.14) £ ¤ In the notation above, AdJµk (y) , is the adjoint of the Jacobian matrix (matrix of cofactors transposed) and does not depend upon the determinant of the transformation. The adjoint matrix may be algebraically determined even when the determinant of the matrix is zero. Hence the pullback is well defined, even if an inverse does not exist. The contravariant density, or current, C µ (x), is like the charge current density of electromagnetism. It is suggested herein that these two rules should be used as foundations for transformations of objects which would be considered to be tensors, if the mappings are constrained to be diffeomorphisms. 9.1.3 Summary: Pair forms and Impair forms Retrodiction and functional substitution leads to two well defined species of differential form objects: Pair (even) forms with antisymmetric coefficients, which pullback relative to the Transpose of the Jacobian of the mapping, and Impair (odd) forms

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with antisymmetric coefficients objects that pullback via Adjoint of the Jacobian matrix. Remark 62 Impair forms are sensitive to orientations, while Pair forms are not. Both objects are not dependent upon an invertible differential mapping, hence can be used to study continuous topological evolution. However, when the maps from an initial state of independent variables to a final state of independent variables are unimodular det =1 diffeomorphisms, then the coefficients of the pair forms behave as covariant tensors, and the coefficients of the impair forms behave as contravariant tensors. If the diffeomorphisms are not such that the determinant of the Jacobian matrix is 1, then the coefficients transform as contravariant tensor densities. • The 2-form F of electromagnetic field intensities is a pair differential form. The integrals of exterior differential forms are invariants of diffeomorphisms which may change the sign of the N-volume (a change of orientation). Such integrals are scalars. • The (N-2)-form G is an impair 2-form density in a space of four dimensions. The integrals of impair differential form densities change sign with respect to diffeomorphisms that change the N-volume orientation. Such integrals are pseudoscalars and can lead to enantiomorphic pairs. 9.2

Why Topology?

The purpose of this monograph is to sensitize the reader to the topological aspects of physical systems. The motivation is to develop a better understanding of the real world of irreversible processes, which if they take place continuously, must involve changing topology. However, before it is possible to make practical use of the concepts of changing topology, it first is necessary to understand what topology itself is all about. It is important to be able to identify topological properties, such that when these topological properties change, under the action of continuous but irreversible processes, the changes will be recognized. Right up front, memorize the following definitions. The implications of these definitions will be clarified in that which follows. 1. A topological property is an invariant of a homeomorphism. 2. A homeomorphism is a map describing a process from an initial to a final state which is both continuous and reversible. Reversible means that the inverse map exists and is also continuous. A homeomorphism need not be differentiable. 3. A topological structure is the specification of enough topological properties on both the initial and final state to permit a determination to be made if a map,

Why Topology?

405

or a process, is continuous from the initial to final state, even though it may not be reversible, and the topology of the initial and final state are not the same! 4. A process or a map is continuous if the limit points of the initial state, relative to the topology of the initial state permute into the limit points of the final state, relative to the topology of the final state. 5. A limit point p of a subset, A, is a point such that every open set of the given topology that contains p contains another point b, of the subset A, but not equal to p. Note that p is not necessarily a point in A. 6. If a homeomorphism and its inverse are both continuous and differentiable, then the map is defined to be a diffeomorphism. Tensor analysis is usually restricted to the study of diffeomorphisms. Simple geometric properties are properties which are invariant with respect to the two special diffeomorphisms of translation and rotation. In this Klein sense, size and shape are geometric properties. 7. Open sets, or their compliments which are closed sets, may be used to define a topology. A given collection of subsets defines an open set topology if the intersection of any pair of elements in the subset collection is also member of the subset collection, and if every union, infinite or not, of elements in the subset collection is also an element of the given subset collection. In essence, topology is based upon the idea of closure under logical union and intersection. A given collection of sets can support many different topologies, just by choosing subset collections that obey the closure rules described above. These concepts are best explained by examples to be found in the section below entitled Point Set Topology. The bulk of the work herein is directed to the study of twice differentiable maps which are continuous but not homeomorphic. There are several ways to approach the concepts of topology. The mathematicians of 1800 to 1900 discovered a number of "global" results and features of mathematical systems that were deformation invariants, but the bulk of the work in topology as we know it now began after 1900. It would appear that the name "topology" became accepted by mathematicians about 1925, although there were then two disjoint schools of topology (point set topology and combinatorial algebraic topology). In the early days these disjoint schools were assumed to be studies of different mathematical topics. It is now known that these different "ways" of getting at topological information are equivalent. Most of the classical approaches of teaching topology, however, are much too stilted to have acceptance by applied scientists and engineers. However, E. Cartan in the early 1900’s developed an extraordinary set of ideas based on two simple extensions of the ordinary calculus. He exploited the notion of

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the Grassmann exterior product, and developed the formalism of exterior algebra. To this (closed) algebraic system he added the concept of the exterior differential. From the notion of the exterior product and the exterior differential almost all of the useful features of differential topology can be constructed. The mathematical science is called the theory of exterior differential forms. Topology is intuitively a non-local, or "global", idea of how interior things, perhaps consisting of many parts, behave synergistically with their environment or exterior, and with their boundary. A topological property may be viewed intuitively as an invariant of a continuous deformation. The concept of physical coherence is intuitively a topological idea in that it corresponds to the notion of a synergetic interaction between parts not at the same point. The classic understanding of the coherence of a crystal is due to a very specialized and refined topology induced on space-time by the presence of matter. The fact that a cylindrical wave guide has a low frequency cut-off, while the coaxial cable can transmit DC current is a topological idea demonstrating the interaction of a system and its boundary. Note that during the discussions above no mention is made of how big or how small the system to be studied is; there has been no geometric definition of scales. This notion that topological properties are independent from scales is the hardest concept to absorb for a scientist non-sensitized to topological thinking. After all, scientists are trained to "measure" something; they want to measure a size or shape. These geometrical hang-ups must be removed from the topological perspective, for if the system is too small, just stretch it out; if the system is to big, just compress it. Physical topology is the study of those properties that are independent from some legislated scale marks on a ruler, or on the number of time ticks of a clock. Physical topology deals with ideas that are in a sense the same on both the scale of the galaxy and on the scale of the atom. This is not to say that the world of microphysics is identical to the world of macrophysics, for when a physical system is constrained to yield geometric invariants, certain topological features will dominate over other topological features, depending on the scales chosen. The coaxial waveguide always has a low-frequency cut-off (a topological idea), but the actual wavelength depends upon the geometric idea of scales. In this book the emphasis will be on the use of Cartan’s theory of differential forms to describe topological features that are independent of scales. Note for those readers with some exposure to the exterior calculus, this means that the concepts of coordinate representations, or metric, or connection, or fiber bundles (with their more geometrical content) are going to be suppressed in favor of the Cartan theory, which does not require such constraints on the base variety of space and time. A natural question arises as to how a "differential form" can be used to describe global information not retrievable from descriptions at a point. The paradox is resolved when it is realized that the differential form is to be considered as an entity "before" a limiting process has taken place. Recall that the limiting process of differential calculus requires some neighborhood constraint to

Why Topology?

407

be specified before the derivative is computed. In fact the limiting process depends upon the specification of a topology. The usual topology assumed is the homogeneous connected Euclidean topology of RN which consists of open sets defined in terms of open balls of domain less than r. Physical systems will require for their description topologies that are not equivalent to RN , although in a small neighborhood in which certain "singularities" have been removed, it is generally assumed that the physical systems can be approximated by RN . Cartan’s differential forms have amazing properties not contained in other mathematical entities: 1. Differential forms on the final state are well behaved with respect to functional substitution of differentiable maps from the initial to the final state, even though these maps are irreversible! This fact is known as the "pullback" and is the cornerstone of the investigations about irreversible processes. By functional substitution, not only is it possible to compute values of a differential form on the initial state from values given on the final state, but also it is possible to compute the functional form of the differential forms on the initial state. The statement is true for both scalar differential forms as well as for differential form densities. Scalar differential forms are defined (retrodictively) on the initial state in terms of the pullback method of functional substitution, which is equivalent to multiplication of the coefficients of the differential form by the transpose of the Jacobian of the map from the initial to final state. Exterior differential form densities also enjoy the pullback property, but now the pullback is equivalent to multiplication of the coefficients by the adjoint of the Jacobian map from initial to final state. Both the transpose and the adjoint of the Jacobian exist even thought the inverse does not. 2. Differential forms can carry information about singularities, and these singularities dictate much of the topological content of the differential form. The singularity to be studied is not of the type that blows up to infinity, necessarily, but instead is the more innocuous zero set. It is the zero points of functions and vector fields under the action of continuous maps that is of predominant interest herein. The zeros of a map become the infinities of the inverse map, but in this monograph the emphasis is on how much can be determined about processes that need not have an inverse. The singular infinities of the Dirac delta function are avoided in this book, if at all possible, but the "singularities" of the zeros are crucial. 3. Differential forms may have pre-images (on the initial state) which are not unique. It is precisely this multi-valuedness of an integral pre-image that allows differential forms to single out the physically interesting characteristics, or wave-fronts, or shock-fronts, or defects of dynamical systems. These topological singularities or defects are point sets (which may or may not be stationary)

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upon which a unique solution to some system of partial differential equations describing the physical system cannot be analytically continued from a nearby neighborhood. As will be determined later, dimension and the number of components are topological properties, and the generation of a defect will correspond to a topological change of dimension, or the creation of multiple components, over some domain. A list is displayed below of a number of physical effects which - to first order - do not depend explicitly on size or shape. therefore, these phenomena must have a topological basis for their explanation. Key topological properties are the number of parts (connectivity), the dimension, and orientability. Concepts of size and shape, metric, or connection can not enter into the explanation of such phenomena, except in an auxiliary way. As the monograph progresses, the details of these topological features will be explained. • Planck’s Radiation Formula

The distribution law is independent from the size and shape of the hot body.

• Coaxial Wave-Guide Propagation

A hollow wave guide is a high-pass filter, but a non-simply connected co-axial cable can pass DC.

• Chaos does not occur in Dimension 2

A system of ODE’s has a PDE equivalent. If the PDE satisfies the Frobenius integrability theorem, according to Darboux there is always a representation in terms of 2 functions.

• Bohm-Aharanov effect and the Flux quantum (based on 1-forms) Flux quanta in superconductors come in integer multiples of ~/2e.

• Gauss Law and the charge quantum (based on 2-forms) RR The integral of the closed D ◦ dS over a bounding surface only depends upon the number, n, of electrons of charge e = 1.6 x 10−19 C in the interior of the closed surface, no matter what the size or shape of the closed surface may be. • Topological Torsion and Topological Spin based on 3-forms)

(The Poincare quanta

The Bohm-Aharanov effect is a one-dimensional period integral. Gauss’s Law is a two-dimensional period integral. There are two three-dimensional period integrals that have been little studied in science, but appear to have significance in non-equilibrium hydrodynamics and plasmas as robust coherent vortex-like helical structures - insensitive to deformations in space-time.

Why Topology?

409

• Quantum Transition Probability (cross ratios are independent from scale) Fermi’s Golden Rule demonstrates that the transition probability is a cross-ratio projective invariant, independent from scale. • Thermodynamic Irreversibility (the Heat 1-form, Q, does not admit an integrating factor) The idea that thermodynamic irreversibility is defined by the failure of the Frobenius Theorem for the 1-form of work implies that the topological dimension of the Action 1-form must be 4. In other words, thermodynamic irreversibility is an artifact of four dimensions. • Thermodynamic Phases

The difference between a vapor and a liquid is that the liquid is apparently connected and the vapor consists of many disconnected parts. The number of components is a topological property. Condensation implies a change in topology takes place. Condensation is a gluing or pasting process which often can be described continuously. Vaporization, on the other hand, is a discontinuous or cutting process, and does not usually permit a continuous description.

• The Law of Corresponding States

In chemistry there is the law of corresponding states, which demonstrates the universality of thermodynamics, independent from the size and configuration of the molecules under consideration.

9.2.1 Geometry and Physics Euclidean Geometry - Invariant Size and Shape - Rigid Body motion Historically, fundamental physical theories have been based on geometrical models, geometrical relationships and geometrical properties of physical objects. In fact the very idea of a physical measurement is associated with the concepts of how big, how far, or how long. These ideas of measurement are intuitively geometrical ideas, and all involve a comparison of something of immediate interest to some legislated standard. However the idea of a comparison is not restricted to pure geometrical concepts of size and shape. An abstract comparison may be viewed as a mapping from a range to a domain, or a transformation from an initial state to a final state. The mapping or transformation may be an element of an equivalence class of mappings, and that class is determined by its invariant properties. Recall that, according to Felix Klein, a (Euclidean) geometrical property is defined to be an invariant of a translation or rotation. A simple observation demonstrates that size and shape are such geometrical properties, and these properties are the fundamental invariants of that branch of physics that deals with rigid body motion.

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Isometric Geometry - Invariant distance - Bending With the advent of tensor calculus, mathematicians relaxed the constraints of pure (Euclidean) geometry a bit in order to include the concept of bending without compression or shear. The fundamental physical invariant of interest became the distance between a pair of points; size is considered as an invariant in such geometries, but shape is not (necessarily) an invariant of the equivalence class of transformations that describe bending processes. The equivalence classes of transformations, based on the property of invariant distance, are called isometries. The pure geometrical constraint of invariant shape is relaxed to include the possibility of bending. The idea of invariant distance has dominated physical theories since the turn of the 20th century. In fact, a new derivative concept, the covariant derivative, was defined in a such a manner that it preserves the concept of distance as an invariant. Displacements via a "covariant derivative" are constrained such that the distance between a pair of points is preserved as an invariant! However, be warned that the prescription of a covariant derivative has, in its foundations, the impossibility of intrinsically describing compressions or shears! Projective Geometry - size, shape, distance and angles are not invariant Take any point p as a perspective point, say in the plane, and draw 4 straight lines emanating from the point. Draw any other straight line that is skew and intersects the lines of perspectivity.

Figure 9.1 Cross Ratios in projective geometry

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Order the intersections as {1,2,3,4}. Next, draw a different skew line (which could be as far away from p as Alpha Centauri). Number the intersections in the same order {1,2,3,4}. In projective geometry, all invariants of a projection are constructed from cross ratios. A projective 3-space requires a set of 4 points (a quadruplet), of which no set of 3 points is coplanar. There are up to 6 distinct cross ratios which come in 3 pairs. One pair falls into the interval, −∞ ↔ 0, one pair falls into the interval, 0 ↔ 1, and the last pair falls into the interval, 1 ↔ +∞. Next construct the six "measured" segments for the blue points on the first line that intersects the 4 projective rays: (1 ↔ 2), (1 ↔ 3), (1 ↔ 4), (2 ↔ 3), (2 ↔ 4), (3 ↔ 4).

(9.15)

Also construct the similar set for the red points {1*,2*,3*,4*} on the second line that intersects the 4 projective rays. Now select any combination of 4 of the 6 segments of the first set to form a dimensionless number, the cross ratio, XRatio, (there are six possibilities): ¶ µ (1 ↔ 2) · (1 ↔ 4) . (9.16) XRatio = (2 ↔ 3) · (2 ↔ 4) It is a remarkable result that if the same construction is used to generate the cross ratio, XRatio , in terms of the second set of intervals, (1∗ ↔ 2∗ ), (1∗ ↔ 3∗ ), (1∗ ↔ 4∗ ), (2∗ ↔ 3∗ ), (2∗ ↔ 4∗ ), (3∗ ↔ 4∗ ). then, XRatio

¶ µ (1 ↔ 2) · (1 ↔ 4) = (2 ↔ 3) · (2 ↔ 4) µ ∗ ¶ (1 ↔ 2∗ ) · (1∗ ↔ 4∗ ) ∗ = . = XRatio ∗ ∗ ∗ ∗ (2 ↔ 3 ) · (2 ↔ 4 )

(9.17)

(9.18) (9.19)

It does not matter where or how the two skew lines are placed. Hence if a physical researcher observes an experiment on his laboratory table, and then observes a similar experiment on Alpha Centauri, and if the measurements have been made in terms of projective cross ratios, the numbers obtained are the same. The fine structure constant on the lab table is the same as the fine structure constant on Alpha Centauri. It is also important to realize that there are 6 possible X-Ratios, and that in general two are found as negative numbers less than zero, two are positive numbers greater than 1, and two are numbers bounded by 0 and 1. For a given value, k, the six distinct cross ratios are defined as X-Ratios k 1−k k/(k − 1) . 1/k 1/(1 − k) (k − 1)/k These six functions are plotted in the Figure 9.2 below.

(9.20)

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Figure 9.2 Cross ratio phases It is important to note that there are usually two X-Ratios that fall in the interval, 0 ⇔ 1. Hence, from a probability point of view there are (usually) two distinct probabilities that are cross ratio invariants. Any value of k (a horizontal line on the graph above) intersects the curves within the X-Ratio region, 0 ⇔ 1, in two places. The exceptions occur at k = [2, 1/2, −1], where the cross ratio pairs are not distinct, but form "doublets". In projective geometry, the exceptional existence of doublets are used to define a homology (see p. 301 in [268]). In three dimensions (the projective plane), the idea of a characteristic equation of a linear transformation having repeated roots has thermodynamic significance. The linear transformation is defined in terms of the Jacobian matrix (for thermodynamics, the Jacobian matrix is computed from the 1-form of Action, A). There are two species of homologies (for which two of the three possible eigenvalues of the Jacobian matrix are the same). In one case, there is a line of fixed points and a point of fixed lines. This case corresponds to the Binodal line of a van der Waals gas. The second case corresponds to two fixed lines and two fixed points. This case corresponds to the Spinodal line of a van der Waals gas. Now back to the meaning of two probability concepts when the root structures are not degenerate. One of these probabilities has a direct correspondence to Fermi’s

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golden rule of quantum transitions. Fermi’s formula for the transition probability between two states, (a, b), in terms of the wave functions, Ψa , and Ψb is, hΨ∗a | |Ψb i hΨ∗b | |Ψa i . (9.21) hΨ∗a | |Ψa i hΨ∗b | |Ψb i As both wave functions are complex numbers, there are 4 functions that can define a perspectivity, and the Hermitian inner products can be used to define the elements of a cross ratio. The reason that atomic transitions appear to be the same on Alpha Centauri as they do on the laboratory table top is due to the fact that Fermi’s golden rule is a projective invariant bounded between zero and unity. Fermi’s Golden Rule:

P =

Remark 63 The question as to the meaning and applicability of the second projective invariant, bounded between zero and one, and a second distinct type of probability theory has yet to be resolved. In a projective dual sense, the values of k come in distinct pairs (also bounded by −∞ ↔ 0, 0 ↔ 1, and 1 ↔ +∞) except for exceptional X-Ratios, XRatio = [2, 1/2, −1]. The projective 3-space is a constrained four-dimensional space that admits mappings between the quadruplets in terms of a 4 x 4 matrix such that the determinant of the mapping is not zero. In addition, the constraint is made to study the set of all matrices with determinant greater than zero, for they define a continuous matrix group: ⎡ 1 1 1 1 ⎤ θ1 θ2 θ3 θ4 ⎢ θ21 θ22 θ23 θ24 ⎥ ⎥ Projective transformation [P] = ⎢ (9.22) ⎣ θ31 θ32 θ33 θ34 ⎦ , det [P] 6= 0 . θ41 θ41 θ43 θ44 ⎡ 1 1 1 1 ⎤ θ1 θ2 θ3 θ4 ⎢ θ21 θ22 θ23 θ24 ⎥ ⎥ Projective matrix group [P+ ] = ⎢ (9.23) ⎣ θ31 θ32 θ33 θ34 ⎦ , det [P] > 0. θ41 θ41 θ43 θ44 The product of all such matrix arrays have the same structure. Hence there is one constraint on the projective matrix of 16 elements that leaves the group dimension =15. The concept of separation is a projective invariant, as are all X-Ratios. Transitive projective geometry - parallelism is preserved by Affine translations The matrix representation of an Affine ⎡ 1 1 θ1 θ2 ⎢ θ21 θ22 [AR ] = ⎢ ⎣ θ31 θ32 0 0

transformation has the format, ⎤ θ13 θ14 θ23 θ24 ⎥ ⎥ , det [A] 6= 0 . θ33 θ34 ⎦ 0 θ44

(9.24)

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In mechanics of deformable media, the matrix is often interpreted as, ⎡

⎤ θ11 θ12 θ13 V 1 ⎢ θ21 θ22 θ23 V22 ⎥ ⎥ [AR ] = ⎢ ⎣ θ31 θ32 θ33 V 3 ⎦ , det [A] 6= 0. 0 0 0 1

(9.25)

where V is the velocity "of translation". In Frame bundle applications, the last column describes how the tangent vectors to a surface vary in the normal direction. These mappings describe shear deformations where parallel planes remain parallel. In projective geometry a special hyperplane "at infinity" can be defined. A typical representation of the "plane at infinity" is given by the column array, |a, b, c, 0i . An Affine transformation preserves the "plane at infinity"; that is, [AR ] ◦ |a, b, c, 0i = |a0 , b0 , c0 , 0i .

(9.26)

Affine transformations preserve parallelism. Intransitive projective geometry - the origin is preserved as a projective fixed point There is another form of a projective transformation that corresponds more to how the normal vector varies in the horizontal direction. It has the format, ⎡

θ11 ⎢ θ21 [AL ] = ⎢ ⎣ θ31 θ41

θ12 θ22 θ32 θ41

θ13 θ23 θ33 θ43

⎤ 0 0 ⎥ ⎥ , det [A] 6= 0 , 0 ⎦ 1

(9.27)

and induces effects that lead to a 3rd Cartan structural equation (see (9.292)). An intransitive mapping has a fixed point often denoted as a polar point of a rotation. The projective matrix representing such nearby distortions has a set of N-1 zeros along the last column. Such matrices form a group, but have different properties from the transitive mappings that describe Affine geometry. An Affine matrix has a representation with N-1 zeros along the bottom row. The origin can be defined as the intersection of 3 hyperplanes. A useful representation can be given in terms of the column vector |0, 0, 0, ai . An affine (transitive) transformation does not preserve the "origin". All points move, except the points at infinity. However, for intransitive projections, the origin is preserved (a fixed point), but the points at infinity are not: [AL ] ◦ |0, 0, 0, ai = |0, 0, 0, a0 i .

(9.28)

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Similarity transformations - parallelism and orthogonality are preserved ⎡ 1 1 1 ⎤ θ1 θ2 θ3 0 ⎢ θ21 θ22 θ23 0 ⎥ ⎥ (9.29) [A] = ⎢ ⎣ θ31 θ32 θ33 0 ⎦ , det [A] 6= 0. 0 0 0 1

The idea that similarity transformations preserve orthogonality and parallelism, limits the generality of the methods used above to describe thermodynamic features associated with a Jacobian matrix. The Cayley-Hamilton theorem does not capture all of the features of the Jacobian matrix. The similarity invariants are useful (and can be put into correspondence with curvatures of a hypersurface) but they do not capture the features of, for example, topological torsion. The eigenfunctions that make up the similarity invariants can be real or complex. When the eigenfunctions are complex the matrix must have antisymmetric components. Homothetic Geometry - Pressure

If the deformation is confined to compressions or expansions, then the distance between a pair of points is no longer invariant, and as both size and shape are not necessarily invariants, the covariant derivative concept becomes obsolete. The question arises as to what invariants might be used to classify such transformations. Such transformations are defined as conformal (projective) transformations that leave invariant the angle between a pair of lines. The next class of transformations are those that are continuous reversible deformations that admit translations, rotations, bending, expansion, and shears. The invariants of such transformations are defined as topological properties. The sign of the determinant is crucial, for if the mapping is such that the determinant is positive, then the orientation of an object is preserved. If the determinant is negative, the orientation is reversed, which is an indication of topological change. 9.2.2 Topological Physics Indeed the geometric method has served physics well, but there are many things in nature that obey rules that are independent from size and shape. For example, the Planck blackbody radiation distribution in frequency is independent from the size, shape, and even chemical makeup of the hot body that is radiating. The simply connected space of a hollow wave guide of any finite size or shape will always have a low frequency cut-off, but the co-axial cable, which is topologically not simply connected, can support DC currents. A physical system is conservative in a thermodynamic sense if the cyclic work vanishes, independent from the length (size and shape) of the process path. The closed surface integral of Gauss’ law does not depend upon the shape or size of the surface but only on the number of charges contained in its interior. A flowing fluid can be in a laminar streamline state which can evolve into a chaotic turbulent state. A measurement with a finite compact apparatus of a infinite

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or non-compact property will always have an uncertainty associated with the measurement. These qualities of nature that do not seem to depend upon size and shape, and seem to be independent of continuous reversible deformations, can be defined and studied as topological properties. The idea is that basic physical properties exist that do not depend upon legislated standards of scales, but are absolute in the sense that they are answers to questions such as: 1. How many parts? 2. Does a solution exist? How many solutions? 3. How many components? A great deal of engineering and physical theories are built around the deterministic geometrical dogma which supposes that, given initial data, the name of the game is to be able to predict the outcome, and predict it uniquely. From a more topological perspective, it will become apparent that unique prediction may become impossible, but deterministic retrodiction can be achieved. Observables as invariants of transformations Concepts that do not depend upon size and shape can still be invariants of an equivalence class of transformations. Again, these invariants, which are not pure geometric invariants, may be used to define an equivalence class of transformations. The issue is how to define and observe these qualities of nature that do not depend upon size and shape. Consider a piece of notebook paper made out of flexible rubber material. The sheet has 3 holes along one side, and can be marked as 1,2,3,4 at its corners, in a prescribed sequence or orientation. Translate the sheet, and ask what are the invariants of the transformation. The answer is: the size, the shape, the number of holes and the orientation sequence, (1,2,3,4), are all invariant properties of the translation. Now rotate the sheet; what stays the same? Again, size, shape, hole count, and orientation stay the same. However, in the case of rotation there exists one other invariant that is not in the class of translations. This additional rotational invariant is the fixed point that defines where the axis of rotation intersects the sheet. Translations are said to be transitive because there is no fixed point, while rotations are intransitive because there must be one fixed point. Recall that by Klein’s definition, the four properties of size,shape, hole count and orientation are geometric properties. Now take the this rubber sheet and deform it by pulling and stretching the sheet. What stays the same? The answer is not the size and not the shape, but the hole count (distorted holes, of course, in the deformed case) and the orientation sequence 1,2,3,4 do stay the same under the deformation. Those properties that stay the same under continuous and reversible deformation are defined to be topological

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properties. Note that topological properties are included in the class of geometrical properties, but the class of geometrical transformations are included in the class of topological transformations. Pure geometric properties will be defined as those properties which are invariant under translations and rotations only. A topological property is defined as an invariant of a homeomorphism, or in more simple terms, a topological property is an invariant of a continuous and reversible deformation, while pure geometrical properties are not. Pure geometrical properties such as size and shape can evolve with respect to homeomorphisms. In this monograph, the process of studying invariants of transformations will be taken one step further, for of physical interest to dissipative systems are those processes that are continuous but not reversible. Pure topological properties are not invariants of continuous but irreversible transformations. As pure geometrical properties evolve with respect to continuous and reversible transformations, pure topological properties evolve with respect to continuous but irreversible processes. For example, if the rim of one of the holes in the rubber sheet was grasped and pulled out of the sheet into the shape of a long trumpet with the rim becoming smaller and smaller until it collapsed to a point that could be glued together, then the topological property of hole count in the rubber sheet would have been changed from three to two during the deformation and gluing process. Note that it was the absolute (integer) number of holes that changed during this process of topological evolution which effectively collapsed one of the holes. It is important to note the topological change is quantized, for you can never have half a hole. The question of how many holes is "absolute", independent from scales, for it is in relation to the integers. What are the invariants of the equivalence class of continuous, but irreversible transformations? Examples of such invariant properties are connectivity, compactness, and most important to this monograph, the concept of closure. Rather than carrying the words "continuous but irreversible" throughout the monograph, a biological concept will be used to define such processes. A continuous but irreversible process will be defined as an element of an equivalence class of transformations, and will be defined as an aging process. Like all transformations, the equivalence class of aging processes will be defined in terms of its invariants. The ability to develop a physical understanding of the aging process must be built upon the observable invariants of such processes, and the dynamical theory of those topological invariants that can change during such processes. This dynamical theory will be called the theory of topological evolution. Physical laws as topological statements The ultimate goal of this monograph is to establish methods of distinguishing topological effects in physics from geometrical ones, to establish laws describing topological properties of matter, and in particular to establish the laws of physical topological evolution. Note that the first step is to go beyond the constraints of geometry and

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study strictly topological properties and the evolution of geometrical properties. The second step is to go beyond the constraints of topology to study the evolution of topological properties. The reader may not realize that he or she has often worked with topological concepts without knowing anything about topology. For example, it was demonstrated herein that the Maxwell theory of electromagnetism, without the geometrical constraint of a Lorentz symmetry group, is a statement about topological properties of space-time. It also will be demonstrated that the first law of thermodynamics is a topological statement of cohomology. The flow of a Navier-Stokes fluid can admit solutions which are examples of an irreversible but continuous topological evolution. 9.3

Cartan’s Exterior Calculus

9.3.1 Introduction After the basic concepts of topology are presented, the next step is to develop a thorough understanding of Cartan’s theory of Exterior Calculus. Cartan developed his exterior calculus long before the word Topology became fashionable, but the key feature of Cartan’s theory is that it transcends the geometrical constraints of tensor calculus and is truly a theory of topology and topological evolution. It was mentioned above that a topological property was an invariant of a homeomorphism. Technically, a homeomorphism is a map from an initial to a final state that has two qualities: 1) it must be continuous, and 2) it must be reversible in the sense that the inverse exists and is continuous. If topological evolution is to take place, then one or both of these qualities must not be true. Of particular interest to the developments in this monograph are those evolutionary processes which are continuous but not reversible. However continuity is not a geometrical idea; it is a concept that does not depend upon size and shape. A major goal will be the development of a useful topological structure, such that it can be decided whether or not a particular process is continuous, or not. Fortunately, the concept of a topological structure can be developed in terms of the Cartan calculus, such that a decision can be made if a process is continuous or not. If the process is determined to be continuous, and if it can be shown that the topological properties change during the process, then the process is an aging process. That is, the process is continuous but irreversible. 9.3.2 The exterior product and the exterior differential Cartan’s methods utilize (what might be unfamiliar) techniques that are described as the "exterior product" with an algebraic symbol, ˆ, and the "exterior differential" with a differential symbol, d, acting on objects, ω p , defined as exterior differential forms of degree (not power) p. These basic concepts will be discussed below, briefly, but are best studied in detail from texts such as that by Harley Flanders [73], Bamberg and Sternberg [12] and by Gockeler and Schucker [83]. I believe the best way to learn about these "new" operations, and the objects upon which they act, is to try a few examples. Flanders is the best text with which to start. About the only thing

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missing in the Flanders presentation is a discussion of the Lie differential acting on p-forms. There are a number of other texts available that discuss exterior differential forms, but many are a bit pedantic or too pompous to be useful at the applied engineering level. Most applications that have appeared in the literature are in the field of general relativity or super-symmetry or super-gravity or string types of theories. Very few texts are to be found which present the Cartan methods as applied to hydrodynamics, electrodynamics, thermodynamics, and other engineering sciences. (Domage!) To reiterate previous statements, recall that the methods of exterior differential forms are important because they carry topological information, and can be used to study topological evolution. My interest in the Cartan methods was cemented when I realized how natural it was to write Maxwell’s equations in terms of differential forms. Moreover it became apparent the Maxwell theory was independent from metric or connection constraints. From this point of view (perhaps initiated by Van Dantzig) electromagnetism is not a geometrical theory, but instead is a topological theory. The PDE’s of the Maxwell - Faraday induction equations, form a nested set in every dimension greater than 3. Experience with electromagnetic theory is very useful, for you can use EM theory to check your developing skills with the Cartan methods. If your application of the Cartan techniques does not replicate well known results in electromagnetism, you have made a mistake. These concepts will be detailed in that which follows. There are two other important operations, besides the exterior product and the exterior differential, which act on differential forms, but these other operations require the specification of a vector field, V , in addition to the differential form, ω. From a physical point of view, differential form(s) may be used to define a physical system, and the vector field may be used to define a thermodynamic evolutionary process. What is remarkable is that this point of view can be used to justify the topological basis of thermodynamics, and to give a non-statistical description of irreversible processes. The two additional operations (with respect to V ) are called the "interior product" with the symbol i(V ), and the Lie differential, L(V ) = i(V )d + di(V ), which combines two operations, the interior product and the exterior differential. The Lie differential is an alternative to the "covariant" differential of tensor analysis, and, like the covariant derivative, will produce tensors from other tensors by means of a differential process. Examples and definitions will be given below. The Lie differential will become the most important tool from a topological perspective, for it permits computations to be made which will distinguish those objects which are topological invariants of a process and those objects which are not. It is remarkable that the Lie differential operating on a 1-form of Action is equivalent to the cohomological statement that defines the first law of thermodynamics. 9.3.3 The exterior algebra The exterior algebra of Cartan is based upon an associative, but not commutative, multiplication rule defined as the exterior (wedge or hook) product of objects defined

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as exterior differential forms. The symbol for the product is ˆ. The structure of the algebra can be built starting from 1-forms on the n-dimensional vector space Λ1 {σ a }, in terms of a basis of 1-forms denoted by {σ a }. The arbitrary 1-form is constructed from the basis elements according to the formula given above, ω 1 = Ak (y) σ k . The addition rule of the algebra is that of vector space addition; you add the coefficients of forms of the same differential exterior products, ω1 = Ak (y) σ k , ω2 = Bk (y) σ k , ω1 + ω2 = {Ak (y) + Bk (y)} σ k .

(9.30) (9.31) (9.32)

Example: Add (3x dx + 4xz dy) and (2y dy + 17y dz), basis = (dx, dy, dz), (3x dx + 4xz dy) + (2y dy + 17y dz) = 3x dx + (4xy + 2y) dy + 17y dz.

(9.33) (9.34)

It is the multiplication rule that is perhaps unfamiliar. The multiplication rules are defined in terms of elements of the basis set: σ a ˆσ b σ b ˆσ b dy a ˆdy b dy b ˆdy b

= = = =

−σ b ˆσ a , 0, −dy b ˆdy a , 0.

(9.35) (9.36) (9.37) (9.38)

These rules are similar to the cross product of Gibbs 3D vector analysis, but the difference is that the exterior product rule extends to n dimensions (the Gibbs cross product does not) and is associative (Gibbs product is not). Asa b c a b c sociative means (σ ˆσ )ˆσ = σ ˆ(σ ˆσ ). In 3D, the Gibbs cross product yields A × (B × C) 6= (A × B) × C. P P Example: Multiply A = Ak σ k times B = Bm σ m = AˆB = C, AˆB = = = = =

{Ak σ k }ˆ {Bm σ m } ...Ak Bk σ k ˆσ k + ...Ak Bm σ k ˆσ m + ...Am Bk σ m ˆσ k ... 0 + ...Ak Bm σ k ˆσ m − ...Am Bk σ k ˆσ m ... {Ak Bm − Am Bk }σ k ˆσ m ... ...C[km] σ k ˆσ m ..

(9.39)

= ...C[km] σ [km] ... = ...CH σ H . Note that AˆB 6= BˆA and that the exterior product of two elements of the vector space Λ1 (σ k ) produce a linear combination of paired basis elements of the form

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(σ k ˆσ m ). Moreover the coefficients of these elements are always antisymmetric under interchange of the paired indices. The antisymmetry and the rules of 1-form multiplication permit the writing of the product of two 1-forms in terms of another vector space, whose basis elements are the antisymmetric pairs (σ k ˆσ m ). This object is defined as a 2-form. It is conventional to rewrite the 2-form so constructed without repeating basis element pairs that are of different sign. That is, Λ2 (σ k ˆσ m ) ⇒ Λ2 (σ [km] ) = Λ2 (σ [H] ). The symbol H = [km] stands for all order pairs where k < m, and is equivalent to the set (12, 13, 14.....23, 24.....34...). It is apparent that the number of ordered singlet basis elements in three dimensions is the same as the number of ordered antisymmetric pairs; but this is only true in 3D. For 4D, the number of singlet basis elements is 4 and the number of ordered antisymmetric pairs is 6. The result of the exterior product is to produce from elements of one vector space of dimension n, another element of a different vector space of dimension n(n − 1)/2. In this limited sense the exterior product is not closed. The exterior multiplicative combination of two objects of the same type (1-forms) does not produce an object of the same type, but instead produces a 2-form. The process of exterior multiplication can be repeated where 2-forms are multiplied by 1-forms to produce 3-forms, and 3-forms are multiplied by 1-forms to produce 4-forms, ultimately building a "closed" algebra. The elements of the closed algebra will consist of classes, or vector spaces with basis element doublets, σ a ˆσ b , classes of triplets, σ a ˆσ b ˆσ c ,... and even n-tuplets of basis vectors, Ω = σ a ˆσ b ˆσ c ˆ...ˆσ n . However, the rules of multiplication are such that the exterior multiplicative combination of more that n basis vectors must vanish. Hence any element of the algebra times another element of the algebra is an element of the algebra, or zero. In this sense, the exterior algebra is closed. The doublets are called 2-forms, the triplets are called 3-forms, and the ntuplets are called n-forms. The multiplication rules demonstrate that each of the p-tuplets has a number of linearly independent elements equal to the possible combinations of n things take p at a time. Each of the p-tuplets forms a vector space basis of dimension equal to the appropriate combinatorial number of Pascal’s triangle.

Pascal’s Triangle:

n=1: 1 n=2: 1 2 1 n=3: 1 3 3 1 n=4: 1 4 6 4 1

The vector subspace dimension of the 0-forms, ω 0 , is 1; the dimension of the 1-forms, ω1 , is n, the dimension of the 2-forms, ω2 , is equal to the number of combinations of n things taken 2 at a time (n(n-1)/2 is equal to 3 for n =3, equal to 6 for n = 4, etc.); the dimension of the (n-1)-forms, ωn−1 , is n, the dimension of the n-forms, ω n , is 1. The dimension of the exterior algebra is the sum of the dimensions of all vector spaces produced by the exterior product; this dimension is equal to 2n .

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As an example consider the exterior algebras up to n = 4. The elements of Pascal’s triangle yield a one-dimensional (scalar) vector space Λ0 for the 0-forms, a four-dimensional vector space Λ1 for the 1-forms, a six-dimensional space Λ2 for the 2-forms, a four-dimensional vector space Λ3 for the 3-forms, and a one-dimensional vector space Λ4 for the (n=4)-forms. From geometrical studies in three dimensions, n=3, the elements of the 4 different vector spaces are called points, lines, surfaces and volumes. From applications to 3D mechanics, the position vector and the momentum vector are from the vector subspace, Λ1 , and their exterior (cross) product is an element of the vector subspace, Λ2 . The fact that angular momentum is an element of a vector space Λ2 is why one never sees angular momentum added to linear momentum (which is an element of a different vector space Λ1 ) in the elementary mechanics text books. From applications to special relativistic physics in four dimensions, n=4, the coefficients of the elements of these five different vector subspaces of the exterior algebra are known as scalars, vectors, tensors, Pseudo-vectors, and Pseudo scalars. p − f orms in 4D 0 1 2 3 4 # of basis elements 1 4 6 4 1 name Scalars Vectors Tensors P-Vectors P-Scalars

(9.40)

The Cartan-Grassmann exterior algebra consists of a vector space of 2n (= 16 in 4D) components with n+1 (= 5 in 4D) different vector subspaces. The algebra is technically called a graded algebra. The exterior algebra is closed with respect to multiplication, for all possible products of the algebra reside within the algebra of 2n dimensions (or are zero). The exterior product of a p-form and a q-form produces a (p+q)-form, or zero if p+q > n. In every case, the higher p-forms can be constructed from sums of products of the singlets. All elements of the algebra can be constructed from linear combinations of the primitive n basis elements, σ k and their products. A nice feature of the exterior algebra (besides being closed) is that the definitions of symbolic operations can be described entirely in terms of 0-forms and 1-forms, when the collective index is used. Every p-form can be rewritten in terms of symbolic coefficients (0-forms) and basis elements (p-forms from vector spaces of different dimensionality of course) with a format similar to that of a 1-form. For example, for n = 4, there are 6 elements of the vector subspace of two forms. That is, there are 6 independent non-zero pairs of the 4 basis 1-forms σk from Λ1 that can be used as the basis elements of Λ2 , namely the set {σ 1 ˆσ 2 , σ 1 ˆσ 3 , σ 1 ˆσ 4 , σ 2 ˆσ 3 , σ 2 ˆσ 4 , σ 3 ˆσ 4 }. If these basis pairs are given a new symbolism as {σ 12 , σ 13 , σ 14 , σ 23 , σ 24 , σ 34 },then the general 2-form (for n=4) can have the expansion coefficient - basis representation given by the formula, F = A12 σ 12 + A13 σ 13 + ... + A34 σ 34 = AH σ H ,

(9.41)

where H is the collective index described above for ordered pairs. This formula for the 2-form F in four dimensions looks like a vector formula for a 1-form, but in

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space of 6, not 4, dimensions; it is just that the index labels are different. All of the distinct basis combinations will be completely antisymmetric in their indices (p > 1). For example, H could be the set of triples [i1, i2, i3] with i1 < i2 < i3, for the vector space of 3-forms, H could be the set of quadruples [i1, i2, i3, i4] with i1 < i2 < i3 < i4, for the vector space of 4-forms; etc. With this collective index notation, combinatorial rules of multiplication and differentiation developed for 1forms can be applied directly to higher order p-forms. This technique, where a 2-form expansion in 4D was used as a 6-vector, was applied (intuitively?) more than 60 years ago by Arnold Sommerfeld in his studies of electromagnetic systems. The 3 components of E and the 3 components of B formed the components of the 6-vector. (See Sommerfeld’s volumes on Lectures on Theoretical Physics.) A similar but little used 6-vector composed of the acceleration and vorticity can be developed for a fluid. It is not clear whether Sommerfeld knew the theory of exterior differential systems at the time of his 6-vector development. Example: In 3D, exterior multiply the 1-form, A, times 2-form, B to produce the 3-form C = AˆB, C = (Ax dx + Ay dy + Az dz)ˆ(Bx dyˆdz + By dzˆdx + Bz dxˆdy) = (Ax Bx + Ay By + Az Bz )dxˆdyˆdz = C.

(9.42) (9.43)

Note that this product of a 1-form and (n-1=3-1=2)-form produces a n-form with a coefficient that looks the same as the Euclidean inner product of two ordinary vectors A ◦ B. Recall that the 1-form has n components and the (n-1)-form has n components. The Euclidean inner product result is valid in all dimensions, n. Example: In 3D, exterior multiply the 1-form, A, times 1-form, B to produce the 2-form D = AˆB, D = (Ax dx + Ay dy + Az dz)ˆ(Bx dx + By dy + Bz dz) = (Ax By − Ay Bx )dxˆdy +(Ay Bz − Az By )dyˆdz +(Az Bx − Ax Bz )dzˆdx = D.

(9.44)

(9.45)

Note that the result has coefficients equivalent to the Gibbs cross product of two vectors, A × B. 9.3.4 The Exterior Differential The exterior differential is a definition of a differential process acting on p-forms, ω p . The operation takes the p-form into a (p+1)-form. Hence, like the exterior product, the exterior differential generates a vector in a different vector subspace of the exterior algebra, d(ωp ) ⇒ ωp+1 . (9.46)

Other properties of the exterior differential will be described by the rules for distributing the operator over a product of 1-forms (note the order of factors and the minus

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sign modification of the Leibniz rule for the differential of a product of scalars), d(AˆB) = dAˆB − AˆdB,

(9.47)

d(d(ωp )) ⇒ 0.

(9.48)

d(ωp ˆω q ) = dω p ˆω q − (−1)p ω p ˆdω q .

(9.49)

and For a product of a p-form and a q-form, it follows that,

The epitome of the exterior differential is the concept of the total differential of a scalar function, which is a familiar operation that takes the 0-form, or function ω 0 = θ(y a ), into the 1-form, ω 1 = Ab dy b , d(ω0 ) = d{θ(y a )} ⇒ {∂θ(y a )/∂y b }dy b = Ab dy b = ω 1 .

(9.50)

The function could be constrained such that the set θ(y a ) = constant defines an implicit surface. It follows that in the constrained case, the total differential is also zero, d{θ(y a )} = 0. An object that has zero for the value of its exterior differential is said to be closed (in an exterior differential - not algebraic - sense). For consistency reasons note that the differential basis elements symbolized by dy b are defined to be closed. That is, d(dy b ) = 0. The exterior differentials of an arbitrary basis, σk , are NOT necessarily zero, but d(σ k ) is closed for dd(σ k ) = 0. The exterior differential of a 1-form is defined as, dω 1 = = = =

d(Ab dy b ) = (dAb )ˆdy b + Ab d(dy b ) (∂Ab /∂y e dy e )ˆdy b + 0 (∂Ab /∂y e − ∂Ae /∂y b ) dy e ˆ dy b F[eb] dy [eb] = F[H] dy [H] .

(9.51)

The collective index notation permits the formula defining exterior differentiation to be generalized, dω p = d(AH dy H ) = (dAH )ˆdy H . (9.52) So to compute the exterior differential of any p-form, first compute the exterior (= total differential) of the scalar coefficients dAH and then exterior multiply the result into the remaining base elements of the form dy H , component by component. Also note that the special 1-form with gradient coefficients, ω1 = d{θ(y a )}, has an exterior differential equal to, dω = = = =

(9.53) d(d{θ(y a )}) 2 a b c c b 2 a c b b c ...{∂ θ(y )/∂y ∂y }dy ˆ dy + ... + {∂ θ(y )/∂y ∂y }dy ˆ dy ...{∂ 2 θ(y a )/∂y b ∂y c − ∂ 2 θ(y a )/∂y c ∂y b }dy c ˆ dy b 0,

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for C2 functions. Hence this special 1-form ω 1 is a closed 1-form, assuming the coefficient functions are twice differentiable. However, as the example 1-form ω 1 has a unique primitive function, θ(y a ), whose exterior differential creates ω1 = dθ, the 1-form ω is said to be not only closed, but also exact. The same concepts hold for all p-forms. A p-form is closed if its exterior differential vanishes, and the p-form is exact if it is constructed by means of the exterior differential operation acting on some (p-1)-form. There are differential forms that are closed but not exact, and those that are neither exact nor closed. The importance of closed and exact, or closed but not exact, p-forms is that they carry topological information about the domain of definition. For example, in a two-dimensional surface every hole is associated with a unique 1-form that is closed, but not exact. The number of closed but not exact 1-forms on a domain counts the topological number of holes. This fact is the basis of the Bohm-Aharanov idea in EM theory, and is at the foundation of the theory of flight in terms of the Joukowski transformation. Suppose that the given exterior differential p-form is expressed in terms of a non-integrable basis set σ H . Then the exterior differential formula becomes, dω p = d(AH σ H ) = (dAH )ˆσ H + AH (dσ H ).

(9.54)

Now it must be recognized that the second term (dσ H ) is not necessarily zero. Such complications arise when the Frame matrix generates 1-forms (σ k ) which are not closed. The basis Frame in that case is not uniquely integrable. Remember that the exterior differential has to be applied to products of 1forms in terms of a modified Leibniz rule, that alternates in sign for every other odd factor. For example, the exterior differential of the product of two 1-forms is, d(σ 1 ˆσ 2 ) = dσ 1 ˆσ 2 − σ 1 ˆdσ 2 .

(9.55)

Example: Compute the exterior differential of the function θ(y a ) = (y1)2 + (y2)2 + (y3)2 − 1. ω = dθ(y a ) = 2(y 1 dy 1 + y 2 dy 2 + y 3 dy 3 ).

(9.56)

Note that the zero set of the function describes a spherical 2-surface, and the coefficients of the deduced 1-form describe the normal field orthogonal to the tangent vectors on the surface. Direct computation demonstrates that ω is closed, as, dω = 2(dy 1 ˆdy 1 + dy 2 ˆ dy 2 + dy 3 ˆ dy 3 ) = 0.

(9.57)

Example: Compute the exterior differential of (Ax dx+Ay dy+Az dz)ˆ(Bx dx+ By dy + Bz dz). d(AˆB) = dAˆB − AˆdB = d(C[mn] dy [mn] ) = d((C[mn] )ˆdy [mn] .

(9.58)

The object which is the result of the exterior differentiation of the 2-form constructed by the product is a 3-form with completely antisymmetric indices. The modified

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Leibniz rule for products is required to make the two ways of computing the resultant 3-form compatible. The operation of the exterior differential acting on an arbitrary 1-form is defined as, d(Ak (y a ) σ k ) = {d(Ak (y a )} ˆ σ k + Ak (y a )ˆ{dσ k } = {∂Ak (y a )/∂y b }dy b } ˆ σ k + Ak (y a )ˆ{dσ k }.

(9.59)

Consider the case where the arbitrary 1-forms are known to the ¯ kto ¤ related ® be£ linearly k a ¯ differentials by means of the linear (Frame) formulas, σ = Fa (y) |dy i . Then, d(Ak (y a ) σ k ) = {∂(Ak Fak )/∂y b }dy b ˆ dy k ba /∂y b − ∂ A bb /∂y a }dy b ˆ dy k = {∂ A = Fb[ab] (y)dy b ˆ dy k ,

where This formula,

ba (y) = (Ak (y)Fak (y)). A

ba dy a ) = Fb[ab] (y)dy b ˆ dy k = Fb[ab] (y)dy [ab] = Fb[H] (y)dy [H] , d(ω 1 ) = d(A

(9.60) (9.61)

(9.62)

is valid on the initial state, or variety {y a , dy a ), whether the Frame matrix has an inverse or not. The coefficients of the 2-form correspond to the antisymmetric components of a "curl" (when n = 3). However, the exterior differential procedure generalizes to spaces of higher dimension. For differential forms expanded in terms of non-closed basis 1-forms, the exterior differential has two terms. The first term is just the exterior product of the total differential of the coefficient function(s) and the remaining factor of non-closed basis forms, while the second term is the exterior product of the functions and the exterior differentials of the non-closed basis forms. The operation of the exterior differential acting on a p-form follows the same formulas, using the collective ordered index, H, d(AH (y a ) σ H ) = {d(AH (y a )} ˆ σ H + AH (y a )ˆ{dσ H }.

(9.63)

Lets examine more carefully the situation for the exterior differential of a 1-form expanded in terms on non-closed basis 1-forms. The outcome of the exterior differential process is to produce a 2-form, which can be expanded in terms of products of 1-forms. For any particular basis 1-form, σ k , the differential is a 2-form, and as such it can be expanded in terms of the paired basis elements, σ [mn] . That is, dσ k = Λk[mn] (y e )σ [mn] = Λk[12] σ [12] + ... + Λk[34] σ [34] = Λk[12] σ 1 ˆσ 2 + ... + Λk[34] σ 3 ˆσ 4 .

(9.64)

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Hence the exterior differential of a 1-form where the basis σk is not integrable is given by the formula, d(Ak σ k ) = dAk ˆ σ k + Ak Λk[mn] σ [mn] . (9.65) When the basis forms σ k are closed in a differential sense, then the coefficients Λk[mn] vanish. How this relates to Affine and Topological torsion will be discussed below, along with the topic of anholonomic coordinates. The n basis 1-forms must be linearly independent otherwise the dimension of the vector space Λ1 {σ k } is not n. The implies that the exterior product of the n 1-forms σk are such that the n-form so constructed is not zero. ¯ ®For£basis 1-forms ¤ σ k constructed from a Frame matrix according to the formula ¯σ k = Fak (y) |dy a i, the non-zero property for the n-fold product implies that the Frame matrix has a non-zero determinant, ω n = σ 1 ˆσ 2 ˆ...ˆσ n = det[F ]dy 1 ˆdy 2 ....ˆdy n 6= 0.

(9.66)

Such domains are either positive or negative and are therefore said to be orientable. Example: Compute the exterior differential of the 1-form A = dz+ydx−xdy. dA = ddz + dyˆdx − dxˆdy = 0 − 2(dxˆdy).

(9.67)

Example: The Gradient: Compute the exterior differential of the general 0-form θ(x, y, z). dθ(x, y, z) = ∂θ/dx dx + ∂θ/dy dy + ∂θ/dz dz = grad θ · dr.

(9.68)

The coefficients form the gradient of the scalar function (3D). Example: The Curl: Compute the exterior differential of a general 1-form A = (Ax dx + Ay dy + Az dz).

dA = {∂Ay /∂x − ∂Ax /∂y}dxˆdy +{∂Az /∂y − ∂Ay /∂z}dyˆdz +{∂Ax /∂z − ∂Az /∂x}dzˆdx.

(9.69) (9.70) (9.71)

The coefficients form the components of the "curl A" in 3D. Example: The Divergence: Compute the exterior differential of the 2form in 3D. V = Udyˆdz − V dzˆdx + W dxˆdy. If V = [U (x, y, z), V (x, y, z), W (x, y, z), then,

(9.72)

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dV

= ∂U/∂x dxˆdyˆdz + ∂U/∂y dyˆdyˆdz + ∂U/∂z dzˆdyˆdz −∂V /∂x dxˆdzˆdx − ∂V /∂y dyˆdzˆdx − ∂V /∂z dzˆdzˆdx +∂W/∂x dxˆdxˆdy + ∂W/∂y dyˆdxˆdy + ∂W/∂z dzˆdxˆdy, = ∂U/∂x dxˆdyˆdz + 0 + 0 −0 − ∂V /∂y dyˆdzˆdx − 0 +0 + 0 + ∂W/∂z dzˆdxˆdy, = {∂U/∂x + ∂V /∂y + ∂W/∂z}dxˆdyˆdz = div(V)dxˆdyˆdz.

(9.73)

(9.74) (9.75) (9.76)

Note that these algebraic ideas do not depend upon the existence of a norm or a metric. Example: Derivation of the Maxwell Faraday induction equations. The Maxwell-Faraday induction equations are a set of partial differential equations that are logically deducible starting with the ordered sequence [1, 2, 3, 4]. Next assume the existence of "ordered coordinate" variables given the symbols [x, y, z, t]. Next assume the existence of an ordered set of functions of the coordinate variables, with symbols [Ax , Ay , Az , φ]. From these beginnings the Maxwell - Faraday equations follow as a consequence of the Exterior Calculus of Cartan. Construct the 1-form from the ordered set of functions and variables, A = Ax dx + Ay dy + Az dz − φdt.

(9.77)

curlE + ∂B/∂t = 0

(9.78)

Next construct the 2-form F = dA. Then construct the 3-form ddA which must vanish, ddA = dF ⇒ 0. In 4D the 3-form has 4 coefficient functions of partial derivatives that must vanish. These PDE’s correspond in format to the 4 Maxwell Faraday equations, with 3D symbols, divB = 0.

The symbols are defined in terms of the coefficient functions of the 1-form (of potentials) as, E = −grad φ − ∂A/t ,

B = curl A.

(9.79)

Now the choice of symbol functions and coordinate functions was completely arbitrary, but the format of the PDE’s that satisfy ddA = dF ⇒ 0 are always the same relative to the ordering process. Experimentally, the logical equations of Maxwell - Faraday have been exploited in electromagnetic applications. However, the SAME formulas (different symbols) are applicable to hydrodynamics (as well as other physical systems of interest). Surprisingly, little has been done with the induction equations in hydrodynamics. These are not analogies. These are consequences of the logic of the Exterior Calculus and have universal applicability.

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9.3.5 The Interior Product The interior product is an operation on p-forms that requires a direction Vector field, V. The interior product lowers the degree of a p-form, changing a p-form into a (p1)-form. The interior product of a Vector direction field and a zero form (function) is defined to be zero. The symbol for the interior product herein is taken to be i(V ). The interior product of a Vector field and an exact basis element equal to the differential of a coordinate dy a is not zero, but is defined to be equal to the ath component of V. Hence the fundamental definitions can be written as, i(V )θ(y a ) = 0 ,

i(V )dy a = V a .

(9.80)

It follows that the inner product with respect to the vector field V acting on a 1-form, A = (Ax dx + Ay dy + Az dz), is given by the expression, i(V )A = i(V )(Ax dx + Ay dy + Az dz) = (Ax V x + Ay V y + Az V z ).

(9.81)

An additional rule is required to take care of the antisymmetries of differential forms. That is for the product AˆB of two 1-forms, the interior product with respect to V becomes, i(V ){AˆB} = (i(V )A)ˆB} − Aˆ(i(V )B), (9.82) and i(V )i(V )A = 0,

(9.83)

similar to the modified Leibniz rule for the exterior differential. Other expressions can be worked out for higher p-forms using these rules: i(V )i(V )ω p = 0,

i(V )i(W )ω p 6= i(W )i(V )ω p .

(9.84)

Example: Compute the interior product of J = [J x , J y , J z ] in 3D, with the 3-form volume element, V ol = dxˆdyˆdz. i(J)V ol = J x dyˆdz − J y dxˆdz + J z dxˆdy.

(9.85)

i(V ){i(J)V ol} = i(V ){J x dyˆdz − J y dxˆdz + J z dxˆdy} = (J y V z − J z V y )dx +(J z V x − J x V z )dy +(J x V y − J y V x )dz.

(9.86)

Example: Compute the interior product of V = [V x , V y , V z ] with the 2-form i(J)Volume.

(9.87)

Note that the construction (in 3D) of the double interior product generates coefficients equal to the cross product of the two different vector fields, J and V , and the double interior product with the same vector is zero.

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The Lie Differential

The Lie differential with respect to a vector field generates a p-form ϑp from a p-form ω p . It is constructed from the raising operator d and the lowering operator i(V ). The general formula is, ωp ⇒ ϑp :

L(V ) ω p = i(V )dωp + d(i(V )ωp = ϑp .

(9.88)

Marsden has called this Cartan’s Magic formula. The reason is that most of the equations of mechanics can be put into this form or derived from its construction. For example, those processes V which are "Hamiltonian" processes are those V such that i(V )dωp is exact. It is also remarkable that this formula is equivalent to the first law of thermodynamics. Consider a 1-form of Action, A, that presents a physical system (this will be done in detail in later sections). Then consider a vector field V that represents an evolutionary process. Define the 0-form (scalar function) of internal energy as U = i(V )A, the 1-form of Work as W = i(V )dA, and the output 1-form ϑp as Q. Then Cartan’s Magic formula becomes, L(V ) A = i(V )dA + d(i(V )A = W + dU = Q.

(9.89)

which is to be recognized as the first law of thermodynamics for a physical system A undergoing an evolutionary process V . This result will be exploited in later sections. Example: Compute the Lie differential with respect to V = [F, V, 1] acting on the 1—form, A = pˆdq − Hˆdt: L(V ) A = = = = =

i(V )dA + d(i(V )A) i(V ){dpˆdq − dHˆdt} + d(pV − H) i(V ){dpˆdq − ∂H/∂p dpˆdt − ∂H/∂q dqˆdt} + d(pV − H) F dq − V dp − F (∂H/∂p)dt − V (∂H/∂q)dt + dH + d(pV − H) (9.90) F (dq − ∂H/∂pdt) − V (dp + ∂H/∂qdt) + d(pV ).

Note that the RHS of the equation above is a perfect differential for all evolutionary vector fields with components V = [F, V, 1], if the two bracket factors vanish. Therefore, vector fields that are generated from the partial derivatives of H(p, q, t) according to the formulas, (dq − ∂H/∂pdt) ⇒ 0 ⊃ V = ∂H/∂p, (dp − ∂H/∂qdt) ⇒ 0 ⊃ F = −∂H/∂p,

(9.91) (9.92)

which produce a 1-form of heat Q which is closed, dQ ⇒ 0. Such processes (vector fields) are defined to be Hamiltonian vector fields (processes). Hamiltonian dynamics is the (constrained) domain of much of theoretical mechanics. The domain is

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constrained, as the 3-form QˆdQ ⇒ 0. Such processes are then always thermodynamically reversible. Later on, irreversible processes for which QˆdQ 6= 0 will be studied. One notes for Hamiltonian processes, L(V ) H = = = =

i(V )dH F ∂H/∂p + V ∂H/∂q + ∂H/∂t F V − V F + ∂H/∂t ∂H/∂t.

(9.93)

If H is independent from time, then H is an evolutionary invariant. In mechanics, the function H is typically defined to be equal to be the sum of kinetic and potential energy, H = p2 /2m+ϕ(x), so that time independent Hamiltonian processes "conserve energy". Even if the Hamiltonian is a function of time, Hamiltonian processes are thermodynamically reversible, as QˆdQ = 0. 9.3.7 The Pullback with examples First consider differential forms whose coefficients would be covariant tensor fields, if the Jacobian matrix has an inverse. A typical representation is a 1-form written in terms of the variables on the final state as, A = Aµ (xν ) dxµ = hAµ (xν )| ◦ |dxµ i .

(9.94)

Consider the differentiable non-linear map, φ : dφ :

y k ⇒ xµ = φµ (y j ), £ £ ¤ ¯ ¤ ® dy k ⇒ dxµ = ∂φµ (y j )/∂y k dy k = Jkµ (y j ) ◦ ¯ dy k .

(9.95) (9.96)

Substitute these formulas into the expression for the differential 1-form expressed in terms of the independent variables on the final state, ¯ £ ­ ¤ ¯ ® (9.97) A = hAµ (xν )| ◦ |dxµ i = Aµ (φν (y j ))¯ ◦ Jkµ (y j ) ◦ ¯ dy k ¯ ¯ k® ­ j ¯ ¯ = Ak (y ) ◦ dy . (9.98) ¯ ­ The coefficients Ak (y j )¯ are well defined functions on the initial state, with arguments in terms of the initial state variables. Now if the map from initial to final state is such that the Jacobian is an invertible matrix, then the coefficient variables, ¯ £ ¯ £ ¤ ­ ¤ ­ hAµ (xν )| ◦ Jkµ (y j ) = Aµ (φν (y j ))¯ ◦ Jkµ (y j ) = Ak (y j )¯ , (9.99) is equivalent to the transformation rules of a covariant tensor field in classic tensor analysis, (9.100) pullback Rule : Aµ ∂xµ /∂y k = Ak .

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Remark 64 The pullback coefficients are not tensor equivalents when the Jacobian matrix is not invertible. However, the pullback of scalar differential forms is always well defined in terms of the language which uses the Jacobian matrix transpose. The transpose always exists even though the inverse does not. Example: The pullback of a 1-form. Consider the example map from three dimensions to three dimensions, for which an inverse Jacobian does not exist: {X, Y, Z} ⇒ {x, y, z} = {XY, Y 3 , X}. {dX, dY, dZ} ⇒ {dx, dy, dz}, {Y dX + XdY, 3Y 2 dY, dX} . £ µ j ¤ ∂φ (x )/∂X k , Jacobian [Jkµ (X)] ⎤ ⎡ Y X 0 = ⎣ 0 3Y 2 0 ⎦ . 1 0 0 µ det [Jk (X)] = 0.

(9.101) (9.102) (9.103) (9.104)

Given : on final state A = y dx − xdy + dz, Substitute : φ and dφ = Y 3 (Y dX + XdY ) − XY 3Y 2 dY + dX, pullback : to initial state = (Y 4 + 1)dX − (2Y 3 X)dY + 0dZ.

(9.107) (9.108)

φ dφ

: : = =

Coef ficients : on final state hAµ | = [y, −x , +1]. pullback : to initial state ­ ¯ Ak ¯ = [(Y 4 + 1), −(2Y 3 X), 0], = hAµ | ◦ [Jkµ ] , ⎡ ⎤ Y X 0 = hy, −x , +1| ◦ ⎣ 0 3Y 2 0 ⎦ , 1 0 0 ¯ ­ = yY + 1 , yX − x3Y 2 , 0¯ , ¯ ­ = (Y 4 + 1), −(2Y 3 X), 0¯ .

(9.105) (9.106)

(9.109) (9.110) (9.111)

(9.112) (9.113) (9.114) (9.115) (9.116) (9.117)

The example demonstrates the fact that the coefficients of the differential forms do not behave as tensors with respect to the non-invertible, but differentiable map. Yet, everything is well defined, functionally, with respect to the pullback operation.

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Example: Pullback of a 2-form. For the same map given in the first example, consider the 2-form below: Given on f inal state (9.108) F = Fxy dxˆdy + Fyz dyˆdz + Fzx dzˆdx = dA. Substitute = Fxy (3Y 3 dXˆdY ) − Fyz (3Y 2 )dXˆdY + Fzx (XdXˆdY ), to yield the pullback to the initial state, = {Fxy (3Y 3 ) − Fyz (3Y 2 ) + Fzx (X)}dXˆdY. Coef ficients on f inal state hFµν | = hFxy , Fyz , Fzx | . pullback to initial state ¯ ¯ ­ ­ F XY ¯ = {−Fyz (3Y 2 ) + Fzx (X) + Fxy (3Y 3 )}, 0, 0¯ .

Example: bian.

(9.118) (9.119) (9.120)

(9.121) (9.122)

Pullback of a contravariant tensor density and the adjoint Jaco-

Now consider the volume element 3-form, V ol3 := dxˆdyˆdz = (det[J])dXˆdY ˆdZ,

(9.123)

and the Current (N-1)-form density, C = i(C µ )V ol3 = C x dyˆdz − C y dzˆdx + C z dxˆdy.

(9.124)

The adjoint matrix to the Jacobian (for which no inverse exists) is, ⎤ ⎡ 0 0 0 0 0 ⎦. [J]Adj = ⎣ 0 2 −3Y X 3Y 3

(9.125)

Substitution yields,

C = −C x (3Y 2 )dXˆdY + C y (XdXˆdY ) + C z 3Y 3 dXˆdY. = {−(3Y 2 )C x + (X)C y + 3Y 3 C z }dXˆdY. Z

= C dXˆdY,

from which it is apparent that ¯ X ¯ x + ¯ C + ¯ C ¯ ¯ ¯ Y Adj ⇐ [J] ◦ ¯¯ C y . ¯ C ¯ Z ¯ Cz ¯ C

(9.126) (9.127) (9.128)

(9.129)

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Note that these results are valid for the case where the inverse Jacobian does not exist. For those cases where the inverse matrix does exist, it is apparent that the Current coefficients do not transform as a contravariant tensor, but instead transform as a contravariant tensor density. Multiplication of both sides of the preceding equation by the Jacobian matrix yields, k

k

[Jkµ ] ◦ |C (Y j )i = Jkµ C ⇒ ∆ · C µ (x) = (det [Jkµ ]) · C µ (x).

(9.130)

If the coefficients transformed as a contravariant tensor, then the preceding formula would have been written as, k Jkµ C ⇒ C µ (x). (9.131) Tensor densities can be sensitive to the sign and magnitude of the determinant of the mapping. 9.3.8

Some Topological Features

The concepts of intersections, closure, and limit points are fundamental topological concepts that have relationships to the Cartan Calculus. In certain situations the exterior product exhibits properties of intersection operator, and the exterior differential exhibits properties of a limit point operator. More formally, given a domain with two exact 1-forms in 3D, the exterior product of the two exact 1-forms (if not zero) represents the points of intersection of the two implicit surfaces generated by the two functions whose gradient coefficients make up the components of the two exact 1-forms. Example: The exterior product and the concept of intersection. Consider two 1-forms created by applying the exterior differential to two distinct functions α(x, y, z) and β(x, y, z). The coefficients of dα = grad(α) ◦ dr form the gradient field, grad(α), which is perpendicular to the implicit surface α(x, y, z) = 0. Similarly, dβ = grad(β) ◦ dr, implies that the gradient coefficients, grad(β), are perpendicular to the implicit surface, β(x, y, z) = 0. If the two implicit surfaces intersect, then exterior product of the two 1-forms create a 2-form, J = dαˆdβ, which is not zero. The components of the 2-form, J, can be interpreted as a contravariant vector in 3D, which is tangent to the points in common (intersections) that make up the intersection of the two surfaces. For n = 3, the number of components of a 2-form are 3, and are in agreement with the 3D cross-product formulas of Gibbs. Example: The exterior differential is a limit point generator. From another point of view, it is possible to deduce a topological structure from a given 1-form A on the domain. It is possible to show that the exterior differential, relative to this Cartan topology, acts as a generator of the limit points of the given topology. This is given further credence from the physical idea that the divergence (an application of the exterior differential) of the D field in electromagnetism has finite values that terminate on charges. That is, the Faraday lines of D come from limit

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points of positive charge and wind up on limit points of negative charge. However, the concept that the exterior differential d is a limit point operator is more formal, and has a basis in Kuratowski’s closure operator. Example: The Lie differential can be used to select topological invariants of a process. The Lie differential with respect to a vector field V may be construed as a convective propagator describing the flow of the points of a p-form down the flow lines generated by V. If the p-form is integrated over a domain of such flowing points, then it is possible to ask if the integral is an invariant of the flow. Moreover it is possible to ask if the flowing points are distorted and deformed, does the integral over the deformed points equal the integral over the non-deformed points. If it is true that the value of the integral is unchanged by continuous deformation, then the integral must represent a topological property. To deform the flowing points is easy enough; just multiply the original vector field V by a function of (say) λ(x, y, z, t). The function λ does not change the flow lines generated by V, but it does deform the points that make up the flowlines by stretching or compression along the flow lines. Then V ⇒ λV and L(V ) ⇒ L(λV ) , and the Lie differential becomes a deformation operator. If it can be shown that if, I I I L(λV ) A = i(λV )dA + d(i(λV )A) = 0, (9.132) H for any function λ then the closed integral A is a deformation invariant of the H process. Note that the second integral always vanishes, d(i(λV )A) = 0, as the integrand is an exact perfect differential. For the first integral to vanish for arbitrary deformation parameter, λ, the integrand must be zero. This leads to the conclusion that, if

i(λV )dA = λi(V )dA = 0 any λ, I then A = deformation invariant.

(9.133)

Hence ifH the Work 1-form is zero, W = i(V )dA ⇒ 0 then the closed integral of the Action A is a topological property (of that process). Cartan has shown that a necessary and sufficient condition for a process to be a Hamiltonian process, is that the closed integral of the Action should be a topological invariant of the process. Example: The first law of thermodynamics is related to a topological statement of Cohomology. A non-exact closed p-form Q is defined to be Cohomologous to another nonexact closed p-form, W, if the difference between the two p-forms is exact. This means that the integrals of the two different p-forms over any closed integration path (cycle or boundary) are the same. For non-exact, closed, 1-forms of Heat, Q, and Work, W , the cohomological statement is the germ of the First Law, Q − W = dU.

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It was noted above that the Lie differential with respect to a vector field (process) acting on a physical system described by a 1-form of Action, is essentially a Cohomological statement of the first law. The Lie differential is a Cohomological generator. Example: Thermodynamic Isolation and Frobenius integrability. In thermodynamics, it is recognized that there are isolated, closed, and open systems. These words are also used to describe topological properties. A set is topologically isolated if it has no intersection with its limit points. This result translates to AˆdA = 0 for a given 1-form and its induced Cartan topology. The constraint of isolation is also equivalent to the Frobenius idea of unique integrability. That is, when AˆdA = 0, there exists a unique function whose gradient (or surface normal) is proportional to the given coefficients of the given 1-form. Caratheodory’s statements about inaccessible states is a statement related to the concept of isolation and connectivity to an equilibrium system. When AˆdA = 0 (no matter what the dimension of the coordinate space happens to be) there exists a transformation to a domain of two independent functions that will describe the properties of the 1form. That is, the 1-form can be written as φdχ, and its coefficient functions are proportional to a gradient, dχ. The problem becomes essentially a two-dimensional problem. The property of isolation is a topological property, hence if a process causes AˆdA 6= 0 to change to AˆdA = 0, or from a state where AˆdA = 0 to a state where AˆdA 6= 0, a topological change has take place. In hydrodynamics, all streamline flows satisfy AˆdA = 0. Hence turbulent flows must involve domains where AˆdA 6= 0. The transition to (from) turbulence from (to) a state of non-turbulence must involve topological change. It should be mentioned that with respect to diffeomorphic transformations, or more simply those transformations that preserve pure geometrical properties, the differences between contravariant and covariant concepts cannot be distinguished. With respect to an aging process involving topological change, the behavior of the two concepts is observably different. 9.4 9.4.1

Closure and Continuity Closure

An important topological idea to be used in this monograph is the idea or concept of closure. The idea of closure is an invariant of a continuous but irreversible process. From set theoretic ideas, the elementary idea of closure means that any pair of elements of a subset can be combined by a rule such that the resultant is still an element of the subset. Closure is perhaps the most fundamental property of a group. Elements of a vector space can be added together such that each sum is an element of the set of all basis elements multiplied by real numbers. The process of addition is closed. However, if two polar elements of a vector space are multiplied together by the method of the Gibbs cross product of engineering science, the resultant axial vector

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is not an element of the original subset of polar vectors. The Gibbs product is not closed. No engineer would ever add a torque to a force, or a linear momentum vector to an angular momentum vector, because they are not vectors of the same species. In the Cartan formalism, a different concept of closure is defined in terms of what is called a differential ideal. A differential ideal is the union of a system of differential forms and their exterior differentials. For 1-forms, Chapter 6 details how to construct a topology in terms of the closure of a 1-form, A, the exterior differential of dA, the closure of A (equal to the union of A and dA, A∪dA), the exterior product (similar to the concept of set intersection) of A and dA (formally written as A^dA), and the closure of the exterior product A^dA (equal to the union of AˆdA and dAˆdA, AˆdA∪dAˆdA). Relative to this "Cartan topology", the exterior differential becomes an operator that creates the topological limit sets of each subset of the topology (see Chapter 6). At this beginning level, the thing to remember is that limit sets are topological properties, and they can be determined for a particular exterior differential p-form by constructing the exterior differential of the particular p-form. This differential process is much easier to compute than is the integration process. In this manner, global (integral) topological information appears at a local (differential) level. Chapter 6 describes these and many other details of the Cartan topology. Key observables in the understanding of the aging process are related to the concepts of closure and connectivity of the non-equilibrium states. Experimental methods to observe "closure" concepts must be devised if the notion of topological evolution is to be made practical. These notions may sound abstract and not useful, but when it is realized that the production of defects in a physical system, and the change of phase from solid to liquid, are exhibitions of topological evolution, then the ideas become more concrete. The topological methods employed herein can be used to determine when a physical system is in an equilibrium or non-equilibrium state. The topological methods employed herein can be used to distinguish thermodynamically irreversible from reversible evolutionary processes. The topological methods employed herein can be used to describe the irreversible dissipative decay processes from open systems into excited stationary states far from equilibrium, and the further decay from excited non-equilibrium states into equilibrium ground states. 9.4.2

Continuity

An intuitive idea of continuity is built on the notion of a single valued function, or transformation, without breaks. The formal∗ topological definition [141] of a continuous transformation between a set X with topology T 1 to a set Y with a topology T 2 states that the transformation is continuous if and only if the inverse image of open sets of T 2 are open sets of T 1. The important point is that the ∗

These formal defintions are most rapidly learned in terms of point set topology, and are presented in the appendix in more detail.

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topologies T 1 and T 2 need not be the same for a continuous transformation. A space is said to have a topological structure if it is possible to determine if a transformation on the space is continuous [79]. There exists another more useful method of defining continuity which does not depend explicitly on being able to define open sets and their inverse images. This second method of defining continuity is based on the concept of closure, discussed in the preceding subsection. The formal topological closure of a set can be defined in (at least) two ways: 1. The closure of a set is the union of the interior and the boundary of a set. 2. The closure of a set is the union of the set and its limit points. The first definition of closure is perhaps the most common, and is often exploited in geometric situations, where a metric has been defined and a boundary can be computed easily. The second definition of closure is independent from metric and is the method of choice in this monograph, both for defining continuity and establishing a topological structure. In terms of the concept of closure, a transformation is continuous if and only if for every subset, the image of the closure of the initial subset is included in the closure of the image of that subset [141]. Another way of stating this idea is: 3. A map is continuous iff the limit points of every subset in the domain permute into the closure of the subsets in the range. It will be demonstrated that relative to the Cartan Topology base on a 1-form of Action, the exterior differential becomes equivalent to a limit point generator. Hence the Cartan Calculus gives an easy method for utilizing the concept of limit points. 9.5

Point Set Topology

In order to establish a foundation for topological evolution, an introduction to topological ideas and definitions is presented in terms of point set methods for which the topological concepts can be exhibited in terms of simple examples. This expose of topology given in this monograph will not be complete, and will not cover all of topological theory. Only those parts of topology that I consider to be necessary and useful for the development of physical and engineering applications will be presented. A conventional introduction to topology often starts with a metric topology, but herein the concept of a metric is purposely avoided, as the idea of a metric is the essence of those geometrical qualities of size and shape. The conventional procedure is to develop the topological ideas in terms of a space with a Euclidean or some Riemannian metric. Then the topological concepts are shown to be independent of the choice of metric. However, the notion of a metric is not needed, and the point set approach takes that point of view that the metric is just extra baggage that can often confuse the issues. Perhaps the best way to learn basic ideas about topology is through the study of point set topology. The concepts and definitions can be illuminated by means of

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examples over a discrete and small set of elements. The early champions of point set topology were Kuratowski in Poland and Moore at UT-Austin. For a long time Point Set topologists were isolated from the Combinatorial Topologists. In fact the name topology, evidently, was introduced about 1925, about the time that it was recognized that topology had many equivalent expositions. One of the best books for rapid assimilation of point set topology is the Schaum’s Outline Series, "General Topology" by S. Lipschutz [141]. Review Chapter 1, then skip to Chapters 5, 6, and 7. In that which follows, four different topologies will be defined over a set of five elements, {a,b,c,d,e}. Then the topological definitions of Open sets, Closed Sets, Limit Points, Closure, Boundary, Interior, Exterior, and the concept of Continuity will be defined and exemplified for each of the four point set topologies. It should become apparent that the same sets of points can have different topologies imposed as a set of constraints on the same elements. It is the topology that allows the concepts of boundary, closure and limit points to be define. In that which follows, the applications of these ideas will be done for systems of differential forms, rather than systems of points. 9.5.1 Closed and Open Sets Consider a set of elements {a, b, c, d, e} and a combinatorial process which is symbolized, for example, by the brackets (ab) or (ade). Construct all possible combinations, and include the null set, 0. Define X = (abcde). Now from the set of all possible combinations, it is possible to select many subset collections. Certain of these subset collections have the remarkable property of logical closure. As an example, consider the subset collection, or class of subsets, given by, T 1(closed) = {X, 0, a, b, (ab), (bcd), (abcd)}. (9.134) Note that the intersection of (a) with (ab) is a, (a ∩ ab = a), which is an element of the collection, and the intersection of (ab) with (bcd) is b which is also an element of the collection. In fact, the intersection of every element of T 1(closed) with every other element of T 1(closed) produces one of the original seven elements of the collection, T 1(closed). In other words, the process of set intersection acting on any number of elements is closed with respect to T 1(closed). Now also note that the union of any two elements of the set is also contained within the set. The idea that a closed algebra can be built upon the notions of union and intersection, and that this algebra be a division algebra, is at the heart of the theory of logic. This idea of logical closure with respect to arbitrary intersection and finite union is said to define a topology, T 1(closed), of closed sets. Definition: A topology T 1(closed) on a set X is a collection or class of subsets that obeys the following axioms: 1. A1(closed): X and the null set 0 are elements of the collection.

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2. A2(closed): The arbitrary intersection of any number of elements of the collection belongs to the collection. 3. A3(closed): The arbitrary union of any pair of elements of the collection belongs to the collection. The elements of the collection, T 1(closed), are defined to be "closed" sets. The compliments of the closed sets are defined as "open" sets. The open sets of the topology are the collection of subsets given by, T 1(open) = {0, X, (bcde), (acde), (cde), (ae), e}.

(9.135)

It is important to note that the same set of all combinations of subsets can support many topologies. For example, the subsets of the collection, T 2(closed) = {X, 0, (bcde), (cde), (de)},

(9.136)

are closed with respect to both logical intersection and union. Hence T 2(closed) is a different topology built on the same set of points, X. The open sets of this topology are, T 2(open) = {0, X, a, (ab), (abc)}. (9.137) The compliments of closed sets are defined to be "open" sets and they too can be used to define a topology. A subset can be both open, or closed, or both, or neither, relative to a specified topology. For example, with respect to the topology given by the closed sets, T 3 = {X, 0, a, (bcde)},

(9.138)

(bcde) is both open and closed, and the set (bc) is neither open nor closed. The topology of closed sets given by the collection, T 4(closed) = {X, 0, (bcde), (abe), (be), (a)},

(9.139)

has its dual as the topology of open sets T 4(open) = {0, X, a, (cd), (acd), (bcde)}.

(9.140)

Note that this topology, T 4, is a refinement of the topology, T 3, in that it contains additional closed (or open) sets. Remarks: In the definition of a topology when the number of elements of the set is not finite, the logical intersection of open sets is restricted to any pair, and the logical union of closed sets to restricted to any pair. There are many other ways to define a topology, but the concepts always come back to the idea of logical closure.

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9.5.2 Limit Points The next idea to be presented is the concept of a limit point. A standard definition states that a point p is a limit point of a subset, A, iff every open set that contains p contains another point of A. Note that p is an element of X and need not be an element of A. Given a subset, A, each point of X must be tested to see if it is a limit point of A relative to the topology specified on the points. If A is a singleton, it can have no limit points, for there are no other points of A. It follows that the limit points of a limit point (a singleton) is the null set. If the limit point of A consists of the singletons or points symbolized by dA, then d(dA) = 0. The set of limit points as a collection of singletons, {a, b, c..} will be denoted by dA, where the union of all limit points will be denoted by symbol Aulp . The symbol d may be viewed as a limit point operator; the symbol d when applied to a set, A, means that each point p of the domain is tested against the specified topology to see if another point of A is included in each open set of the topology. Consider the subset A = (ab) and the topology given by T 1(open). Now test each point relative to the collection T 1(open), T 1(open) = {0, X, (bcde), (acde), (cde), (ae), e}.

(9.141)

The point a is not a limit point of (ab) because the open set (acde) which contains a does not contain b; dA = 0 at a. The point b is not a limit point of (ab) because the open set (bcde) does not contain a; dA = 0 at b. The point c is not a limit point of (ab) because the open set (cde) does not contain either a or b; dA = 0 at c. The point d is not a limit point of (ab) because the open set (cde) does not contain either a or b; dA = 0 at d. The point e is not a limit point of (ab) because the open set (cde) does not contain either a or b; dA = 0 at e. In other words, the subset (ab) has no limit points in the topology given by T 1. The limit point set of ab, designated in this monograph as Aulp (ab), is given by, Aulp (ab) = {0},

(9.142)

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the empty set, as dA = 0 at all points of X. Now make the same tests with regard to the same subset A = (ab), but this time relative to the topology given by T 2(open), T 2(open) = {0, X, a, (ab), (abc)}.

(9.143)

The point a is not a limit point of (ab) because the open set (a) is a singleton; dA = 0 at a. The point b is a limit point of (ab) because both the open sets that contain b also contain a; dA 6= 0 at b.

The point c is a limit point of (ab) because the open set X, the only open set that contains c, contains a and b which are points of A; dA 6= 0 at c.

The point d is a limit point of (ab) because the open set X, the only open set that contains d, contains another point of A; dA 6= 0 at d.

The point e is a limit point of (ab) because the open set X, the only open set that contains e, another point of A; dA 6= 0 at e.

Hence, the points b, c, d and e are limit points of A = (ab) relative to the topology T 2(open). The limit set of Aulp (ab) = (bcde). (9.144) With respect to the topology of T 4(open), T 4(open) = {0, X, a, (cd), (acd), (bcde)},

(9.145)

test for limit points of the set (ab): The point a is not a limit point of (ab) because the open set a is a singleton; dA = 0 at a. The point b is not a limit point of (ab) because the open set (bcde) does not contain a; dA = 0 at b. The point c is not a limit point of (ab) because the open set (cd) which contains c does not contain a or b; dA = 0 at c.

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The point d is not a limit point of (ab) because the open set (cd) which contains d does not contain a or b; dA = 0 at d. The point e is a limit point of (ab) because the open set (bcde) which contains e contains another point b of A; dA 6= 0 at e. The limit set of (ab) relative to T 4(open) becomes Aulp = {e}.

Note that the set of limit points as a collection, or a class of sets, may or may not have limit points. If the limit set is a singleton, then the limit points of the set of limit points is the null set. However, consider the limit set Aulp = {bcde} of the set (ab) relative to the topology T 2. Then the limit points of Aulp are the points (b, c, d, e). In other words dAulp 6= 0 necessarily, but ddAulp = 0. 9.5.3 Closure The closure of a set is defined to be the union of the set and its limit points. Note that a closed set contains its limit points, if any exist. In the examples given above the closure of the set A = (ab) relative to the topology T 1 is equal to the union of A = (ab) and its limit points, which is the null set, e = A ∪ Aulp = (ab). A

(9.146)

e = A ∪ Aulp = (ab) ∪ (bcde) = (abcde), A

(9.147)

e = A ∪ Aulp = (ab) ∪ (e) = (abe). A

(9.148)

Note that A = (ab) is a closed set, and has no limit points relative to the topology T 1(open). However, the closure of A relative to the topology T 2(open) is,

which is the whole set. When the closure of a subset is the whole set X, the subset is said to be dense in X relative to the specified topology. The closure of (ab) relative to the topology T 4(open) is,

Note that the closure of a subset is equal to the smallest closed set that contains the subset. Every closed set is its own closure. A closed set may or may not have limit points, but if it does have limit points they are contained within the (closed) set. 9.5.4 Continuity Now comes a major issue of this section. Continuity of a transformation is defined relative to the different topologies that may exist on the initial and final states. Let the set of points X with topology T 1(open) be mapped into the set of points Y with the topology T 2(open). Then the map is continuous iff the closure of every subset of the initial state relative to T 1 is included in the closure of the image of the final state relative to the topology T 2. Another test for continuity is given by the statement that the inverse image of every open set of Y relative to T 2 is an open set of X relative to the topology T 1.

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Later on, the first definition will be used to prove that any topology built on subsets of exterior forms with C2 (twice differentiable) coefficients will be continuously transformed by evolutionary processes that are generated by the Lie convective derivative with respect to C2 vector fields. For the present, the second definition will be used in terms of simple point set topological systems. As a first example consider the transformations given on X to Y by the following diagram:

Figure 9.3 Continuous, Discontinuous and Homeomorphic maps 9.5.5 Interior When emphasis is placed on open sets rather that closed sets, other ideas come to the forefront. In particular, the concept dual to the notion of closure is the concept of interior. While closure asks for the smallest closed set that covers any specified subset, the idea of interior asks for the largest open set included in the specified subset. The interior of a set can be empty (for there may be no open sets other than the null set contained within the specified set)!

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For example, the subset (ab) has no interior relative to the topology T 1(open). However, the interior of (ab) is itself, (ab), relative to the topology T 2(open), because (ab) is an open set in this topology! Relative to the topology T 4(open), the interior of (ab) is the singleton, (a):

IntA = {0} relative to T 1, IntA = (ab) relative to T 2, IntA = (a) relative to T 4.

(9.149) (9.150) (9.151)

The set (abe) has an interior (ae) relative to T 1(open) and an interior (ab) relative to the topology T 2(open). 9.5.6 Exterior The exterior of a specified set is the interior of the compliment of the specified set. The compliment of (ab) is the set (cde) which has the interior (cde) relative to T 1(open) and has no interior relative to the topology T 2(open). Relative to the topology T 4(open), the exterior of (ab) is the set (cd). ExtA = {cde} relative to T 1, ExtA = (0) relative to T 2, ExtA = (cd) relative to T 4.

(9.152) (9.153) (9.154)

9.5.7 The Boundary The points that make up the boundary of a subset are union of those points that are not included in the interior or the exterior. However, the union of the points that make up the boundary may have subsets that are not connected. Consider a solid disk. The points that make up the rim of the disk forms its boundary. Now punch a hole in the disk. The collection of points that make up the outer rim and the inner hole now form the boundary of the disk. The two sets of boundary points are not connected. Similar to the limit point operator, d, a boundary operator, ∂ (some books use δ), may be defined in terms of a procedure, such that when ∂ is applied to the set A, it implies that a test is made at each point p to see if p is an element of the interior or of the exterior of the selected subset A. If the test fails then ∂A 6= 0 and the point is a boundary point. If the point p is an element of the exterior or interior of A, then ∂A = 0 at the point p. The boundary of A, or bA, of the set A, is defined as the union of all boundary points. As a first example, consider again the set A = (ab) and the T 1 topology. The set A = (ab) has no interior, but the exterior of (ab) is the set (cde), and therefore the boundary set, Aboundary , consists of the union of the points (a, b). In this first example, then,

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but It follows that,

Aulp = {0},

e Aboundary = (ab) ⊂ A. A ∩ Aulp = 0, A ∩ Aboundary 6= 0.

(9.155) (9.156)

(9.157) (9.158)

The boundary set exists even though the limit set does not! Relative to T 2(open), the set (ab) has an interior set (ab), no exterior set, but a boundary set is Aboundary = (cde). In this case the boundary is included in the closure, Aulp = (bcde),

and,

(9.159)

e = (abcde), Aboundary = (cde) ⊂ A

(9.160)

A ∩ Aulp = 6 0, A ∩ Aboundary = 0.

(9.161) (9.162)

Although all boundary points are limit points, there exist limit points that are not elements of the boundary. Relative to T 4(open), the set (ab) has an interior set (a), and exterior set (cd) and a boundary set (be): Aulp = (e), Aboundary = (be),

(9.163) (9.164)

A ∩ Aulp = 0, A ∩ Aboundary 6= 0.

(9.165) (9.166)

It is apparent that the boundary points contain limit points, but there are boundary points which are not limit points!. In all cases, note that the union of the interior and the boundary is equal to the union of the set and its limit points. The boundary is always included in the closure, but the boundary may contain points which are not limit points. e = IntA ∪ Aboundary = A ∪ Aulp . A

(9.167)

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These examples point out that there exist certain correspondences between limit points and boundaries, but they are not necessarily the same concept. Much of current physical theory has emphasized the boundary and open set point of view, while in this monograph the emphasis is on the limit points and closure point of view. It will become evident that these concepts are at the heart of the differences between contravariant and covariant concepts in physical theories, an idea that ultimately expresses itself in the differences between the particle or wave perspective of physics. In topology, these notions are at the heart of the differences between Homology and Cohomology (which will be discussed in detail later). In this monograph, the cohomological point of view is emphasized. If a set has the property that its intersection with its limit set is empty, then the set is said to be isolated. This idea of isolation, whereby A ∩ Aulp = 0, can be translated into the Cartan statement, AˆdA = 0. The physical significance of the topological concept of isolation will be correlated with the Caratheodory statement of the existence of inaccessible states in a thermodynamic system, and to the notion of Frobenius complete integrability for a laminar, non-chaotic flow. The concept of isolation is a topological property. and its compliment is a necessary condition for chaos. The observation of a flow transforming from a laminar state (isolated) to a turbulent state (non-isolated) is an observation of topological evolution. It should be mentioned that with respect to diffeomorphic transformations, or more simply those transformations that preserve pure geometrical properties, the differences between contravariant and covariant concepts cannot be distinguished. Further note that the existence of a metric implies that the contravariant concepts can be converted into covariant concepts, and their possible differences are masked into an alias-alibi format; that is, there are no measurable differences between the two concepts. However, with respect to an aging process, the behavior of the two concepts is observably different. The differences between the behavior of contravariant and covariant concepts may be interpreted as the existence of topological evolution. Note that although all of the symbols used above are familiar in the realm of electromagnetism, the topological results and formulas obtained apply to any set of symbols, representing an arbitrary physical system, for example a fluid. The Faraday Maxwell equations are universal ideas on continuous physical systems of C2 functions. Computational examples in Maple code are given in Volume 6, "Maple programs for Non-Equilibrium systems" [278], or may be found on the Cartan website. http://www22.pair.com/csdc/pdf/maxwell.pdf 9.6 9.6.1

Mappings, Basis Frames, and Connections From the point of view of topology

Exterior differential forms are objects that are well behaved (in terms of functional substitution) with respect to differentiable mappings which need not be invertible. In this sense their properties go well beyond the constrained domain of tensors, which

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are defined relative to maps with inverse, and where both map and inverse are differentiable (that is, diffeomorphisms). Exterior differential forms are of two types. They consist of coefficient functions and differentials that behave as scalars with respect to functional substitution, or they behave as scalar densities. I find that the use of vector arrays and matrix arrays of exterior differential p-forms to be superior to the use of tensor notation. These arrays are naturally covariant with respect to diffeomorphisms, and do not suffer from the plague of too many indices. Moreover, certain subtle non-abelian effects appear in a natural way. Many of the ideas have a foundation in differentiable mappings, which will be emphasized, at first. Then the fundamental ideas will be extended to the non-integrable cases. 9.6.2

The Jacobian matrix as a Basis Frame

Consider a differentiable non-linear map, φ, from a domain {y a } to a range {xk }. Differentiation of the mapping functions leads to a¯ linear ® map, dφ, between differentials a k ¯ on the initial state, |dy i , and the final state, dx . In symbols,

Non-linear φ : {y a } ⇒ {xk } = φk (y) 1 ≤ k ≤ n, (9.168) ¯ k® £ k £ k ¤ ¤ a a a a i. Linear dφ : |dy i ⇒ ¯dx = ∂φ (y)/∂y ◦ |dy i = Ja (y) ◦ |dy (9.169)

£ ¤ The differential mapping is linear in the differentials, and the matrix Jka (y) so generated is the Jacobian matrix of partial differentials of the coordinate mapping functions. The determinant of the Jacobian matrix is not zero over the domain of diffeomorphic constraint. Therefore on that domain of non-zero determinant, the Jacobian matrix can be used as a (global) vector Basis Frame of functions. At first, the mapping functions, φ, will be assumed to be at least C2 differentiable. For demonstration purposes partition the Jacobian matrix Basis Frame into columns of "horizontal vector components", ekα , in a (n-1)D subspace, and a vertical vector, nk . The superscript index k represents the (row) components of the column vectors, where the subscript index α represents a particular column vector. The columns of the Jacobian matrix form a linearly independent set of n basis vectors when the determinant of the matrix is not zero: ⎡

e1α ⎢ e2α [Jka (y)] = ⎢ ⎣ ... enα

e1β e2β ... enβ

... ... ... ...

ekα = ∂φk (y)/∂y α , nk = ∂φk (y)/∂y n .

⎤ n1 /λ £ ¤ n2 /λ ⎥ ⎥ = ek1 ek2 ... nk , ⎦ ... n n /λ

(9.170) (9.171) (9.172)

Consider the Frame matrix with a slightly different but perhaps more familiar

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449

notation. Identify, φn ⇒ t ∼ = t(y 1 , ..y n−1 , τ ), ⎡ 1 eα e1β ⎢ e2α e2β [Jka (y)] = ⎢ ⎣ ... ... ∂t/∂y 1 ∂t/∂y 2

V k (y, t) = ∂φk /∂τ , ⎤ ... ∂φ1 /∂τ ... ∂φ2 /∂τ ⎥ ⎥. ⎦ ... ... ... ∂t/∂τ

(9.173) (9.174)

If the last mapping function (representing time in a dynamical sense) is independent of the spatial variables, such that time is a spatially global concept, then the last row has n-1 zeros, and the Frame matrix becomes a representation of the 13-parameter (in 4D) P-Affine subgroup (see (9.24)) of the projective group, ⎡ 1 1 ⎤ eα eβ ... V 1 (y, τ ) ⎢ e2α e2β ... V 2 (y, τ ) ⎥ ⎥. P-Affine [Jka (y, τ )] = ⎢ (9.175) ⎣ ... ... ... ... ⎦ 0 0 0 ∂t/∂τ The group is a transitive group with no fixed points; all points are translated, including the origin. Note that "time" is not dependent upon the spatial coordinates. It is universally the same at all points, in the sense of Newtonian dynamics. On the other hand, if the coefficients of the partition are such that, ⎡ 1 ⎤ eα e1β ... 0 ⎢ e2α ⎥ e2β ... 0 ⎥, W-Affine [Jka (y, τ )] = ⎢ (9.176) ⎣ ... ⎦ ... ... ... ∂t/∂y 1 ∂t/∂y 2 ∂t/∂y 3 ∂t/∂τ

the Frame matrix is intransitive (see (9.27)) and a different 13-parameter subgroup of the projective group. It differs from the P-Affine subgroup as there is a fixed point (the origin) to be associated with rotations and expansions. The P-Affine group is more appropriate for a Particle contravariant ideas, and the W-Affine group is more for covariant Wave ideas — hence, the specialized notation is appropriate. 9.6.3 The Right and Left Cartan Connections Matrices of 1-forms and the Right Cartan Connection, [C] Now consider a more general case, where a Basis Frame, [B], is defined by some appropriate arguments. This Basis Frame may not be associated with an integrable Jacobian matrix. That is, the infinitesimal map of nearby neighborhoods, ¯ ® £ k¤ Ba ◦ |dy a i = ¯σk , (9.177) ¯ ® may not be integrable. The 1-forms ¯σ k may not be exact differentials, such that a global non-linear extension as defined by equation (9.168), is not possible.

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As the Basis Frame matrix of C2 functions is presumed to have a non-zero determinant, then it has an inverse matrix, [B]−1 , such that, [B]−1 ◦ [B] = [B] ◦ [B]−1 = [I] . The exterior differentiation of the C2 functions that make up the basis frame and its inverse leads to the matrix equation, d [B] = − [B] ◦ d [B]−1 ◦ [B] , d [B] = [B] ◦ [C]. Differential Closure

(9.178) (9.179)

It is apparent that the exterior differential of each of the basis vectors (columns of the Basis Frame) is a linear combination of the original basis vectors. The result is defined as differential closure. The matrix [C], with matrix elements that are differential 1-forms, is defined as the Right Cartan matrix of connection 1-forms, [C] = −d [B]−1 ◦ [B] = + [B]−1 ◦ d [B] .

(9.180)

Note that this concept of a connection and differential closure is algebraic and does not involve an explicit metric constraint. Matrices of 1-forms and the Left Cartan Connection, [∆] It is also possible to define a left Cartan Connection matrix of 1-forms, which will be given the symbol [∆]: d [B] = = [∆] = [∆] 6=

− [B] ◦ d [B]−1 ◦ [B] [∆] ◦ [B] , − [B] ◦ d [B]−1 = +d [B] ◦ [B]−1 , [C].

(9.181) (9.182) (9.183) (9.184)

The right and left Cartan Connection matrices are not the same, but they are related by a similarity transformation defined in terms of the Basis Frame, [B]−1 ◦ [∆] ◦ [B] = [C].

(9.185)

This point is often missed, or ignored, in most tensor dominated treatments. 9.6.4 The vector of Cartan Torsion 2-forms The concept of a Basis Frame [B] with inverse [B]−1 also leads to a formulation of the matrices of connection 1-forms for the inverse Basis Frame, d [B]−1 = [B]−1 ◦ [−∆] = [−C] ◦ [B]−1 .

Mappings, Basis Frames, and Connections

Multiplication of the equation (9.177) [B]−1 yields the formula, ¯ ® |dy m i = [B]−1 ◦ ¯σ k .

451

(9.186)

Another exterior differentiation leads to the expression in terms of the left Cartan connection, [∆], ¯ ® ¯ ® (9.187) d |dy m i = 0 = [B]−1 {d ¯σ k − [∆] ˆ ¯σk }.

By linearity, the bracketed factor on the right, as a vector of 2-forms, must vanish. The bracketed factor is defined as the vector of Cartan Torsion 2-forms, ¯ ® (9.188) Cartan Torsion 2-forms : ¯ΣCar tan _torsion_2_f orms ¯ k® ¯ k® ¯ ® ¯ΣCar tan _torsion_2_f orms = {d ¯σ − [∆] ˆ ¯σ } ⇒ 0. (9.189)

This formula is not (usually) equivalent to the vector of Affine Torsion 2-forms, which is given by the expression in terms of the right Cartan connection, ¯ ® ¯Σaf f ine_torsion_2_f orms = [C] ˆ |dy a i .

(9.190)

For a Cartan space, Cn (Cartan), the Cartan vector of Torsion 2-forms must vanish. However, the vector of Affine Torsion 2-forms does not vanish, necessarily. These concepts are free from metric considerations. This bracket vector that defines the vector of Cartan Torsion 2-forms is another (the second) Cartan equation of structure. The first equation of structure will be defined below in terms of the Cartan matrix (not vector) of curvature 2-forms. Both equations of structure are zero for vector spaces defined by Basis Frames, [B] . Even though the Basis Frame is not a Jacobian matrix of an integrable global mapping, it is possible to study the linear relationship, ¯ ® [B] ◦ |dy a i = ¯Ak , (9.191)

as defining a nearby (infinitesimally close) neighborhood. The notation has been changed slightly from that written in equation (9.177) for purposes will become ¯ kthat ® k ¯ more evident. Each 1-form element, A , of the vector array, A , need not be exact, nor closed, but might be integrable or even non-non-integrable. These are topological of the matrix elements (Pfaffian forms) of the vector array of ¯ properties ® 1-forms, ¯Ak . Each 1-form may be viewed as a 1-form of covector potentials, hence the suggestive notation can utilize concepts developed in the topological theory of electromagnetism, [276]: Ak = Akm dxm .

(9.192)

Exterior differentiation of the nearby neighborhood equation (9.191) leads to ¯ the ® expression for a vector array of exact 2-forms, that define the field intensities ¯F k , constructed from each 1-form of covector potentials,

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¯ ® d([B] ◦ |dy a i) = [B] ◦ [C] ˆ |dy a i = ¯dAk ¯ ® ¯ ® Field intensities ¯dAk = ¯F k .

(9.193) (9.194)

9.6.5 The vector of Affine Torsion 2-forms, |Ga i (Not Cartan Torsion) The vector of "Affine Torsion" 2-forms |Ga i † is given in terms of the right Cartan Connection, [C], by the expression, |Ga i = [C] ˆ |dy a i .

(9.195) a

It is then evident that this vector array of "Affine ¯ ® Torsion" 2-forms |G i is related to the vector array of Field intensity 2-forms, ¯F k , by means of a "constitutive" map, [B]−1 , ¯ ® |Ga i = [B]−1 ◦ ¯F k .

(9.196)

Each element of the vector of 2-forms, |Ga i , can be viewed as the antisymmetric components ¯ k ®of the electromagnetic excitations (D and H). Similarly, the vector of 2-forms ¯F can be viewed as the antisymmetric components of the electromagnetic intensities (E and B).¯ ® ¯ ® If the 2-forms ¯dAk = ¯F k ⇒ 0, it is ¯obvious that the coefficients of Affine ® a k ¯ Torsion are zero, |G i ⇒ 0. If the 1-forms A satisfy the Frobenius integrability theorem, then there exist integrating factors, λ(k) , such that for each 1-form designated by the index, (k), d(λ(k) A(k) ) ⇒ |0i . So multiplication of both sides of the equation (9.191) by a diagonal matrix of integrating factors (if they exist) leads to a new formula, and a new Basis Frame, b [B]: ¯ ® (k) (k) [λdiagonal ] ◦ [B] ◦ |dy a i = [λdiagonal ] ◦ ¯Ak , ¯ E ¯ (k) k a b [B] ◦ |dy i = ¯λ A ,

(9.197)

(9.198)

with an exterior differential,

¯ E b ◦ [C] b ◦ |dy a i = d ¯¯λ(k) Ak = |0i . [B]

(9.199)

b and its modified Hence an algebraic modification of the Basis Frame, [B] ⇒ [B], c b have a zero matrix of affine torsion 2-forms, |Gi, Cartan Connection, [C], †

c = [C] b ◦ |dy a i ⇒ |0i . Modified Affine Torsion |Gi

(9.200)

The symbol |Gi will be used alternately with the symbol |Σi to represent the vector of "Affine Torsion" 2-forms.

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453

¯ k® ¯A have been reduced to zero! Those 1The Affine Torsion 2-forms for integrable ¯ k® ¯ forms A that are not integrable produce irreducible components of Affine Torsion. It is these irreducible Affine Torsion 2-forms that are of interest, in the sense that they are to be associated with 3-forms of charge-current density, and topological spin. 9.6.6 Vectors of 3-form Currents, Topological Spin, and Topological Torsion The Current vector of 3-forms It follows that the exterior differential of each 2-form element in |Ga i leads to the vector of 3-form currents, Current 3-forms :

¯ ® d |Ga i = |J a i = (d [B]−1 )ˆ ¯F k .

(9.201)

Following the lead of Electrodynamics from a topological perspective, [276], it is possible to define the concept of Topological Torsion as the 3-form, Topological Torsion 3-form and the Topological Spin as the 3-form, Topological Spin 3-form

­ k¯ ¯ k® A ¯ ˆ ¯F ,

­ k¯ ¯ k® A ¯ ˆ ¯G .

(9.202)

(9.203)

It becomes apparent that the concept of spin is intimately related to the concept of Affine Torsion, but the relationship is not 1-to-1. In four-dimensional domains, the exterior differential of the Topological Torsion 3-form yields the concept of a Topological Parity 4-form (the second Poincare invariant), ­ ¯ ¯ ® ­ ¯ ¯ ® P (9.204) Poincare II: d Ak ¯ ˆ ¯F k = F k ¯ ˆ ¯F k = 2 (E ◦ B)k . k

Non-zero values of the Second Poincare invariant imply (see the previous chapters) that thermodynamic processesPin such systems can be irreversible, and the dissipation coefficient is related to 2 (E ◦ B)k . In four-dimensional domains, the exterior k

differential of the Topological Spin 3-form yields the first Poincare invariant, ­ ¯ ¯ ® ­ ¯ ¯ ® ­ ¯ ¯ ® d Ak ¯ ˆ ¯Gk = F k ¯ ˆ ¯Gk − Ak ¯ ˆ ¯J k P {(B ◦ H − D ◦ E)k − (A ◦ J−ρφ)k }. Poincare I: =

(9.205) (9.206)

k

If the First Poincare invariant is zero, the Topological Spin 3-form is closed. By deRham’s theorems the integrals (over closed three-dimensional chains) of the closed component of the Topological Spin 3-form are rational. That is, the closed integrals of Topological Spin are quantized, when the First Poincare invariant vanishes.

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¯ k® ¯A and 2-forms of The 3-form of Topological Spin depends on both the Potentials ¯ k® ¯ Affine Torsion, G . The Topological ¯ k ® Spin is zero when the Affine Torsion coefficients are zero, for then there is ¯G = 0 . In order for the Topological Spin to be quantized, the 3-form must be closed, but not exact. There are two cases, either both terms in the expression for the Poincare invariant are individually zero (the photon?) of they cancel one another (the electron?). That is, it is conceivable, but not ¯ k ® that the difference between the magnetic and electric energy density, ­ k ¯necessary, ¯ F ˆ ¯G , (often called ­ ¯the¯ Field ® Lagrangian) could be compensated by the Interaction energy density Ak ¯ ˆ ¯J k . Quantized Topological Spin places constraints on the Affine Coefficients. As it is known that a Frame perturbation can be made to eliminate the Affine Torsion coefficients that are integrable, it follows that Quantized Topological Spin ¯ k® ¯ requires G 6= 0; that is, the Affine¯ Torsion coefficients must be irreducible. This ® constraint implies that the 1-forms ¯Ak are not integrable, which in turn implies that Topological Torsion is not zero. Remark 65 Quantized spin AˆG requires irreducibly that the 3-form be closed, but not exact. Hence the existence of Affine Torsion 2-forms is not sufficient to produce quantized Spin. Irreducible 2-forms of Affine Torsion require that the 3-form of Topological Torsion is not zero. Non-zero Topological Torsion is necessary for irreducible Affine Torsion components. Hence irreducible Affine Torsion is a necessary, but not sufficient, source of quantized Spin. Quantized Spin requires non-zero Topological Torsion, and a zero value of the First Poincare Invariant. These requirements indicate that such physical systems are NOT in thermodynamic equilibrium. 9.6.7 Matrices of Cartan Curvature 2-forms [Φ] Exterior differentiation of the differential closure equation (9.179) leads to the expression, d(d[B]) = d[B]ˆ[C] + [B]ˆd[C] = [B] ◦ { [C] ˆ [C] + [dC] } ⇒ 0.

(9.207)

The bracketed expression leads to the definition of the Cartan Curvature matrix of 2-forms, [Φ]: [Φ] = Matrix of Curvature 2-forms based on [C] [Φ] = { [C] ˆ [C] + [dC] }.

(9.208) (9.209)

In differential geometry, this formula is used to define the matrix of curvature 2-forms based on any connection [Γ]. For the Cartan Connection, [C], deduced from the sole assumption of the existence of a Basis Frame [B], the value of the Cartan Connection matrix of curvature 2-forms is zero. The domain of [B] is said to be "flat" as its Riemann tensor is zero, [Φ] = [0]. The Bracketed function defines Cartan’s First Equation of Structure (for an arbitrary connection).

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455

9.6.8 Matrices of Bianchi 3-forms Exterior differentiation of the matrix of curvature 2-forms (Cartan’s first equation of structure) leads to another Bianchi statement (which has the appearance of a Heisenberg operator): d [Φ] − [dC] ˆ [C] + [C] ˆ [dC] d [Φ] − [J1] + [J2] or [Φ] ˆ [C] − [C] ˆ [Φ] [J1] − [J2]

⇒ ⇒ = =

0, 0; d [Φ] , d [Φ] .

(9.210) (9.211) (9.212) (9.213)

The Bianchi formula is valid for any matrix of curvature 2-forms defined by equation (9.209) for an arbitrary connection, [Γ]. The Bianchi statement is a statement in cohomology theory, in that the difference between two matrices of 3-forms [J1]−[J2], is exact and equal to d [Φ] . The matrix of curvature 2-forms for the Cartan Connection is zero, [ΦCar tan ] = 0. However, it is possible to construct a connection [ΓChristof f el ] based on a quadratic form (a metric) for which the matrix of curvature 2-forms is not zero [ΦChristof f el ] 6= 0. It is these non-trivial matrices of curvature 2-forms that are used to describe the gravitational field. 9.6.9 Decomposition of the Cartan Connection The Cartan Connection as a matrix of 1-forms, can be decomposed into a matrix representing a metric (quadratic form) type of connection, [ΓChristof f el ] , and a residual matrix of 1-forms, [T] , [C] = [ΓChristof f el ] + [T] .

(9.214)

This decomposition leads to a Strong Principle of Equivalence, when the decomposition formula is inserted into the Cartan definition of Curvature 2-forms. 9.7

The Strong Principle of Equivalence

The Cartan matrix of curvature 2-forms can be evaluated by substituting [Γchrist ]+[T] for [Cright ] , to yield: Christoffel Curvature 2-forms = = +Interaction 2-forms = +Residue curvature 2-forms = =

[Φchrist ] [Γchrist ] ˆ [Γchrist ] + d [Γchrist ] , [Γchrist ] ˆ [T] + [T] ˆ [Γchrist ] , [Φresidue ] [T] ˆ [T] + d [T] .

(9.215) (9.216) (9.217)

The sum of these 3 terms must vanish, as the sum equals the Cartan matrix of curvature 2-forms (which vanishes),

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[ΦCar tan ] = [C] ˆ [C] + [dC] = 0.

(9.218)

If (following Einstein’s 1915 theory) the curvature 2-forms, generated by the metric and the associated Christoffel connection, are defined as the Gravitational stress energy, [Φchrist ] = [”Gravitational stress energy”] , (9.219) and if the 2-forms which depend upon the interactions and curvatures of the Residue connection are defined as Inertial stress energy, [”Inertial stress energy”] = −{[Φresidue ] + [Γchrist ] ˆ [T] + [T] ˆ [Γchrist ]},

(9.220)

then the fact that Cartan Curvature 2-forms must vanish leads to the result [”Gravitational stress energy”] = [”Inertial stress energy”].

(9.221)

The equation represents a "Strong Principle of Equivalence", built on the balance between gravitational stress energy and inertial stress energy. The formula does not compare accelerations explicitly, for the sum of [Γchrist ] + [T] is not zero, but is equal to [Cright ] , which is not zero. This result is universally true for any linear connection based upon a Basis Frame of C2 functions to compute the Cartan Connection, [C], and a quadratic metric field, [g] which can be used to computed the Christoffel Connection. 9.7.1 Examples using Maple Many examples and more detail are to found in Chapter 3 of Volume 2 [274]. In addition Maple programs have been constructed to carry out the extensive algebra. These programs will be collected in a CD ROM [278], but at the time of writing, these maple worksheets can be downloaded in pdf file format. The URL’s are: http://www22.pair.com/csdc/download/mapleEP1.pdf Schwarzschild metric only, embedded in the Perturbed Cartan connection, yields reducible Affine Torsion. http://www22.pair.com/csdc/download/mapleEP2.pdf Frame perturbations and irreducible Affine torsion is discussed. 9.7.2 The Particle (Contravariant) Affine Connection As a first example, a particle affine Basis Frame in 4D will be examined. For simplicity, the space-space portions of the Basis Frame will be assumed to be the 3 × 3 Identity matrix, essentially eliminating spatial deformations and spatial rigid body motions. The 4th (space-time) column will consist of components that can be identified with a velocity field. The bottom row will have three zeros, and the B44 component will be described in terms of a function ψ(x, y, z, t), ⎡ ⎤ 1 0 0 −V x ¤ ⎢ 0 1 0 −V y ⎥ £ ⎥ (9.222) BP article_af f ine = ⎢ ⎣ 0 0 1 −V z ⎦ . 0 0 0 ψ

The Strong Principle of Equivalence

457

The projected 1-forms become,

⎡

1 ⎢ 0 ⎢ ⎣ 0 0

0 1 0 0

¯ ® [B] ◦ |dy a i = ¯σ k ,

¯ x ⎤ ¯ ¯ σ + ¯ dx + 0 −V x ¯ y ¯ ¯ σ ¯ dy 0 −V y ⎥ ⎥ ¯ ⇒ ¯¯ z z ⎦◦¯ 1 −V ¯ σ ¯ dz ¯ ω ¯ dt 0 ψ ¯ ¯ dx − V x dt + ¯ ¯ dy − V y dt . = ¯¯ z ¯ dz − V dt ¯ ψdt

(9.223)

(9.224)

(9.225)

The Cartan right Connection matrix of 1-forms is given by the expression, ⎡

0 £ ¤ ⎢ 0 CP article_af f ine_right = ⎢ ⎣ 0 0

0 0 0 0

⎤ 0 −dV x + V x d(ln ψ) 0 −dV y + V y d(ln ψ) ⎥ ⎥. 0 −dV z + V z d(ln ψ) ⎦ 0 d(ln ψ)

(9.226)

The Connection coefficients can be computed and exhibit components of nonzero "P-Affine Torsion ". The vector of "Affine Torsion" 2-forms is: ¯ ® ¯ΣP −Af f ine_torsion = Cˆ |dy m i ' |GP article i , with ¯ ¯ −d(V x )ˆd(t) + V x d(ln ψ)ˆd(t)) + ¯ ¯ −d(V y )ˆd(t) + V y d(ln ψ)ˆd(t)) , |GP article i = ¯¯ z z ¯ −d(V )ˆd(t) + V d(ln ψ)ˆd(t)) ¯ d(ln ψ)ˆd(t) ¯ ® 6= ¯ΣCar tan _torsion .

(9.227) (9.228) (9.229)

If the velocity field is a function of time only, then total differential of the velocity field leads to the classic kinematic concept of accelerations, and the "affine" torsion 2-forms depend only on the potential ψ. If the potential function is such that its total differential is zero, or a function of time (and not dependent on the spatial coordinates), then all of the affine torsion coefficients vanish (for this example). Moreover, under the kinematic assumption, the 3-forms of currents vanish, |Ji = d |GP articel i ⇒ 0.

(9.230)

Now consider the case of topological fluctuations, ∆xk , about Kinematic Perfection. That is suppose,

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dxk − V k dt = ∆xk , ¯ ¯ k À À ¯ ∆xk ¯ σ ¯ ¯ = ¯ or ¯ ω ψdt ¯ ¯ À À ¯ dσ k ¯ d(∆xk ) ¯ ¯ such that ¯ . = ¯ dψˆdt dω

(9.231) (9.232) (9.233)

The algebra of this example based on topological fluctuations about Kinematic Perfection are discussed in the Maple programs mentioned above, and will be published on a CD ROM, [278]. 9.7.3 The Wave (Covariant) Affine Connection The next example utilizes the canonical form of the "wave affine" Basis Frame, which will have zeros for the first 3 elements of the right-most column. For simplicity, the space-space portions of the Basis Frame will be assumed to be the 3 × 3 Identity matrix, essentially eliminating spatial deformations and spatial rigid body motions. The 4th (space-time) column will have three zeros, and the B44 component will be described in terms of a function φ(x, y, z, t), ⎡ ⎤ 1 0 0 0 £ ¤ ⎢ 0 1 0 0 ⎥ ⎥. (9.234) Bwave_af f ine = ⎢ ⎣ 0 0 1 0 ⎦ Ax Ay Az −φ The projected 1-forms become,

¯ ® [Bwave_af f ine ] ◦ |dy a i = ¯Ak ,

(9.235)

¯ ⎤ ¯ ¯ dx + ¯ dx + 1 0 0 0 ¯ ¯ ¯ k® ¯ dy ⎥ ¯ dy ⎢ 0 1 0 0 ¯A , ¯ ⎥◦¯ ⎢ = ⇒ (9.236) ¯ ¯ ⎣ 0 0 1 0 ⎦ ¯ dz ¯ dz ¯ Action ¯ dt Ax Ay Az −φ ¯ ¯ dx + ¯ ¯ ¯ k® dy ¯A .(9.237) = ¯¯ dz ¯ ¯ Ax dx + Ay dy + Az dz − φdt ⎡

Note that the Action 1-form produced by the wave affine Basis Frame is precisely the format of the 1-form of Action used to construct the Electromagnetic field intensities. It will be evident, from that which is displayed below, that the 2-form F = dA can be identified with the coefficients of "W-Affine Torsion". This result does not occur in P-Affine group, but appears naturally in the W-Affine group. The Cartan right Connection matrix of 1-forms is given by the expression

The Strong Principle of Equivalence

459

⎡

⎤ 0 0 0 0 ⎥ £ ¤ ⎢ 0 0 0 0 ⎥. Cwave_af f ine_right = ⎢ ⎣ ⎦ 0 0 0 0 −d(Ax )/φ −d(Ay )/φ −d(Az )/φ d(ln φ)

(9.238)

The Connection coefficients can be computed and exhibit components of nonzero "Affine Torsion ". The vector of "Affine Torsion" 2-forms is given by the expression, ¯ ® £ ¤ ¯ΣW −Af f ine_torsion = Cwave_af f ine_right ˆ |dy m i ' |Gi ¯ ¯ ¯ 0 0 + + ¯¯ ¯ ¯ ¯ 0 0 , = ¯¯ |Gi = ¯¯ 0 0 ¯ ¯ ¯ −d(Action)/φ ¯ −F/φ) ¯ ® 6= ¯ΣCar tan _torsion .

(9.239) (9.240) (9.241)

Remark 66 It is remarkable that, for the wave affine group of Basis Frames, the coefficients of "Affine torsion" of the Right Cartan matrix of connection 1-forms are directly related to the structural format of the electromagnetic field 2-form, F = dA. The exterior differential of the vector of 2-forms |Gi produces the vector of 3-form currents |Ji. The result is, ¯ ¯ ¯ ¯ 0 0 + + ¯ ¯ ¯ ¯ 0 0 = ¯¯ |Ji = ¯¯ . (9.242) 0 0 ¯ ¯ ¯ +dφˆF/φ2 ¯ d(−F/φ) Note that for this example, the current 3-form is closed producing a conservation law, d |Ji = |0i .

(9.243)

The wave affine group of Basis Frames has a connection with Affine Torsion 2-forms that are abstractly related to the forces per unit charge of electromagnetic theory (the E and B fields). The particle affine group of Basis Frames has a connection with Affine Torsion 2-forms which are directly related to Accelerations of massive particles. 9.7.4 The Schwarzschild Metric embedded in a Basis Frame In this example, the isotropic form of the Schwarzschild metric will be incorporated into a Cartan Connection. The Schwarzschild metric is a diagonal metric of the form,

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(2 − m/r)2 (δs) = −(1 + m/2r) {(dx) + (dy) + (dz) } + (dt)2 , (9.244) 2 (2 + m/r) p 2 2 2 with r = (x) + (y) + (z) . (9.245) £ ¤ If at the point, {xk }, there exists a covariant metric field, g(at_x) , then it should have an expression at {y a } in terms of the congruent pullback mapping, ¤ £ ¤ £ e ◦ g(at_x) ◦ [B]. (9.246) g(at_y) = [B] 2

4

2

2

2

Hence the square of the (non-integrable) infinitesimal line element, δs, becomes a differential invariant: £ ¤ e ◦ g(at_x) ◦ [B] ◦ |dy m i , (9.247) (δs)2 = hdy n | ◦ [B] n m (9.248) = hdy | ◦ [gnm (y)] ◦ |dy i , ¯ k® ­ j¯ ¯ ¯ (9.249) = σ ◦ [gjk (x)] ◦ σ .

If the covariant metric, gjk (x), at {xk } is diagonal, then it can be expressed as the product of three diagonal matrices, [gjk ] = [fdiagonal ] ◦ [η] ◦ [fdiagonal ] .

(9.250)

The matrix [η] is a diagonal Sylvestor signature matrix of elements ±1, and the diagonal matrices [fdiagonal ] are related to the square roots of the magnitudes of the diagonal metric [gnm ]. The square of the line element becomes: (δs)2 = hdy n | ◦ [gnm ] ◦ |dy m i , b T ◦ [η] ◦ [B] b ◦ |dy m i , = hdy n | ◦ [[B] b = [fdiagonal ] ◦ [B]. [B]

(9.251) (9.252) (9.253)

For the isotropic Schwarzschild example, gjk = [fe] ◦ [η] ◦ [f ] , ⎡ α 0 0 0 ⎢ 0 α 0 0 where f = ⎢ ⎣ 0 0 α 0 0 0 0 β

and

and

(9.254)

⎤

⎥ ⎥, ⎦

α = (1 + m/2r)2 , ⎡ −1 0 0 ⎢ 0 −1 0 η = ⎢ ⎣ 0 0 −1 0 0 0

(2 − m/r) β= , (2 + m/r) ⎤ 0 0 ⎥ ⎥. 0 ⎦ 1

(9.255)

(9.256)

(9.257)

The Strong Principle of Equivalence

461

Bottom line: The effects of a diagonal metric [gjk ] can be absorbed into a re-definition of the Frame matrix, b = [f ] ◦ [B]. [B]

(9.258)

{y a } = {r, θ, ϕ, τ } ⇒ {xk } = {x, y, z, t}, φk : [r sin(θ) cos(ϕ), r sin(θ) sin(ϕ), rcos(θ), τ ] ⇒ [x, y, z, t].

(9.259) (9.260)

To complete the example, at first consider the map φk from spherical to Cartesian coordinates,

The Jacobian of the map can be utilized as a Basis Frame matrix which is an element of the F-Affine group, ⎡

sin(θ) cos(ϕ) r cos(θ) cos(ϕ) −r sin(θ) sin(ϕ) ⎢ sin(θ) sin(ϕ) r cos(θ) sin(ϕ) r sin(θ) cos(ϕ) [B] = ⎢ ⎣ cos(θ) −r sin(θ) 0 0 0 0

⎤ 0 0 ⎥ ⎥. 0 ⎦ 1

(9.261)

This Jacobian Basis Frame matrix will be perturbed by multiplication on the left by the diagonal matrix, [f ] . The perturbed Basis Frame becomes, b = [f ] ◦ [B], [B] ⎡ α sin(θ) cos(ϕ) αr cos(θ) cos(ϕ) −αr sin(θ) sin(ϕ) ⎢ α sin(θ) sin(ϕ) αr cos(θ) sin(ϕ) αr sin(θ) cos(ϕ) = ⎢ ⎣ α cos(θ) −αr sin(θ) 0 0 0 0

Use of the congruent pullback formula based on the connection yields, b transpose ] ◦ η ◦ [B], b [gmn ] = [B ⎡ −α2 0 0 0 ⎢ 0 −α2 r2 0 0 = ⎢ ⎣ 0 0 −α2 r2 sin2 (θ) 0 0 0 0 +β 2

⎤

⎥ ⎥, ⎦

⎤

(9.262)

0 0 ⎥ ⎥ . (9.263) 0 ⎦ β

(9.264) (9.265)

which is the isotropic Schwarzschild metric in spherical coordinates. It actually is more than the stationary Schwarzschild metric when the coefficients, α,and β, are assumed to be dependent upon both r and τ .

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b as a matrix of 1-forms relative to the Basis The Cartan (right) Connection [C] b becomes, Frame [B] d −1 ] ◦ d[B], b = [B b [C] ⎡ −2mdr/rγ −rdθ sin2 (θ)rdφ 0 ⎢ dθ/r δdr/γ − cos(θ) sin(θ)dφ 0 b = ⎢ [C] ⎣ dφ/r cot(θ)dφ cot(θ)dθ + δdr/γ 0 0 0 0 4mdr/(γδ) γ = (2r + m), δ = (2r − m).

⎤

(9.266)

⎥ ⎥ , (9.267) ⎦ (9.268)

Surprisingly, for the Basis Frame perturbed by a massive object, the affine torsion terms are not zero, and can be evaluated as, ¯ ® ¯Σaf f ine_torsion_2−f orms ' |Gi , ¯ ¯ 0 + ¯ ¯ (2m/rγ)(dθˆdr) . |Gi = ¯¯ ¯ (2m/rγ)(dφˆdr) ¯ (4m/γδ)(drˆdτ )

(9.269) (9.270)

The idea is that the arbitrary Basis Frame without metric can be perturbed algebraically to produce a new Basis Frame that absorbs the properties of a metric constraint. ¯ ® Another remarkable feature is that the 1-forms ¯σ k constructed according to the formula, ¯ ® b ◦ |dy a i = ¯σ k , [B] (9.271)

are all integrable. The symbol |dy a i stands for the set [dr, dθ, dϕ, dτ ] (transposed b is the "perturbed" Schwarzschild metric. This means into a column vector), and [B] that there exist integrating factors for each σ k such that a new Basis Frame can be constructed algebraically. Relative to this new Basis Frame, the vector of affine torsion 2-forms is zero! The Coriolis acceleration which is related to the 2-form of affine torsion 2-forms has been eliminated.! Incroyable! Formidable! Of course this is impossible if any of the 1-forms, σk , is of Pfaff dimension 3 or more. The Basis Frame then admits Topological Torsion, which is irreducible. 9.7.5 The Basis Frame and Physical Vacuums The creation of a Cartan Connection in terms of a Basis Frame of C2 functions has applications to the theory of Physical Vacuums [243]. Details of these concepts with applications to Cosmology can be found in [274]. The idea is that a "Physical Vacuum" is different from an "Ideal Vacuum". Both descriptions are "voids" in the sense that the Riemann tensor (computed from the Cartan Connection formula) vanishes. On the other hand the "Physical Vacuum" is not "empty" in that

Partitioned internal structure of Cartan Basis Frames

463

it can exhibit Topological substructures such as irreducible topological torsion and topological fluctuations. In this sense, the Physical Vacuum is a non-equilibrium thermodynamic system. The Ideal Vacuum is a Physical Vacuum that has come to a state of thermodynamic equilibrium. 9.8

Partitioned internal structure of Cartan Basis Frames

The partition of the (arbitrary) basis frame B in terms of the associated (often called horizontal, interior, coordinate or transversal) vectors, ea , and the adjoint (often called normal, exterior, parametric or vertical) field, n, leads to a corresponding partition of the Right Cartan matrix, C, of connection 1-forms, ⎡ 1 1 ⎤ Γ1 Γ2 ... γ 1 ⎢ Γ21 Γ22 ... γ 2 ⎥ ⎥ (9.272) d [B] = [B] ◦ [C] = [B] ◦ ⎢ ⎣ ... ... ... ... ⎦ , h1 h2 ... Ω ⎡ 1 1 ⎤ ⎤ ⎡ 1 1 eα eβ ... n1 /λ Γ1 Γ2 ... γ 1 ⎢ e2α e2β ... n2 /λ ⎥ ⎢ Γ21 Γ22 ... γ 2 ⎥ ⎥ ⎥◦⎢ (9.273) = ⎢ ⎣ ... ... ... ... ⎦ ⎣ ... ... ... ... ⎦ . h1 h2 ... Ω enα enβ ... nn /λ

Note that the Right Cartan matrix [C] is, in general, an element of the General Linear group, but can inherit properties from the group structure of the Basis Frame [B]. For example, if the Basis Frame matrix is orthogonal, then the Cartan matrix is antisymmetric. When n is even and equal to 4, the Cartan matrix [C] can be an element of the symplectic group of 6-parameters (dimensions), and the Frame matrix [B] is an element of the orthogonal affine group with a fixed point, which has 4-parameters (dimensions). In individual basis vector notation, the differentials of the column partitions of the frame matrix can be written as: dea = em Γm k + nha , m dn = em γ + nΩ,

(9.274) (9.275)

further demonstrating the concept of differential closure; i.e., the differentials of basis vectors are composed of linear combinations of themselves. However, it is important to note that in general the differentials of the interior vectors are not linear combinations of only interior vectors, and the differentials of exterior vectors also are not linear combinations of only interior vectors. Special cases occur when the coefficients ha ⇒ 0, and/or γ m ⇒ 0. When ha ⇒ 0, the Right Cartan Connection matrix of 1-forms is an element of the P-Affine group. When γ m ⇒ 0, the Right Cartan Connection matrix of 1-forms is an element of the W-Affine group. When both ha ⇒ 0 and γ m ⇒ 0, the Right Cartan Connection matrix of 1-forms is a member of the F-Affine subgroup of 10-parameters.

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Topology and the Cartan Calculus

As mentioned above, the partition of the (arbitrary) basis frame B in terms of the associated (often called horizontal, interior, coordinate or transversal) vectors, ea , and the adjoint (often called normal, exterior, parametric or vertical) field, n, leads to a corresponding partition of the Right Cartan matrix, C, of connection 1-forms, ⎡ 1 1 ⎤ Γ1 Γ2 ... γ 1 ⎢ Γ21 Γ22 ... γ 2 ⎥ ⎥ (9.276) d [B] = [B] ◦ [C] = [B] ◦ ⎢ ⎣ ... ... ... ... ⎦ , h1 h2 ... Ω ⎡ 1 1 ⎤ ⎤ ⎡ 1 1 1 eα eβ ... n /λ Γ1 Γ2 ... γ 1 ⎢ e2α e2β ... n2 /λ ⎥ ⎢ Γ21 Γ22 ... γ 2 ⎥ ⎥ ⎥◦⎢ (9.277) = ⎢ ⎣ ... ... ... ... ⎦ ⎣ ... ... ... ... ⎦ . h1 h2 ... Ω enα enβ ... nn /λ

Note that the Right Cartan matrix [C] is, in general, an element of the General Linear group, but can inherit properties from the group structure of the Basis Frame [B]. For example, if the Basis Frame matrix is orthogonal, then the Cartan matrix is antisymmetric. When n is even and equal to 4, the Cartan matrix [C] can be an element of the symplectic group of 6 parameters (dimensions), and the Frame matrix [B] is an element of the orthogonal affine group with a fixed point, which has 4-parameters (dimensions). In individual basis vector notation, the differentials of the column partitions of the frame matrix can be written as, dea = em Γm k + nha , m dn = em γ + nΩ,

(9.278) (9.279)

further demonstrating the concept of differential closure; i.e., the differentials of basis vectors are composed of linear combinations of themselves. Note that the Cartan matrix is P-Affine if hk = 0, and W-Affine if γ m = 0, and F-Affine if both hk = 0, and γ m = 0. Another classic case [73] is given by the assumptions that the Cartan Connection is an element of the (N 2 −(N −1)−1 = 12)parameter subgroup (in 4D) of the form, ⎡

Γ11 ⎢ Γ21 dB = B◦ ⎢ ⎣ ... h1

Γ12 Γ22 ... h2

⎤ ... γ 1 ... γ 2 ⎥ ⎥ , with ... ... ⎦ ... 0

hk = −γ k (a constraint).

(9.280) (9.281)

In this system, the N-1 "tangent" vectors ek may be considered as residing in an N-1 hypersurface of the N-dimensional space, and the vector n is the local normal to this hypersurface.

Partitioned internal structure of Cartan Basis Frames

465

The Cartan matrix, C, is a matrix of differential 1-forms which can be evaluated explicitly from the functions that make up the basis frame, if they admit first partial derivatives. Moreover, the differential of the position vector can be expanded in terms of the same basis frame and a set of Pfaffian 1-forms, ¯ a À ¯ ¯ a À À ¯ dy ¯ $ ¯ dy −1 , (9.282) dR = I ◦ ¯¯ = B ◦ B ◦ ¯¯ = B ◦ ¯¯ ω dτ dτ ¯ À ¯ $ is a vector of 1-forms that can be computed explicitly. Note where the vector ¯¯ ω that, ¯ a À ¯ ¯ a À ¯ À À ¯ dy ¯ $ ¯ σ ¯ −1 ¯ dy ¯ ¯ ¯ B ◦¯ 6= B ◦ ¯ . (9.283) =¯ =¯ ω A dτ dτ By the Poincare lemma, it follows that,

and

¯ ¯ À ¯ $ ¯ ¯ + F ◦ ¯¯ ddR = dFˆ ¯ ω ¯ À ¯ ¯ $ ¯ = F ◦ {Cˆ ¯¯ + ¯¯ ω

d$ dω d$ dω

À

,

À } ⇒ 0,

d(dB) = dBˆC + BˆdC = B ◦ {C^C + dC} ⇒ 0.

(9.284) (9.285)

(9.286)

For the arbitrary, partitioned, Basis Frame matrix and the Cartan matrix with equivalent relative partitions, the Poincare lemma breaks up into linearly independent factors, each of which must vanish. The results are:

ddR = 0 = e{d |$i + [Γ]ˆ |$i − ωˆ |γi} +n{dω + Ωˆω + hh| ˆ |$i}, dde = 0 = e{d[Γ] + [Γ]ˆ[Γ] + |γi ˆ hh|} +n{d hh| + Ωˆ hh| + hh| ˆ[Γ]}, ddn = 0 = e{d |γi + [Γ]ˆ |γi − Ωˆ |γi} +n{dΩ + ΩˆΩ + hh| ˆ |γi}. 9.8.1

(9.287) (9.288) (9.289)

The Structural equations

By reasons of linear independence, each of the curly bracket factors must vanish, leading to certain important results on the interior domain (coefficients of e). These

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Topology and the Cartan Calculus

results are: Cartan’s first Structural Equation ¯ ¯ ωˆγ 1 + ¯ |Σi = d |$i + [Γ]ˆ |$i = ωˆ |γi ≡= ¯¯ ωˆγ 2 , ¯ ωˆγ 3 with |Σi = the interior torsion vector of dislocation 2-forms.

(9.290)

Cartan’s Second Structural Equation ⎤ ⎡ 1 γ ˆh1 γ 1 ˆh2 γ 1 ˆh3 [Φ] = d[Γ] + [Γ]ˆ[Γ] = − |γi ˆ hh| = ⎣ γ 2 ˆh1 γ 2 ˆh2 γ 2 ˆh3 ⎦ ,(9.291) γ 3 ˆh1 γ 3 ˆh2 γ 3 ˆh3 with [Φ] = the matrix of interior curvature 2-forms.

A new Structural Equation ¯ ¯ Ωˆγ 1 + ¯ |Ψi = d |γi + [Γ]ˆ |γi = Ωˆ |γi = ¯¯ Ωˆγ 2 , ¯ Ωˆγ 3 with |Ψi = the exterior torsion vector of disclination 2-forms.

(9.292)

The first two equations (9.290), (9.291) are precisely Cartan’s equations of structure (on an affine domain). It is the last equation (9.292) of exterior disclination 2-forms, d |γi + [Γ]ˆ |γi = Ωˆ |γi = |Ψi, that appears to be a new equation of structure valid on a projective domain, when Ω 6= 0. The 1-form Ω, defined as the abnormality 1-form, can be interpreted in terms of a combined expansion and rotation. The components |γi can be interpreted in terms of rotations, while the components hh| can be interpreted in terms of translations. |Ψi physically seems to represent a different kind of "torsion" which is perhaps associated with W-Affine Basis Frames. I intuit that this "torsion" is more to be associated with rotations and expansions about a fixed point. Perhaps this could be a better description of disclination defects. Recall that Kondo [119] has developed the theory of dislocation defects based on P-Affine Basis Frames, |Σi . Remark 67 A remarkable result (to me) of this construction is the fact that the matrix of interior curvature 2-forms, [Φ] , can be constructed in two ways. The classical method utilizes differential processes {d[Γ] + [Γ]ˆ[Γ]} , while the second method is purely algebraic {− |γi ˆ hh|}. The order of partial derivatives contained in the algebraic (exterior) expression for the interior curvature {− |γi ˆ hh|} is one less than the classic expression built on the connection coefficients, {d[Γ] + [Γ]ˆ[Γ]}.

Partitioned internal structure of Cartan Basis Frames

467

Exterior differentiation of the matrix of interior curvature 2-forms yields, d[Φ] = −d |γi ˆ hh| = (− |dγi ˆ hh|) + (|γi ˆ hdh|) = 0,

([Γ]ˆ |γi ˆ hh|) − (Ωˆ |γi ˆ hh|) − (|γi ˆ Ωˆ hh|) − (|γi ˆ hh| ˆ[Γ]) = 0.

(9.293)

(9.294)

The fundamental result is that the matrix of 2-forms that forms the interior curvature matrix is closed! There are also three equations of structure on the exterior domain (coefficients of n) which are given by the constructions: dω + Ωˆω = − hh| ˆ |σi , d hh| + Ωˆ hh| = − hh| ˆ[Γ], dΩ + ΩˆΩ = θ = − hh| ˆ |γi ,

(9.295) (9.296) (9.297)

where θ represents the exterior curvature 2-forms. Constraints can be imposed upon the Frame matrix, limiting the generality and application. When the normalization factor, λ, is chosen in such a way as to force the determinant of the transformation to be unity (or a constant), the abnormality 1-form Ω becomes zero. This single constraint on the determinant can be interpreted as reducing the general Cartan connection matrix to a projective Cartan matrix. In such cases, the disclination 2-forms, |Ψi, vanish. If the arbitrary Frame matrix is locally constrained such that the Cartan connection matrix is an element of the orthogonal structure group, then Ω vanishes, and the Cartan matrix, becomes antisymmetric, with hh| = − |γi . There are two types of Affine Cartan matrices. The first type is an element of the matrix group where |γi = 0. The second type of affine transformation is an element of the matrix group where hh| = 0. Often the structural group is chosen as a Lie group. A purpose of this section was to prove constructively the existence of |Ψi, a vector of "exterior" torsion 2-forms which, it is suggested herein, should be put into correspondence with disclination defects, rotational shears and coherent structures in hydrodynamics. This vector is zero on Euclidean orthonormal or affine manifolds. |Ψi physically seems to represent a different kind of "torsion" which has a correspondence with disclination defects. Recall that Kondo has developed the theory of dislocation defects based on |Σi . 9.8.2

Exterior Algebraic and Interior Differential Curvatures

As mentioned above, a remarkable result of this construction is the fact that the matrix of interior curvature 2-forms, [Φ] , can be constructed in two ways. This result can be used to demonstrate that the matrix of curvature 2-forms, [Φ], is closed.

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Topology and the Cartan Calculus

Exterior differentiation of the matrix of interior curvature 2-forms yields, d[Φ] = −d |γi ˆ hh| = (− |dγi ˆ hh|) + (|γi ˆ hdh|), = ([Γ]ˆ |γi ˆ hh|) − (|γi ˆ hh| ˆ[Γ] −(Ωˆ |γi ˆ hh|) − (|γi ˆ Ωˆ hh|)), ⇒ 0.

(9.298) (9.299) (9.300)

The fundamental result is that the matrix of 2-forms that forms the interior curvature matrix is closed! It is this fact that leads ultimately to the idea relating the Euler characteristic and the components of the curvature 2-forms. It is important to note that due to the partition, the exterior curvature (in this example, scalar valued) 2-form θ = − hh| ˆ |γi with, dθ = − hdh| ˆ |γi + hh| ˆ |dγi , = +Ωˆ hh| ˆ |γi + hh| ˆ[Γ]ˆ |γi − hh| ˆ[Γ]ˆ |γi + hh| ˆΩˆ |γi , ⇒ 0,

(9.301) (9.302) (9.303)

is also closed. As Ω is a 1-form for a single exterior vector, then the 2-form θ is exact. The exterior curvature 2-forms can generate a Maxwell-Faraday system of PDE’s. Both the exterior and the interior curvature 2-forms can be matrix valued depending upon the partition of the Frame. Each curvature matrix exhibits a set of similarity invariants deduced from the coefficients of the Cayley-Hamilton characteristic polynomial. It would appear therefore that there are two species of Chern characteristic classes that can be constructed from the Cayley-Hamilton polynomial similarity invariants. If (in the example) the projective Cartan matrix is constrained to be Euclidean, then Ω = 1, and both h = 0, and γ = 0. Hence both the interior and the exterior curvature vanish. Indeed, then both types of torsion 2-forms vanish. On the other hand, if the Cartan matrix is antisymmetric (as it must be for an orthonormal frame matrix) then Ω = 0, and γ = −h. Hence, the exterior curvature vanishes, and |Ψi = 0, but the domain could support interior curvature and dislocation torsion 2-forms, |Σi 6= 0. If the Cartan matrix is left affine, then h = 0, Ω = 1. The interior and exterior domains are flat, but the structure could admit both forms of torsion 2-forms. The moral of this section is that there usually is more than one way to do things. In the case of a projectivized line bundle over a variety, the Cartan method of computing the significant quantities is equivalent to the methods of fiber bundle theory, but it is much simpler to use and easier to interpret in physically useful ways for engineering applications. 9.8.3 Non-Integrable Frames It is possible to start with a set of differentiable functions on an initial state, and impose a Basis Frame of functions (on the initial state) that are representations of

Partitioned internal structure of Cartan Basis Frames

469

some group. Given the exact differentials |dy a i on the initial state, the vector of 1-forms projected to the final state can be determined, by means of the formulas, ¯ ® £ ¤ |dy a i ⇒ ¯σ k (y) = Bka (y) ◦ |dy a i . (9.304)

The concept of a right Cartan connection follows as before, even though the mapping functions from initial to final state are not known. (The dimension of the set σ k could be different from the dimension of {y a }.) An alternative point of view is that a linear combination of the differentials £ k¤ a on the initial state | i such that the linear mapping Ba acting on the vector of ¯ ® one forms, | a i , produces perfect differentials ¯dxk on the final state. This point of view is assumed in the Flanders book. ¯ ® £ ¤ (9.305) | a i ⇒ ¯dxk = Bka ◦ | a i . ¯ ® The two vector sets of linear combinations of differentials, | a i and ¯σ k on the initial state are not the same, even when the dimension of the initial and final states are ¯ k® £ the ¤ a ¯σ are determined by the linear map Bka . same. Given the initial state |dy i , the ¯ ® Given the final state ¯dxk , the | a i are determined by the inverse of the linear map, £ k ¤−1 Ba . If the two spaces are not of the same dimension then the inverse of the linear map need not exist. It is important for generalizations that the concept of the initial state or domain (and its coordinate functions, y a ) be kept¯ distinct ® £ from ¤the final state or range (and its coordinate functions, xk ). The format ¯σ k = Bka (y) |dy a i will be emphasized herein. £ ¤ The Linear mapping Bka (y) so generated is the Jacobian matrix of partial differentials of the coordinate (or vector space) mappings. As an example, review the concept of spherical or cylindrical coordinates mapped into Cartesian space. (Maple programs have been developed giving most of the details in terms of symbolic mathematics. See Volume 6, "Maple programs for Non-Equilibrium systems", or http://www22.pair.com/csdc/pdf/mtpertu5.pdf. ¯ ® ¯ ® At first, in this presentation, the dimension of the vector space ¯σ k = ¯dxk will be assumed to be n, the same dimension as the space of independent variables. Then the range of the index a is, a = {1...n}, and the range of the index k is, k = {1...n}. This restriction will be relaxed later during the development of a more general theory. The n × n Jacobian matrix of the n mapping functions establishes the vector space ideas as a linear mapping, and gives the primitive realization of what is to be known as a Frame matrix (of functions on the initial state). Suppose that another function, say Ψ(y a ), is given in terms of the initial variety, {y a }. Then its total differential is given by the expression, X ¯ ® dΨ(y a ) = {∂Φ(y a )/∂y b } dy b = Ab (y a )dy b = hAb (y a )| ◦ ¯dy b . (9.306) b

The object on the right is an example of an exterior differential 1-form, ω1 , with coefficient functions Ab (y a ) and basis elements, dy b . (From here on the sum convention

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Topology and the Cartan Calculus

on up-down symbols - the index b in the formula above - will be presumed, without P the use of the symbol.) The coefficients, by construction in this example, Ab (y a ) = {∂Ψ(y a )/∂y b },

(9.307)

form the components of a covariant gradient vector field. In the Cartan theory of differential forms these concepts are extended to situations where the differential basis elements σk , of the vector space Λ1 {σ k } can be written in terms of some arbitrary matrix (of functions on the initial state) acting on the differentials of the independent variables in a linear way, ¯ k® £ k ¤ ¯σ = Ba (y) |dy a i . (9.308) ¤ £ In other words, in the Cartan extension, it is not assumed that the linear map Bka (y) is necessarily a Jacobian matrix of some non-linear coordinate mapping, nor is it assumed that the matrix is even similar to£the Jacobian matrix of a coordinate mapping. ¤ This more general matrix of functions, Bka (y) , will be defined as a basis Frame and is the cornerstone of Cartan’s development of the Repere Mobile. Given such a matrix of functions a key question revolves about the determination of the solubility of the Frame. Given a Frame, does there exist a unique set of mapping functions φ from which the Frame is determined to be the Jacobian matrix dφ of the mapping? If not, is it possible that there exists a non-unique solution set to the problem? The question of non-unique integrability of the Frame matrix is the basis of what is called Affine and Topological Torsion. Torsion appears when the basis Frame (or its equivalence class) is NOT integrable. For applications, how the Frame may be related to specific physical problems is of key importance. The early development of the Frenet-Serret-Cartan Frame for a point moving along a space curve indicates that it is possible to construct the Frame from differentials of the mapping function with respect to a parameter along a space curve. That is, the velocity, acceleration and the rate of change of acceleration can be used to build a Frame matrix of a point moving along a space curve in three dimensions. These things are physical, measurable, and applicable quantities. The same idea can be generated for continuous media such as a fluid. The velocity field, the vorticity field, and the helicity field of the fluid become the analogs of the Frenet-Serret differentiations. For those ubiquitous cases (or better said, on those restricted domains) where the Frame has an inverse, then the Frame matrix is an element of the General Linear group. Often in particular applications the Frame matrix is constrained to be an element of an equivalence class of "admissible" Frames by assuming the Frame belongs to some sub-group of the GL group. In the Frenet-Serret case, a usual restriction constrains the 3D Frame matrix elements such that the Frame is a member of the special orthogonal (orthonormal) group. The columns of the basis Frame matrix are orthogonal unit vectors. This constraint is used to create the concepts of arclength,

Partitioned internal structure of Cartan Basis Frames

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curvature, and torsion of the 3D space curve. These "intrinsic" properties of the space curve are the similarity invariants of all equivalent Frames (that is, all Frames that are members of the orthonormal group). These intrinsic (often called invariant) properties of the equivalence class are computed by means of the coefficients of the Cayley-Hamilton theorem. From a physics point of view, all observers who may use different elements or representations of the orthonormal group for reference systems will be able to express their views in terms of a common set of qualities, the similarity invariants. All equivalent observers will agree that the values of the similarity invariants are the same. Restrictions to particular subgroups are often called "gauge theories". It is important to note that certain (normal) subgroups (such as the orthonormal subgroup) cannot distinguish between left-handedness and right-handedness (chirality), but other equivalence classes of subgroups can. It would seem that this ability to distinguish a chiral property is of value to the study of biological systems, where most biological molecules appear to be left or right handed. The moral (or warning) of this paragraph is that the common orthonormal system of basis vectors (i, j, k) of engineering practice must be modified to handle chiral distinctions. It is important to be reminded of the idea of a similarity transformation. Given a matrix [M] and a transformation matrix [F ], the matrix [N] is said to be similar to [M] , if [M] ⇒ [N ] = [F ]−1 ◦ [M] ◦ [F ] . (9.309) When the Cayley-Hamilton polynomial is constructed for [M] and [N] the coefficients of the polynomials are the same (if the matrices are "similar"). Two of the important similarity invariants are the trace of [M] and its determinant. In differential geometry, these ideas will be used to define curvature properties of manifolds. In the FrenetSerret-Cartan theory of the orthonormal subgroup, the similarity invariants lead to the concepts of arclength, curvature and torsion. The zero sets of the similarity invariants have particular physical importance. In the thermodynamics of a van der Waals gas, the Cayley-Hamilton polynomial based upon the Gibbs function is a cubic polynomial with the surface shape of a swallowtail. The critical point is where all three similarity invariants vanish. The Spinodal line of phase instability is where the quadratic similarity coefficient (the Gauss curvature of the swallowtail surface) vanishes. The similarity equation can be rewritten in a manner that does not require the immediate computation of an inverse, [F ] ◦ [N] = [M] ◦ [F ] .

(9.310)

δ[M] = [M] ◦ [B] ± [B] ◦ [M] ,

(9.311)

This equation can be used to test if [N ] is similar to [M] . A special situation occurs if the matrix [N] is the same as [M] . This situation places a constraint on the equivalence class of matrices that can be used for the transformations [B] . Suppose that [N] = [M] ∓ ∆[M], then the differential similarity equation becomes,

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and is suggestive of the Heisenberg matrix operator format (the transformation matrix [B] plays the role of the "Hamiltonian" operator). These similarity formats will reappear below when the matrix of connection 1-forms is discussed. Note that the column vector array of differential basis elements, |dy a i, transforms as a contravariant tensor in the Jacobian case, where a coordinate mapping is available. This property can be extended if the Frame matrix of functions has a non-zero determinant, for then the columns of the Frame matrix can be used as a basis set for contravariant vectors in the initial space (domain or state). £ ¤ This basis argument does not depend upon the fact that matrix elements Bka (y) form a Jacobian (i.e., integrable) ¯system. demonstrated that it is this lack of ¤ be ® £ kIt will k a ¯ unique integrability for the σ = Ba (y) |dy i that leads to the concepts of affine Torsion and Topological Torsion, two topics that will be discussed in great detail in subsequent sections. At present, given a basis of 1-forms, construct arbitrary exterior differential 1-formsDfrom the as a row ¯ matrix product of arbitrary coefficient functions¯ arranged ® a ¯ k b vector, Ak (y )¯ , and the basis set arranged as a column vector, ¯σ , ¯ ¯ ® D a ¯ ¯ k b bk (y) σ k . ω = Ak (y )¯ ◦ σ = A 1

(9.312)

Note that if the coefficient functions are chosen to be a covariant vector array (and that is why the index is a lower index on the Ak (y)), then the differential 1-form ω is a scalar invariant of "coordinate transformations". The coefficient functions, however, do not have to be a gradient array. The covariant constraint implies that if (as assumed), ¯ k® £ k ¤ ¯σ = Ba (y) |dy a i , (9.313) then

¯ D £ ¤ bk (y a )¯¯ = hAb | ◦ Bbk −1 , A

(9.314)

£ ¤−1 £ ¤ where Bbk is the inverse matrix of functions to the frame matrix, Bka (y) . These are the rules of classical tensor analysis defining what is meant by contravariant and covariant vectors of ordered sets of components with respect to special transformations (defined as diffeomorphisms). The differential form so constructed in terms of tensor coefficients is then independent from a "choice of coordinate system", ¯ ¯ ® D £ ¤−1 £ k ¤ 1 a ¯ ¯ k b ω = Ak (y )¯ ◦ σ = hAb | ◦ Bbk ◦ Ba ◦ |dy a i . (9.315)

For physicists and engineers what this implies is that laws of physics written in terms of differential forms are independent of the observer’s choice of a "reference system". A "reference system" is defined as an element of an equivalence class of differentiable mappings. The most common equivalence class usually accepted is the class of diffeomorphisms, which implies that the mapping, φ, and the linear mapping, dφ,have

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473

inverses, and the inverse mapping is differentiable. Such diffeomorphic mappings are constrained subsets of other mappings known as homeomorphisms. Homeomorphisms (and therefore diffeomorphisms) preserve topology from initial to final state, and therefore cannot be used to describe topological evolution. (Bummer.) Sometimes the equivalence class of reference systems is even further constrained. For example, the acceptable class of reference systems known as inertial frames of reference in the physics of special relativity is constrained to be the Lorentz equivalence class. Sometimes such constraints throw the baby out with the wash. For example, General Relativity is designed to admit all diffeomorphisms as the equivalence class of frames of reference; Special Relativity admits only elements of the Lorentz equivalence class, which is a subset of all diffeomorphisms. The Lorentz equivalence class consists of those matrices, [L], for which the Minkowski line metric is preserved. That is, ⎡ ⎤ −1 ⎢ ⎥ £ ¤ −1 ⎢ ⎥ = [η] = L−1 ◦ [η] ◦ [L] . (9.316) ⎣ ⎦ −1 1

There is a further subclass of Lorentz matrices, with matrix elements which are constants, and which are use in special relativistic (non-accelerated) applications. (This special subclass turns out to be "affine" torsion free, so that left-handed and righthanded chirality species evolve in the same way.) However, there are Lorentz matrices that preserve the Minkowski metric that are not composed of constant elements. Such matrices admit accelerations, and also admit Affine Torsion coefficients‡ . That is, the system of 1-forms generated by a Lorentz transformation of non-constant elements is not necessarily uniquely integrable, and therefore admit different behavior for chiral systems. (This difference in behavior distinguishes Optical Activity from Faraday Rotation in electromagnetic systems.) It should be realized that differential forms have the tensor-like property that if the differential form is zero in one coordinate system of reference, then it is zero in all other diffeomorphically equivalent systems, no matter what constraints are applied to limit the elements of the equivalence class of diffeomorphisms. In addition, if a differential form is zero on the final state, then its pullback to the initial state is also zero with respect to continuous but not homeomorphic, and therefore not diffeomorphic maps. 9.9

Distributions and the Adjoint Field

Although the emphasis in this monograph is on concepts that are independent from the choice of metric or connection§ , it is useful to demonstrate how a 1-form of Action, ‡

See http://www22.pair.com/csdc/pdf/lorentz.pdf The evolutionary processes of primary interest herein are those described by operating on differential forms with the Lie differential with respect to a direction field. §

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Topology and the Cartan Calculus

A, may be used to generate a compatible frame field [B] and a Cartan connection [C] on the variety. The symmetry features of [B] lead to metric ideas, and certain antisymmetry features of [C] lead to the concept of Affine torsion (which is not the same as Topological torsion, or Frenet torsion). The concept of a differential connection also leads to the famous geometrical structural equations of Cartan, which are different from the topological structure concept utilized in this monograph. The topological structure concepts used herein are independent from the choice of connection or metric. The details of the refined topological features of subspaces based upon the constraints of a global vector Basis Frame will be discussed in Volume 5 of this series [277]. One objective is to search for Frame Fields that accommodate a given 1form of Action. It should be realized from the outset that Frame fields are not uniquely determined by a given 1-form of Action. When a Frame field exists, the differential connections [C] which generate differential closure can be placed into equivalence classes, determined by the group properties of the matrices involved. The investigation of the properties of these various group equivalence classes has become known as the study of "gauge theories", and the method enjoys great popularity at present. However, the choice of a gauge group in physics is often just that, a choice of constraint made by guessing, followed by attempts to put the constrained results into correspondence with physical properties and measurements. In this monograph, the focus is on those topological features that can be put into correspondence with experiment, and yet are independent from a specific choice of connection and/or metric. Even though a Frame field is not necessarily unique, and goes beyond the primitive topological concepts that do not depend upon metric or connection, an algorithm for producing a Frame field will be discussed in the next subsection. From a given 1-form, A, there are two important types of procedures that can be used to construct a useful Frame field. One procedure is differential, and is related to parametric surface theory. The second procedure is algebraic, and is more closely related to implicit surface theory. The algebraic procedure will be discussed first. 9.9.1 The implicit algebraic Frame field To construct the Frame field from a given 1-form, note that at a regular point, {x}, of an N-dimensional space, any given 1-form, A, will admit N-1 linearly independent vector direction fields, V(x). Each vector direction field has N component functions, V µ , to be determined algebraically from the following formula, algebraic orthogonality Aµ V µ = 0.

(9.317)

The collection of N-1 vectors orthogonal to the 1-form are called elements of a distribution direction field, for multiplication of each vector field, V, by any non-zero function, 1/λ(xµ ), is also a solution to the algebraic orthogonality equation, Aµ (V µ /λ(xµ )) = 0 if Aµ V µ = 0.

(9.318)

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475

This independence from scale is typical of projective geometries, and homogeneous functions. One possible (algebraic) construction, using the given⎡ functional coefficients, An 0 ... A1 /λ ⎢ 0 An ... A2 /λ Aµ , of the 1-form, A, yields a Frame matrix of the form, [B] = ⎢ ⎣ ... ... ... ... −A1 −A2 ... An /λ The first N-1 columns satisfy the algebraic orthogonality constraint, which implies that the last column vector is proportional to the adjoint of the matrix of N-1 columns. The determinant of the Frame field is given by the expression: det [B] = (An )(n−2) {(A1 )2 + (A2 )2 + ... + (An )2 }/λ2 .

(9.319)

If the rescaling factor, 1/λ, is chosen such that the determinant is unity over the domain of support of A, then on that domain the Frame field is globally defined and always has an inverse. The N-1 vectors (direction fields of N components) which satisfy the orthogonality relations are defined to be a basis of the "associated", or "horizontal" vectors relative to the given 1-form, A. Note that in the construction above, the coefficient An appears to have a privileged position. However, in spaces of odd topological (Pfaff) dimension, a canonical Darboux ([164]) format indicates that there is one coefficient (presumed to be An ) that is equal to unity. The differential 1form then has the canonical format, A = pµ dqµ +1ds. For even topological dimensions, the canonical format is A = pµ dq µ + Hdt = Ldt + pµ (dqµ − V µ dt), where H is an independent function. Note that the classical Hamiltonian constraint that H ⇒ H(p, q, t) reduces the topological dimension 2n+2 to 2n+1. The Cartan connection matrix for a Frame field constructed in an implicit algebraic manner can admit certain antisymmetries of subspace that have been defined as Affine translational torsion. The parametric method described below, will not produce a connection with affine translational torsion of subspaces. 9.9.2 The parametric differential Frame field If a parametric mapping of N functions in terms of N-1 parameters is given, ξ α ⇒ xk = X k (ξ α ) (1 ≤ k ≤ N) (1 ≤ α ≤ N − 1),

(9.320)

then the N-1 associated vectors can be defined differentially. That is, the partial derivatives of the N mapping functions with respect to the N-1 parameters can be used to form the first N-1 columns (associated vectors) of the matrix, [M]. ⎡ 1 1 ⎤ eα eβ ... 0 ⎢ e2α e2β ... 0 ⎥ ⎥ (9.321) [M] = ⎢ ⎣ ... ... ... ... ⎦ , enα enβ ... 0 ekα = ∂X k (ξ β )/∂ξ α .

(9.322)

⎤

⎥ ⎥. ⎦

476

Topology and the Cartan Calculus

The adjoint vector direction field to this N-1 system of associated vectors can be interpreted as a "normal or vertical" direction field via the algebraic orthogonality relations. Given the N-1 associated vectors, the adjoint vector, n, can be constructed algebraically by adding a column of zeros to the N by N-1 matrix [M] of contravariant associated vectors, ekα . The component index k ranges from 1 to N and the index α ranges from 1 to N-1, ⎡ 1 1 ⎤ eα eβ ... 0 ⎢ e2α e2β ... 0 ⎥ ⎥ (9.323) [M] = ⎢ ⎣ ... ... ... ... ⎦ . enα enβ ... 0

The determinant of [M] is zero, but there always exists an of a column of N-1 by N-1 sub-determinants, ⎡ 0 0 ⎢ 0 0 transpose of the Adjoint of [M] = ⎢ ⎣ ... ... 0 0

adjoint matrix consisting ... ... ... ...

⎤ n1 n2 ⎥ ⎥. ... ⎦ nn

(9.324)

The parametric method permits the creation of the (orthogonal) Adjoint 1-form given the N-1 distribution vectors, while the implicit method permits the creation of the N-1 orthogonal distribution vectors from a given 1-form, that is adjoint to the vectors of the distribution. The adjoint direction field, n, exists algebraically whether or not the distribution of N-1 vectors, ekα , span a simple hypersurface. By construction via the orthogonality constraint, the coefficients of the given 1-form Aµ are in effect proportional to the adjoint direction field. As discussed in the previous subsection, in the more simple situations the coefficients of a differential 1-form, A, can be viewed as a representation of the normal field to a hypersurface. In all cases the coefficients of a differential 1-form can be viewed as being an adjoint direction field. Perhaps even more remarkably, it is possible to scale the adjoint direction field (hence the differential 1-form) by a function λ such that the determinant of the N x N matrix, ⎡ 1 1 ⎤ eα eβ ... n1 /λ ⎢ e2α e2β ... n2 /λ ⎥ ⎥, (9.325) [B] = ⎢ ⎣ ... ... ... ... ⎦ enα enβ ... nn /λ is globally equal to a constant. The procedure thereby defines a Frame of N basis vectors everywhere over the N-dimensional domain of support of the 1-form, A. It follows that exterior differentials of each of the basis vectors of the Frame are linear combinations of the set of the basis vectors. That is, the exterior differential process acting on the basis vectors of the Frame is closed. The process of exterior differentiation acting on elements of the set creates objects that remain within the

Distributions and the Adjoint Field

477

set. Although this parametric procedure is similar to the implicit method described previously, the parametric method never generates a Frame with a connection that supports translational Affine torsion of subspaces. When acting on p-forms, the exterior differential carries a p-form from one vector space into a (p+1)-form in a different vector space. The concept of a connection constrains the differential process to transport a initial vector of one vector space into a final vector in the same vector space. Both vectors have the same basis. 9.9.3 Projective Frames In each of the "adjoint" methods given above, the orthogonality conditions are in effect 2(N-1) constraints on the general N2 variables of a Frame matrix. A determinantal constraint of the type det [B] = 1 adds one more constraint condition. Quadratic (metric) symmetry features imply that the symmetric product of the Frame fields, constructed by the adjoint procedure above, yields a matrix with a fixed point, ⎡ 1 ⎤ gα gβ1 ... 0 2 ⎢ 2 ⎥ f ◦ [B] = ⎢ gα gβ ... 0 ⎥. (9.326) [B] ⎣ ... ... ... ... ⎦ 0 0 ... det[Bbk ]/An−2 n The coefficients of a projective frame normally would have only one constraint. The utility of the "adjoint" procedure is that quadratic geometric metric properties of the tangent space can be decoupled from the geometric properties of the "adjoint" or "normal" space with an appropriate choice of 1/λ. Remarks In three dimensions, the Gibbs cross product of engineering vector calculus is considered to be a "vector" for it has the same number of components as the gradient. Yet it has different behavior under transformations of the basis, and is therefore called a "pseudovector", or an axial vector. In the exterior calculus, the exterior product of the two 1-forms, with components proportional to covariant tensor of rank 1, creates a 2-form with covariant components of rank 2. Only in constrained geometries, such as Euclidean 3-space, do 2-forms have any resemblance to the Gibbs cross product (a rule which fails in dimension n > 3). The pseudo-vector is an object that behaves like a contravariant tensor density of rank 1. Such objects are usually defined as "currents". In general, there are two species of differential forms that are often dual to one another, and are well behaved with respect to functional substitution and the pullback (p.115 [148]) operation. As discussed earlier, the two species are exterior differential p-form scalars and exterior differential (N-p)-form densities or currents. The scalar species pulls back (meaning that the form is well defined with respect to functional substitution and the Jacobian matrix of the mapping) by using the features of the transpose of the Jacobian matrix. The density species pulls back by using the features of the adjoint of the Jacobian matrix. Of course for orthogonal systems,

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Topology and the Cartan Calculus

these concepts are degenerate, for the inverse and the adjoint and the transpose of the Jacobian matrix are the same. Recall that at a point it is always possible to define a vector basis in terms of an orthogonal system (use the Gram-Schmidt process), but the possibility of extending, or mapping, the property of orthogonality smoothly and uniquely (without singularities) from one neighborhood to another neighborhood in a global sense requires that the mapping process be constrained to be an element of the orthogonal group. Such constraints apply nicely to rigid body motion, but fail to describe the deformation of a solid. Hence the reader is advised that the automatic or indiscriminate use of orthonormal basis frames will not yield a complete understanding of nature. If the neighborhoods can be connected by a singly parameterized vector field, then these concepts are at the basis of the Frenet-Serret moving frame analysis (and kinematics). Cartan extended these ideas to domains that are not so simply connected, and developed the notion of the moving basis Frame, which he called the Repere Mobile. A particular choice of a Basis Frame usually implies the selection of an equivalence class of matrices (of functions), as well as the problem of defining the "origin". The intuitive assumption is that the "origin" can be uniquely defined as a point of reference. However, another possibility is that the origin need not be unique, and might itself be a member of some other equivalence class, and might even incorporate fluctuations. 9.10 9.10.1

Intersection, Envelopes and Topological Torsion Introduction

In physical systems the existence of an envelope has its most well-known example in the form of Huygen’s principle. A wave front (in 3D) is the envelope of multiple expanding spherical surfaces whose multiple origins reside on some initial surface. Another example used in sophomore physics lectures is the propagation of a plane wave through a pair of slits, where the superposition of the waves coming from each slit forms a composite envelope wave function. The theory of Cherenkov radiation also involves the theory of an envelope. In simple terms of surface theory, the envelope is a point set where two or more surfaces have a common tangency. Herein attention is focused on the fact that the envelope is to be associated with the concept of non-uniqueness for at each point on the wave front, there exists not only the wave front surface but also the spherical wavelet surface. The concept of non-uniqueness implies that a parametric point of view of describing a surface with its unique range is not applicable. This observation focuses attention on implicit representations of curves and surfaces, where non-uniqueness is admissible. In the theory of implicit surfaces, the criteria of uniqueness - and therefore the existence of a parametric representation - is related to a differential constraint on the neighborhoods in the form of a Pfaffian equation (a 1-form set equal to zero) defining properties of the surface. Locally at a point p, the coefficients of the 1-form define a direction

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479

field orthogonal to the surface. If the 1-form, A, satisfies the Frobenius criteria of unique integrability, AˆdA = 0, then the surface can be uniquely established in the sense that a normal field can be defined by at most two functions, one perhaps giving its scale, and the other its direction field. The direction field is a vector with components defined in terms of the partial derivatives of a unique function. That is, N = φdψ. In these cases the Pfaff dimension of the 1-form, A, is 2, and the Topological Torsion 3-form AˆdA is zero. On the other hand, if AˆdA 6= 0, the Pfaff dimension of the 1-form A is 3 or greater, and if there are solutions, non-uniqueness is to be expected. Topological torsion is not exactly the same as the Frenet torsion of space curve, (which is a parametric, not implicit, concept) nor exactly the same as the more subtle Affine torsion of a connection, but like these concepts Topological Torsion is an artifact of three (irreducible) dimensions or more. First, a few examples of envelopes will be given to demonstrate how the existence of topological torsion is related to the concept of non-uniqueness. 9.10.2

A Family of Curves in the Plane ((2+1)-space)

As mentioned above, the basic idea of an envelope is that there is a non-uniqueness criteria lurking somewhere. First consider the concept of a implicit curve in the plane given as the "global" zero set of a function F (x, y) of two variables, (x, y). It is important to note that the implicit curve (in the plane) can consist of multiple components and branches. A common parametrization may not exist. Some components may be disjoint; some components may have points of tangency; some components may have intersections with other components, and themselves. No direction (of motion) is defined a priori on any particular curve component by the implicit function equation. Each separated disjoint component can be locally represented by a parametrization, say t, that defines an orientation (or direction of motion), but the local parametrization for one component need not be the same for a different component. A family of non-directed (non-oriented, but orientable) curves may be constructed on the plane if the implicit function is a function of one or more other parameters, such as σ, λ,.... Then, for example, in the case of a single (family) parameter, σ, the global zero set of F (x, y, σ) = 0 defines an implicit 2-surface in the (2+1)-space of variables {x, y, σ}, with an induced differential Pfaffian equation, or 1-form set equal to zero. dF ≡ (∂F/∂x)dx + (∂F/∂y)dy + (∂F/∂σ)dσ = Fx dx + Fy dy + Fσ dσ ⇒ 0, = A + Fσ dσ.

(9.327) (9.328) (9.329)

For certain points p each of the coefficients, Fx , Fy , Fσ of the differential dF will vanish, thereby satisfying the criteria that dF = 0. This set of points are defined as "critical points" and are to be excluded from the domain of support of the 1-form F.

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Topology and the Cartan Calculus

For an explicit choice of the family parameter, σ = constant, the implicit function defines a curve of perhaps multiple components which may be viewed as the intersection of the surface F (x, y..σ) = 0 and the plane defined by the value of σ in the space {x, y..σ}. In the formula for dF such a case would correspond to the case that dσ = 0. Next consider the case where Fσ = 0 but dσ 6= 0. Then again, A(x, y, σ, dx, dy) = Fx dx + Fy dy.

(9.330)

If AˆdA 6= 0, the Pfaff dimension of A can be maximal and equal to 3. If maximal, the 2-form dA establishes a contact odd-dimensional manifold structure, and has 1 real eigenvector of eigenvalue zero, for the matrix of the coefficients of dA is completely antisymmetric. In mechanics this null eigenvector would be defined as an extremal field, and has a Hamiltonian generator (see Chapter 2.3.1). The other two eigenvectors of dA are complex, with complex conjugate eigenvalues. The two complex eigenvectors are isotropic for the sum of squares of the three eigenvector components vanish. Such "complex isotropic" vectors have been defined by Cartan to be "Spinors" [44]. It is known that such Spinors generate conjugate minimal surfaces [124]. The criteria that F = 0 and Fσ = 0 is used to define the concept of an envelope, where the second condition, Fσ = 0, permits the family parameter, σ, to be solved as one or more functions of σ k (x, y..) (at points where Fσσ = (∂ 2 F/∂σ2) 6= 0). Substitution of each functional form for σ k into F (x, y, ..σ k (x, y)) defines the various components of the Envelope Implicit "surface" function in the subspace (x,y,..). At other points it may be true that the spatial differential components, A, cancel the parametric variations, Fσ dσ. For such cases, the Pfaff topological dimension of the spatial 1-form, A = Fx dx +Fy dy, is at most 2. In such cases, there exist vector fields, V = [V x , V z , 1] such that dx − V x (x, y, σ)dσ = 0 and dy − V y (x, y, σ)dσ = 0 satisfy the constraint, dF = A + Fσ dσ = = −Fx V x (x, y, σ)dσ − Fy V y (x, y, σ)dσ + Fσ dσ = {−Fx V x − Fy V y + Fσ 1}dσ ⇒ 0.

(9.331) (9.332) (9.333)

Such possibilities defines curves of singular points computed from the kinematic relations. With respect to the direction fields V = [V x , V z , 1], the Function F is defined as a first integral. 9.10.3 Singular and stationary points It is important to distinguish between the singular point sets of the implicit function and the stationary points that may exist on the selected surface of the family, F (x, y..σ) = 0. The stationary points are where the differentials dx, dy, dσ vanish. The singular points are where the induced differential form, dF, vanishes, although

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481

the differentials do not. As mentioned above, the critical points are where all of the coefficient functions Fx , Fy , Fσ vanish (as the intersection of three implicit surfaces). For points which are not critical points, it is possible to find n-1 (n=2 in this example) differential directions for which the 1-form dF vanishes. These n-1 directions define the tangent space to the implicit surface. If all components of the direction field vanish simultaneously, then Cramer’s rule implies that the determinant of the Jacobian matrix of the direction field (Fx , Fy , Fσ ) must vanish at certain points, Rcritical (x, y, σ). The condition can be expressed by the fact that (n=3)-form must vanish, Θ = dFx ˆdFy ˆdFσ = ∆(x, y, σ)dxˆdyˆdσ ⇒ 0 at a singular point.

(9.334)

Note that the zero set of the function ⎡

⎤ Fxx Fxy Fxσ ∆(x, y, σ) = det[J(gradF )] = det ⎣ Fyx Fyy Fyσ ⎦ ⇒ 0, Fσx Fσy Fσσ

(9.335)

defines the "surface of singular points" of the function, F (x, y, s). The two (possibly multiple component) surfaces, ∆(x, y, σ) = 0, and the selected surface, F (x, y, σ) = 0, have intersections at points when dF ˆd∆ 6= 0. In three dimensions this object (a 2-form) has components proportional to the Gibbs cross product of the direction field normal to the implicit surface and the direction field normal to the surface of critical points. At points where the intersection 2-form vanishes, the two surfaces can either be disjoint, or have a point of contact. Hence logical intersection of the critical points includes points of surface intersection and points of surface contact. The problem of finding the critical points is a global issue, but given a point it is possible to test to see if it is a critical point using (local) differential methods. If the 3-form Θ never changes sign, there is no implicit surface of critical points. If Θ is zero, then a surface of critical points exists, but this surface may not have intersection with the surface F (x, y, σ) = 0. An intersection exists producing a curve of critical points when the 2-form dF ˆd∆ is not zero. 9.10.4 Envelopes Envelopes are also related to the logical intersection of two surfaces generated by the family in the sense that two surfaces have a common set of points that are tangent to both surfaces. To find an envelope is more difficult than to determine whether or not the envelope exists. In two dimensions, there exist neighborhood directions constraining the displacements dx, dy, ds such that the differential form dF vanishes in those selected (not all) directions. The covariant components of the 1-form dF define the normal direction field to the implicit 2-surface F (x, y, s) = 0. Displacements orthogonal to the normal field satisfy the equation, dF = 0. Note that the zero set of the implicit function creates a surface in three-dimensional space, {x, y, s}, not a curve. In order to determine a space curve, a second surface must be described, and

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the logical intersection of this second surface and the first surface yields the "space curve". The logical intersection implies that the contact be tangential in the sense that the normals to each surface are collinear, or the contact can be such that the two surfaces have normals which are not collinear. For example, the simple intersection of the surface F (x, y, s) = 0 with the plane s = 0 determines a curve in the x, y plane, but the normals to the plane are not necessarily collinear with the normals to the 3D surface. As s varies a family of such curves are produced which may have multiple components. It is assumed that the members of the family can be projected to the {x, y} plane. For a point p on the surface F (x, y, s) = 0 (in {x, y, s} space), the neighborhood directions that cause dF to vanish are directions orthogonal to the surface normal at the point p. These directions determine the tangent plane to the surface at p. A second surface in (2+1)-space can be determined from the original implicit function by differentiation with respect to the family parameter and setting the resulting function to zero, Fs (x, y, s) = ∂F/∂s = 0. The intersection of these two surfaces produces a tortuous curve of perhaps several segments (components) in the space {x, y, s}. The first surface constraint F (x, y, s) = 0 leads to the differential 1-form, dF as, dF ≡ (∂F/∂x)dx + (∂F/∂y)dy + (∂F/∂s)ds = Fx dx + Fy dy + Fs ds,

(9.336)

and the second surface constraint Fs (x, y, s) = 0 leads to the differential 1-form, dFs as, dFs ≡ (∂ 2 F/∂s∂x)dx + (∂ 2 F/∂s∂y)dy + (∂ 2 F/∂s∂s)ds = Fsx dx + Fsy dy + Fss ds. (9.337) A necessary condition that the two surfaces F (x, y, s) = 0 and Fs (x, y, s) = 0 have a simultaneous solution is established by the requirement that the exterior product of the two 1-forms dF and dFs does not vanish. On the sets F (x, y, s) = 0 and Fs (x, y, s) = 0, this requirement reduces to the constraint, dF ˆdFs = {Fx Fsy − Fy Fsx }dxˆdy + Fss {Fx dx + Fy dy}ˆds 6= 0.

(9.338)

If Fss 6= 0, then it is possible to solve for s from the equation of the second surface, Fs (x, y, s) = 0. Use this value to eliminate the family parameter in the first equation of the surface. The result is a Function of {x, y} only, that defines a curve in the {x, y} plane independent from the family parameter. This curve is defined as the 2D envelope. This intersection has a component in the {x, y} plane if the first factor does not vanish. Note that (for fixed s) the critical points of (F = 0) ∩(Fx = 0) ∩ (Fy = 0)

Intersection, Envelopes and Topological Torsion

483

must be excluded. In more simple language, the critical points are where the tangent vector to the surface vanishes, and the points of interest for self intersection and envelopes is where the normal vector to the surface is zero. The three components of this 2-form on (2+1)-space form the components of a contravariant vector, J, which is tangent to the curve of intersection. If all three components vanish, then the two surfaces do not intersect. In particular, if, {Fx Fsy − Fy Fsx } = 0 and Fss = 0,

(9.339)

there is no intersection and no singularity. If, [{Fx Fsy − Fy Fsx } = 0 and Fss 6= 0],

(9.340)

[{Fx Fsy − Fy Fsx } 6= 0 and Fss = 0],

(9.341)

[{Fx Fsy − Fy Fsx } 6= 0 and Fss 6= 0],

(9.342)

dF ˆdFs ˆdFss 6= 0.

(9.343)

or there is a singularity, but no envelope. If both

then there is a curve which is an envelope of the family of curves. Note that the envelope condition implies that the primitive function, F, is non-linear in the family parameter, s. The process can be continued. A cuspoidal point of regression can be determined when the three functions F, Fs , and Fss satisfy the equation,

9.10.5 A Family of Surfaces in (3+1)-space The basic idea of the preceding section extends to higher dimensions. An implicit function Φ(x, y, z, σ) = 0,does not determine a surface in 3-space, but instead determines a hypersurface in 4-space. For a family of surfaces in three dimensions {x, y, z}, with a family parameter, σ, the criteria for the logical intersection of Φ(x, y, z, σ) = 0 and ∂Φ(x, y, z, σ)/∂σ = Φσ (x, y, z, σ) = 0 becomes, dΦˆdΦσ = {Φx Φσy − Φy Φσx }dxˆdy + {Φy Φσz − Φz Φσy }dyˆdz + {Φz Φσx − Φx Φσz }dzˆdx + Φσσ dΦˆdσ 6= 0.

(9.344) (9.345)

The first three terms are to be recognized as the components of the cross product, (9.346) ∇(x,y,z) Φ × ∇(x,y,z) Φσ .

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Topology and the Cartan Calculus

The argument is that when either {∇(x,y,z) Φ × ∇(x,y,z) Φσ 6= 0

and Φσσ = 0},

(9.347)

or {∇(x,y,z) Φ × ∇(x,y,z) Φσ = 0

and Φσσ 6= 0},

(9.348)

then the family has an intersection singularity. When both ∇(x,y,z) Φ × ∇(x,y,z) Φσ 6= 0 and Φσσ 6= 0, then there is a surface envelope. velopes.

(9.349)

Only non-linear family parameters produce en-

9.10.6 The edge of regression The process can be extended to find an edge of regression. In this case it is assumed that the three zero sets Φ(x, y, z, σ) = 0, Φσ (x, y, z, σ) = 0 and Φσσ (x, y, z, σ) = 0 have a common solution. The criteria for solubility for an edge of regression requires that the 3-form, which is the exterior product of all three differentials, does not vanish, dΦˆdΦσ ˆdΦσσ 6= 0. (9.350) The spatial components of this expression require that, (∇(x,y,z) Φ × ∇(x,y,z) Φσ ) • (∇(x,y,z) Φσσ ) 6= 0,

(9.351)

for the existence of an (cuspoidal) edge of regression. 9.10.7 Examples of Envelopes of families of surfaces Spheres moving along x-axis - the cylindrical canal surface Consider the function, Φ = (x − σ)2 + y 2 + z 2 − 1,

(9.352)

with a zero set which represents a family of unit spheres with centers at σ = ct moving along the x-axis. Φσ = ∂Φ/∂σ = −2(x − σ), Φσσ = +2. (9.353) The 2-form dA = −dΦˆdΦσ ⇒ {4zdxˆdz + 4ydxˆdy} at dσ = 0, is non-zero, and Φσσ 6= 0. From another point of view, ∇(x,y,z) Φ × ∇(x,y,z) Φσ = 0i − 4zj + 4yk. therefore the necessary conditions for the existence of an envelope are valid. Solving for σ from Φσ = 0 and substituting in Φ = 0, leads to the equation of the envelope, y 2 + z 2 − 1 = 0.

(9.354)

The envelope is a cylinder of radius 1, with the x-axis as the axis of rotational symmetry. The 3-form dΦˆdΦσ ˆdΦσσ vanishes so there is no edge of regression.

Intersection, Envelopes and Topological Torsion

485

Expanding spheres moving along the x-axis — the Mach cone Consider the function, Φ = (x − kσ)2 + y 2 + z 2 − σ 2 ,

(9.355)

with a zero set which represents a family of expanding spheres of radius σ with centers at kσ moving along the x-axis. When k > 1 the translational speed exceeds the expansion speed (of, say, sound, where σ = ct): Φσ = ∂Φ/∂σ = −2k(x) + 2(k2 − 1)σ,

Φσσ = +2(k2 − 1).

(9.356)

The 2-form dA (dA = −dΦˆdΦσ ⇒ {4zkdxˆdz + 4ykdxˆdy} for dσ = 0) is non-zero, and Φσσ 6= 0. Therefore the necessary conditions for the existence of an envelope are valid. Solving for σ from Φσ = 0 and substituting in Φ = 0, leads to the equation of the envelope, (k2 − 1)(y 2 + z 2 ) − x2 = 0,

(9.357)

which is a cone (the Mach cone), with a symmetry axis as the x-axis, and an aperture, tanθ =

p 1/(k2 − 1).

(9.358)

The 3-form dΦˆdΦσ ˆdΦσσ vanishes so there is no edge of regression. Concentric Spheres Consider the function, Φ = x2 + y 2 + z 2 − σ 2 ,

(9.359)

with a zero set which represents a family of unit spheres with variable radii, σ = ct, and centered on the origin. Φσ = −2σ,

Φσσ = −2.

(9.360)

The 2-form, dA = −dΦˆdΦσ = 0, for dσ = 0. therefore the necessary conditions for the existence of an envelope are not valid. The family of surfaces do not intersect as, ∇(x,y,z) Φ × ∇(x,y,z) Φσ = 0.

(9.361)

The 1-form A is integrable. The 3-form dΦˆdΦσ ˆdΦσσ vanishes so there is no edge of regression.

486

Topology and the Cartan Calculus

Spheres with a common point of tangency on the x-axis Consider the function, Φ = (x − σ)2 + y 2 + z 2 − σ 2 ,

(9.362)

with a zero set which represents a family of spheres of various radii and with centers along the x-axis. Φσ = ∂Φ/∂σ = −2x, Φσσ = 0. (9.363)

The 2-form dA = −dΦˆdΦσ 6= 0 for dσ = 0. therefore the necessary condition for the intersection singularity exists, but the subsidiary condition Φσσ 6= 0 is not satisfied. The singularity is the point where all the spheres have a common tangent, {x = 0, y 2 + z 2 = 0}. The envelope does not exist because the subsidiary condition Φσσ 6= 0 is not valid. Spheres with a common circle of intersection Consider the function, Φ = (x − σ)2 + y 2 + z 2 − (a2 + σ 2 ),

(9.364)

with a zero set which represents a family of spheres with centers along the x-axis. Φσ = ∂Φ/∂σ = −2(x), Φσσ = 0. The 2-form dA = −dΦˆdΦσ 6= 0 for dσ = 0. Therefore, the necessary condition for the intersection singularity exists, but the subsidiary condition Φσσ 6= 0 is not satisfied. However, the singularity exists as the circle of radius a in the x=0 plane, {x = 0, y 2 + z 2 = a2 }. The envelope does not exist because of the subsidiary condition Φσσ 6= 0 is not valid. 9.10.8 The Jacobian cubic characteristic polynomial Recall that the Jacobian matrix of a vector field in 3D always generates a polynomial equation of the cubic format. The Jacobian matrix is defined as the matrix of partial derivatives, [J(V m )] = [∂V m /∂xn ] = [Jm n ].

(9.365)

The cubic polynomial associated with the Jacobian matrix is called the CayleyHamilton characteristic polynomial, and always exists for any 3x3 matrix, Characteristic Polynomial of [J] ⇒ σ 3 − XM σ 2 + YG σ − ZA = 0.

(9.366)

The polynomial equation represents a family of implicit (possibly multi-component) surfaces, with the "family parameter" σ playing the role of the eigenvalues for the matrix, [J]. The coefficients of the polynomial, {XM , YG , ZA }, are called similarity invariants, for they remain the same for all similarity transformations of the [J] . These similarity invariants can be used as a set of intrinsic coordinates that are universal relative to similarity transformations.

Intersection, Envelopes and Topological Torsion

487

For a given vector field, V(x, y, z), the similarity invariants are functions of (x, y, z) and can be computed from the formulas: XM (x, y, z) = trace [Jm n ], adjoint , YG (x, y, z) = trace [Jm n] m ZA (x, y, z) = det [Jn ] .

(9.367) (9.368) (9.369)

From classic theory, the position vector in the space of intrinsic coordinates has components, [XM , YG , ZA ], XM (x, y, z) = σ 1 + σ 2 + σ 3 YG (x, y, z) = σ 1 σ 2 + σ 2 σ 3 + σ 3 σ 1 ZA (x, y, z) = σ 1 σ 2 σ 3 ,

(9.370) (9.371) (9.372)

where σ 1 , σ 2 , σ 3 are the eigenvalues of [Jm n (x, y, z)] . To repeat, these functions can be viewed as intrinsic coordinates or universal functions on the variety (x, y, z), and are invariant under similarity transformations of the Jacobian matrix, [J]. (For ease of notation, the subscripts on the capital letters, X, Y, Z,will be suppressed in that which follows.) Consider the cubic polynomial equation in standard format, Φ(x, y, z; σ) = σ3 − Xσ 2 + Y σ − Z = 0.

(9.373)

Define a second function defined by the first partial derivative of Φ with respect to the family parameter, Φσ (x, y, z; σ) = ∂Φ/∂σ = 3σ 2 − 2Xσ + Y.

(9.374)

Similarly, define a function defined by the second partial derivative Φ with respect to the family parameter, Φσσ (x, y, z; σ) = 6σ − 2X. (9.375) In terms of the intrinsic similarity variables, X, Y, Z, interpreted as intrinsic coordinates, the analysis that follows is universal for all 3x3 Jacobian matrices. The function σ is the eigenvalue parameter, and in thermodynamics can be associated with the molar density. It is of interest to determine if the family of cubic polynomial surfaces has an envelope. The envelope will be determined by the simultaneous solution of the two hypersurfaces, Φ(X, Y, Z; σ) = 0, and Φσ (X, Y, Z; σ) = 0. The envelope in general is an implicit two-dimensional surface in 3D with possibly many components; but for the cubic polynomial in intrinsic coordinates, the envelope will consist of two components. The envelope surface is independent from the choice of the parametrization, σ. The envelope is regular (without creases) on domains where Φσσ (X, Y, Z; σ) 6= 0. The envelope can exhibit singular properties, such as crease, where the tangential

488

Topology and the Cartan Calculus

contact of two surface portions forms a cuspoidal contact with a "curve" defined as the edge of regression. In thermodynamics, it is this type of curve (the edge of regression) that is associated with the spinodal line of the Gibbs surface of a van der Waals gas. The curve of intersection contact is associated with the binodal line for the Gibbs surface of a van der Waals gas.

Figure 9.4 Curves of Surface Intersection and Tangential Contact For the case of the cubic polynomial, solving for σ from Φσ = 0 leads to two roots, σ + and σ − expressing the values of the family parameter which will satisfy the envelope condition. In terms of intrinsic coordinates, there are two possibilities, σ ± = (X ±

√ X 2 − 3Y )/3.

(9.376)

Recall that in general the envelope surface does NOT exist if Φσσ (x, y, z; σ) = 0. Substitution of σ± into the primary cubic implicit surface function, Φ = 0, leads to two envelope equations, Φ+ = 0, and Φ− = 0, depending on which root was chosen for the substitution. For every cubic polynomial there are two envelope sheets. Each envelope implicit surface function, Φ± , is independent of the parameter, σ. These surfaces (in terms of universal similarity coordinates) are given by the formula, ¡ ¢3/2 Φ± = {−2/27X 3 ∓ 2/27 X 2 − 3Y + XY /3 − Z}. (9.377)

Intersection, Envelopes and Topological Torsion

489

It was a surprise to find that the product of the two functions Φ+ · Φ− leads to a new function which is precisely the historical Cardano-Tartaglia function (to within a numeric factor). The Cardano function is given by the formula, Cardano = Φ+ · Φ− = −4(X 2 − 3Y )3 + (2X 3 − 9XY + 27Z)2 .

(9.378)

This derivation of the famous Cardano formula is remarkable in that it is based upon the theory of envelopes in an intrinsic space of 3 coordinates and 1 family parameter. Recall that the Cardano function determines the root structure of the cubic polynomial (see p. 99, [79]). If the Cardano function is positive, then the cubic polynomial has one real solution and two conjugate complex solutions. If the Cardano function is negative, then there are three real distinct roots. If the Cardano function vanishes, then there are three real roots, and two coincide (the classical homology). The zero set of the Cardano function determines the degenerate root structure, where, in this case, the two envelope functions Φ± meet in a cuspoidal tangential contact. The degeneracy condition in effect defines the edge of regression for the envelope of two surface components. The position vector to the curve that defines the edge of regression can be established by first solving the equation Φσσ (X, Y, Z; σ) = 0, for X, Xedge = 3σ edge = 3u.

(9.379)

The parameter σ along the edge of regression is defined as u. Then, substitute this value for X(u) in the equation, Φσ (X, Y, Z; σ) = 0 to find a value for Y (u) in terms of the parameter, Yedge = −3u2 + 2Xedge u = 3u2 . (9.380) Substitution of the two values for X(u) and Y (u) into the equation, Φ(X, Y, Z; σ) = 0, yields a values for Z(u), Zedge = u3 − Xedge u2 + Yedge u ⇒ u3 ,

(9.381)

thereby defining the position vector to the curve of tangential contact in [X, Y, Z] space in terms of a single parameter, u, Redge_of _regression = [Xedge , Yedge , Zedge ] ⇒ [3u, 3u2 , u3 ].

(9.382)

Differentiation of this position vector to the curve of tangential contact (edge of regression) with respect to the family parameter u leads to the "velocity" expression, dRedge_of _regression = [3, 6u, 3u2 ]du = V(u)du.

(9.383)

The formula can be put into the Frenet Format, by defining the arclength parameter, s , in the classic way, p √ T(s) = V(u)/ V ◦ V = [1, 2u, u2 ]/{ (1 + 4u2 + u4 )}. (9.384)

490

Topology and the Cartan Calculus

The Frenet Torsion and the Frenet curvature for the space curve (defined as the edge of regression of the cubic polynomial) in the intrinsic coordinates can be evaluated as:

Edge of Regression Frenet Curvature Frenet Torsion

p = 2/3 (u4 + u2 + 1)/(1 + 4u2 + u4 )3/2 , = (1/3)/(u4 + u2 + 1).

(9.385) (9.386)

An Electrodynamic Perspective of the Cubic Polynomial From another point of view, construct the 2-form from the exterior product of the two differentials that make up the envelope, Fenvel = dΦˆdΦσ ⇒ {1dY ˆdZ + 2σdZˆdX + σ2 dXˆdY } +{2σ(−Xσ + Y )dXˆdσ + (3σ 2 − Y )dY ˆdσ +(−6σ + 2X)dZˆdσ}.

(9.387)

The 2-form Fenvel has an image in the spatial domain for constant values of σ. This 2-form is globally closed (dFenvel = 0), and therefore can be put into correspondence with 2-form representing the "EM" field, Fenvel − dAenvel = 0. In terms of the coefficients of the 2-form, Fenvel , the functions that define E and B can be written as, B = [1, 2σ, σ 2 ], E = [2σ(−Xσ + Y ), (3σ 2 − Y ), (−6σ + 2X)].

(9.388) (9.389)

Note that on the edge of regression, the E field goes to zero, but the B field is finite. If the parameter σ is given the significance of "time", then the B field is explicitly "time" dependent, of zero spatial divergence, and of zero spatial curl. The Maxwell - Faraday induction law holds, such that,

curlE + ∂B/∂σ = 0, [0, −2, −2σ] + [0, 2, 2σ] = 0.

(9.390) (9.391)

By direct computation it becomes evident that, globally, Fenvel ˆFenvel = 2E ◦ B ⇒ 0.

(9.392)

The implication is that the thermodynamic system of the cubic polynomial is in effect a closed, not an open system, and the system need not be in equilibrium. The generating 1-form, Aenvel , is not of Pfaff dimension 4, but is either 2, or 3. If the

Intersection, Envelopes and Topological Torsion

491

Pfaff dimension of Aenvel is 3, then the system is not in equilibrium. Note that given the 2-form Fenvel does not permit a unique determination of the generating 1-form, Aenvel , hence closed but not necessarily exact 1-forms can be added to any specific 1-form, Aenvel , and yet the 2-form Fenvel remains unchanged. The Pfaff dimension (if odd) depends upon these "gauge" additions. The space curve that represents the edge of regression of the cubic polynomial is defined by "curl" of some vector potential, Aenvel and that curl is equivalent to the magnetic B field from an EM point of view. From a hydrodynamic point of view, the curl field is equivalent to the fluid velocity. The E field has no component in the direction of B, and no component along the z-axis, when Φσσ ⇒ 0. This last implicit function constraint is a necessary requirement for the existence of an "edge of regression". It is also possible to construct the 3-form, J, as,

J = dΦˆdΦσ ˆdΦσσ = i(2[3, 2X, Y, 1])dXˆdY ˆdZˆdσ,

(9.393)

which is also globally closed. For the cubic polynomial it is known that the zero set of the Cardano function not only separates the domains for which the eigenvalues are real or complex, but is also the surface upon which there can exist repeated roots. The Cardano function is exactly equal to the Envelope function divided by 4. Hence the zero sets are the same. The edge of regression is precisely such a curve of repeated roots. A plot of the Cardano envelope appears in Figure 9.5 below, The coordinate labels are equivalent to X = 3M, Y = 3G, Z = A. Note the edge of regression and the two Envelope surfaces meeting the edge of regression in a cusp. The Cardano function (or envelope) can be constructed as a tangential developable (a ruled surface) based on the curve whose tangent vector is given by the edge of regression formula, J = λ[1, 2σ, σ 2 ]. A point on the Cardano surface is given by X = R ± λJ with 1 sheet of the envelope determined by positive motion along the edge of regression and the other sheet of the envelope determined by motion in the opposite direction. It is important to note that the linear neighborhood extensions are NOT "time reversal invariant", although the line of the edge of regression is "time reversal invariant". This property of the trajectory neighborhoods is due to the fact the edge of regression has torsion in the sense of Frenet. The linear extended neighborhood in the direction of the tangent, for motion in the positive direction, extends to one sheet of the Envelope, (say Φ+ ). Linear extension along the tangent in the opposite direction of the edge of regression extends along the other sheet of the envelope, (say Φ− ).

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Topology and the Cartan Calculus

Figure 9.5 The Universal Cardano Envelope These concepts have utility in thermodynamics, for the Gibbs equilibrium surface of a van der Waals gas is a function which is cubic in its family parameter. The Spinodal Line of the Gibbs surface is an edge of regression (and is determined by the condition that the Gauss curvature vanish). The Binodal line is a line of self intersection. The critical point is where both the mean curvature and the Gauss curvature of the surface vanish. Thermodynamically, the Spinodal line is the edge of regression of the Gibbs surface for a van der Waals gas. The observation above regarding time reversal invariance implies that motion along the Spinodal line in the direction of the critical point is stable in one direction, but unstable in the other. 9.10.9 The General Theory For a given implicit surface equation, Φ(x, y, z, .., σ) = 0 another way to specify the envelope is to construct the 1-form, Σ, in a space of N+1 variables, Σ = Φx dx + Φy dy + Φz dz... = dΦ − Φσ dσ = d(Φ − σΦσ ) + σdΦσ ,

(9.394)

(which by construction is not explicitly dependent only upon parametric displacement, dσ). This 1-form may not be globally exact, as dΣ = −dΦσ ˆdσ 6= 0, necessarily. In fact, this 1-form, Σ, need not be uniquely integrable, for globally, ΣˆdΣ = −dΦˆdΦσ ˆdσ 6= 0,

(9.395)

Intersection, Envelopes and Topological Torsion

493

necessarily. If the 2-form dΦˆdΦσ = 0 then no envelope exists, and the Topological Torsion of the 1-form vanishes, ΣˆdΣ = 0. In other words, the 1-form Σ defined above does not satisfy the Frobenius criteria of unique integrability, when an envelope exists. Moreover, the space exhibits Topological Torsion. This result is pleasing, for the concept of an envelope intuitively implies non-uniqueness. In each of the examples above, the criteria for an envelope to exist required that the 2-form dΦˆdΦσ does not vanish. The conclusion to be reached is that: Remark 68 The existence of topological torsion is necessary for the existence of an envelope. Hence an envelope implies that the 1-form Σ is of Pfaff topological dimension 3 or more. The envelope is indicative that the thermodynamic system defined by Σ is not an equilibrium system. The necessary and sufficient conditions for an envelope of a family of functions parameterized by σ are given by the exterior differential system, dΦˆdΦσ 6= 0 and Φσσ 6= 0.

(9.396)

The same argument works in higher dimensions. The basic idea is that if for a singly parametrized function, Φ(x, y, z...; σ) on a space of N+1 dimensions, the 1-form Σ = dΦ − Φσ dσ is not necessarily globally integrable, a fact which implies non-uniqueness of the solution to the Pfaffian equation, Σ = 0. The concept of nonuniqueness admits to the possibility of finding an envelope of dimension N-2 which is independent from the parameter, σ. For suppose Φσ = 0 defines a set of dimension N-1 which intersects with the N-1 set Φ = 0, to produce a set of dimension N-2. In order for an envelope to exist, the non-uniqueness argument implies as a necessary condition that the 2-form dΦˆdΦσ cannot vanish, and a sufficient condition for nonuniqueness as ΣˆdΣ = −dΦˆdΦσ ˆdσ 6= 0. This result implies that the condition for the existence of an envelope in three spatial dimensions and one parametric dimension requires that, ΣˆdΣ 6= 0 ⇒ ∇(x,y,z) Φ × ∇(x,y,z) Φσ 6= 0. (9.397) In (3+1)-space, the "envelope" is the two-dimensional surface of logical intersection of the two three-dimensional sets, Φ = 0 and Φσ = 0, subject to the constraint that Φσσ 6= 0. The edge of regression in (3+1)-space The surface function may be non-linear in the parameter, σ, such that it is possible to compute Φ = 0, Φσ = 0, and Φσσ = 0. If Φσσσ 6= 0, then it may be possible (but admittedly difficult) to find a simultaneous intersection of the three N-1 sets to produce in this case a one-dimensional line. For the tangential intersection to be non-empty it is necessary that the 3-form dΦσ ˆdΦσσ ˆdΦ 6= 0.

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Topology and the Cartan Calculus

The convenience of the characteristic polynomial expansion is that it is possible to continue the partial differentiation process until (in intrinsic coordinates) that is possible to solve for the equivalent of the edge of regression, as was done in the preceding subsection.

Chapter 10 REFERENCES AND ADDENDA 1. The notion that the Liouville theorem must fail disallows the utilization of a symplectic geometry approach (better said in 1997: symplectic transformations cannot be utilized to study turbulence, but there exist uniquely defined thermodynamic irreversible processes on the symplectic manifold!), e.g., Abraham, R. (1967) "Foundations of Mechanics" Benjamin, NY 2. Aebischer, B. et. al., (1992) "Symplectic Geometry" Birkhauser Verlag, Basel Boston Berlin. 3. Adkins, C. J., (1968) "Equilibrium Thermodynamics", McGraw Hill, London. 4. Alexandroff, P., (1961) "Elementary Concepts of Topology" Dover, NY, 66. 5. Antonelli, P. L., Ingardern, R.S., and Matsumoto,. M., (1989) "The Theory of Sprays and Finsler Spaces with applications to Biology and Physics", Kluwer, Dordrecht, Nd. 6. Aris, R. (1962) "Vectors, Tensors, and the Basic Equations of Fluid Mechanics", 84. Prentice-Hall. 7. Arnold, V.I., (1989) "Mathematical Methods of Classical Mechanics", 198, Springer Verlag, NY 8. Arnold, V.I., (1981) "London Math. Soc. Lect. Note Series 53", Cambridge University Press, 225. 9. Atiyah, Michael, (1988): "Collected Works by Michael Francis Atiyah - Mathematics - 1988 - 720 pages", 583 - "For this we just have to recall that, intrinsically, the Higgs field is the difference between the Lie derivative and the covariant derivative along the orbit. . . ". 10. Bahcall, J. and Schmidt, M. (1967) Phys Rev Lett. 19, 1294. 11. Baldwin, P. and Kiehn, R. M. (1991) “Cartan’s Topological Structure“, Poster presented at the summer workshop, “Topology in Fluid Dynamics“, Inst. for Th. Phy. UCSB, August 1991. arXiv.org/math-ph/0101033, ..topstru4.pdf. 12. Bamberg, P. and Sternberg, S. (1992) "A course in mathematics for students of physics, 2", Cambridge University press, Cambridge, 775- 780. 13. Barbossa, J. L. and do Carmo, M. (1976), Amer. Journ. Math. 98, 515. 14. Batchelor, G. K. (1967), "Fluid Dynamics", Cambridge University Press, 354. 15. Bateman, H. (1914-1955) "Electrical and Optical Wave Motion", Dover, NY 16. Bellissard, J. (1986) Lect. Notes in Physics, 257, 99.

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modes

do

not

transmit

power

in

a

waveguide.

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193. Kiehn, R.M. (1977), Periods on manifolds, quantization and gauge, J. Math. Phy. 18, 614. (http://www22.pair.com/csdc/pdf/periods.pdf) 194. Sanders, V. E. and Kiehn, R.M. (1977) Dual Polarized Ring Lasers, IEEE J Quant Elec QE-13, 739. 195. The explicit variation of the helicity density was reported by R.M.Kiehn as a reason for the transition from laminar to non-laminar flow in the NASA-AMES research document NCA-2-0R-295-502 (1976). Also see Kiehn, R. M. (1977) J. Math. Phy. 18, 614; and Kiehn, R. M. (1975) Intrinsic hydrodynamics with applications to space-time fluids., Int. J. Engng Sci. 13, 941-949. 196. The importance of the (N-1)-form A^H (now written as A^G) was first anticipated in: Kiehn, R. M. and Pierce, J. F. (1969) An Intrinsic Transport Theorem, Phys. Fluids, 12, #9, 1971. The concept was further employed in R. M. Kiehn, R. M. (1977) Periods on manifolds, quantization and gauge, J. of Math Phys 18 #4, 614, and with applications described in Kiehn, R. M. (1991) Are there three kinds of superconductivity, Int. J. Mod. Phys B 5, 1779. 197. Kiehn, R. M., (1987) The Falaco Effect as a topological defect was first noticed by the present author in the swimming pool of an old MIT friend, during a visit in Rio de Janeiro, at the time of Halley’s comet, March 1986. The concept was presented at the Austin Meeting of Dynamic Days in Austin, January 1987, and caused some interest among the resident topologists. The easily reproduced experiment added to the credence of topological defects in fluids. It is now perceived that this topological phenomena is universal, and will appear at all levels from the microscopic to the galactic. (http://www22.pair.com/csdc/pdf/falaco85o.pdf), arXiv.org/gr-qc/0101098 (http://www22.pair.com/csdc/pdf/falaco97.pdf and (http://www22.pair.com/csdc/pdf/topturb.pdf) 198. Sterling. et.al. (1987) Why are these disks dark? Phys Fluids 30, 11. 199. Kiehn, R. M. (1988) Torsion in Crystalline Fluids, The Energy Laboratory Newsletter, 18 Univ. of Houston, 6 200. Kiehn, R. M. (1989) Irreversible Topological Evolution in Fluid Mechanics in "Some Unanswered Questions in Fluid Mechanics" ASME- Vol. 89-WA/FE-5, Trefethen, L. M. and Panton, R. L. Eds. 201. First presented at the January 1990 Dynamics Days, conference at Austin, Texas. 202. Kiehn, R.M. (1990) Topological Torsion, Pfaff Dimension and Coherent Structures, in: "Topological Fluid Mechanics", H. K. Moffatt and T. S. Tsinober eds, Cambridge University Press, 449-458. (http://www22.pair.com/pdf/csdc/camb89.pdf) 203. Kiehn, R. M. (1991) Compact Dissipative Flow Structures with Topological Coher-

References and Addenda

505

ence Embedded in Eulerian Environments, in: "Non-linear Dynamics of Structures", edited by R.Z. Sagdeev, U. Frisch, F. Hussain, S. S. Moiseev and N. S. Erokhin, 139-164, World Scientific Press, Singapore. 204. Kiehn, R.M. (1991) Continuous Topological Evolution, arXiv.org/math-ph/0101032 (http://www22.pair.com/csdc/pdf/contevol3.pdf) 205. Kiehn, R. M. (1991) Are there three kinds of superconductivity, Int. Journ. of Modern Physics, 5 #10, 1779. 206. Kiehn, R. M., Kiehn, G. P., and Roberds, B. (1991) Parity and time-reversal symmetry breaking, singular solutions and Fresnel surfaces, Phys. Rev A 43, 5165-5671. (http://www22.pair.com/csdc/pdf/timerev.pdf) 207. Kiehn, R.M. (1991) Some Closed Form Solutions to the Navier Stokes Equations. arXiv/physics/0102002 (http://www22.pair.com/csdc/pdf/nvsol.pdf) 208. Kiehn, R.M. (1991), "Dissipation, Irreversibility and Symplectic Lagrangian systems on Thermodynamic space of dimension 2n+2", (http://www22.pair.com/csdc/pdf/irrev1.pdf) 209. Kiehn, R. M., "Topological Parity and the Turbulent State of a Navier-Stokes Fluid", (http://www22.pair.com/csdc/pdf/topturb.pdf) 210. Kiehn, R. M. (1992), Topological Defects, Coherent Structures and Turbulence in Terms of Cartan’s Theory of Differential Topology, in "Developments in Theoretical and Applied Mathematics, Proceedings of the SECTAM XVI conference", B. N. Antar, R. Engels, A.A. Prinaris and T. H. Moulden, Editors, The University of Tennessee Space Institute, Tullahoma, TN 37388 USA. 211. Kiehn, R. M. (1993), Instability Patterns, Wakes and Topological Limit Sets, in "Eddy Structure Identification in Free Turbulent Shear Flows", J.P.Bonnet and M.N. Glauser, (eds), Kluwer Academic Publishers, 363. 212. Kiehn, R. M. (1993) Talk presented at the EUROMECH 308 meeting at Cortona, Italy. Unpublished. (http://www22.pair.com/csdc/pdf/italy.pdf) 213. Kiehn, R. M. (1995), Hydrodynamic Wakes and Minimal Surfaces with Fractal Boundaries, in "Mixing in Geophysical Flows", J. M. Redondo and O. Metais, Editors CIMNE, Barcelona ERCOFTAC (meeting in 1992) 52-63. 214. Kiehn, R. M. (1997) "When does a dynamical system represent an Irreversible Process" SIAM Snowbird May 1997 poster. 215. Kiehn, R. M., (1999), Coherent Structures in Fluids are Topological Torsion Defects, in J, "IUTAM Symposium on Simulation and Identification of Organized Structures in Flows", N. Sørensen, et al., eds., Kluwer Academic Publishers, Dordrecht,. See (http://www22.pair.com/csdc/pdf/copen5.pdf). Presented at the IUTAM-SIMFLO Conference at DTU, Denmark, May 25-29, (1997). 216. Kiehn, R.M. (1999), Topological evolution of classical electromagnetic fields and the photon, in, "Photon and Poincaré Group", V. Dvoeglazov (ed.), Nova Science Publishers, Inc., Commack, NY, 246-262. ISBN 1-56072-718-7. Also see (http://www22.pair.com/csdc/pdf/photon5.pdf) 217. Kiehn, R. M. (2000), 2D turbulence is a Myth, (invited speaker EGS XXIV General

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Assembly IUTAM, the Hague, 1999 (http://www22.pair.com/csdc/pdf/hague6.pdf) 218. Kiehn, R.M., (2001) Topological-Torsion and Topological-Spin as coherent structures in plasmas", arXiv.org/physics /0102001 (http://www22.pair.com/csdc/pdf/plasmas.pdf) 219. Kiehn, R. M. (2002), Curvature and torsion of implicit hypersurfaces and the origin of charge, Annales de la Foundation Louis de Broglie, vol 27, 411. arXiv.org/gr-qc /0101109 (http://www22.pair.com/csdc/pdf/contevol3.pdf) 220. Kiehn, R.M. (2002), The Photon Spin and other Topological Features of Classical Electromagnetism, in "Gravitation and Cosmology, From the Hubble Radius to the Planck Scale", Amoroso, R., et al., eds., Kluwer, Dordrecht, Netherlands, 197-206. Vigier 3 conference in 2000. (http://www22.pair.com/csdc/pdf/vig2000.pdf) 221. Kiehn, R. M. (2003), Thermodynamic Irreversibility and the Arrow of Time, in "The Nature of Time: Geometry, Physics and Perception", R. Bucher et al. (eds.), Kluwer, Dordrecht, Netherlands, 243-250. (http://www22.pair.com/csdc/pdf/arwfinal.pdf) 222. Kiehn, R. M. (2004), A topological perspective of Electromagnetism, to be published in "Classical electrodynamics: new horizons", Editors Andrew E. Chubykalo and Roman Smirnov-Rueda, Rinton Press, Inc. USA. Also see (http://www22.pair.com/csdc/pdf/topevem.pdf) 223. Kiehn, R. M. (2004) A Topological Perspective of Cosmology, in "Relativity, Gravitation, Cosmology" Edited by Valeri V. Dvoeglazov and Augusto A. Espinoza Garrido. ISBN 1-59033-981-9; QC173.58.R45 2004-06-09 530.11—dc22; 200405627. Published by Nova Science Publications, Inc., NY, USA, 2004, 163 Also see (http://www22.pair.com/csdc/pdf//cosmos.pdf) 224. Kiehn, R. M. (2004?) "A Topological Perspective of Plasmas" to be published. Also see (http://www22.pair.com/csdc/pdf/plasmanew.pdf) 225. see Holder Norms (3D), (http://www22.pair.com/csdc/pdf/holder3d.pdf) Holder Norms (4D), ( http://www22.pair.com/csdc/pdf/holder4d.pdf) Implicit surfaces ( http://www22.pair.com/csdc/pdf/implinor.pdf) 226. see Dissipation, Irreversibility and Symplectic Lagrangian Systems on Thermodynamic Space of Dimension 2n+2 (http://www22.pair.com/csdc/pdf/irrev1.pdf) 227. Kiehn, R. M. (2005), Propagating topological singularities: the photon, in "The Nature of Light: What Is a Photon?"; Chandrasekhar Roychoudhuri, Katherine Creath; Eds,Proc. SPIE 5866, 192-206. 228. R. M. Kiehn (2006), A topological theory of the Physical Vacuum. qc/0206118

arXiv.org/gr-

229. Robertson, H. P. and Noonan, T.W. (1968) "Relativity and Cosmology", Saunders, Phildelphia. 230. Robinson, D.C and Shadduck, W. F. (1996) Fields Institute Communications, 7, 189. 231. Rodriguez, C. C. (1991) From Euclid to Entropy, In Maximum Entropy and Bayesian Methods, W. T. Grandy, Jr. Ed., Kluwer Academic Publishers, 1-6 232. Rohrlich, F. (1965) "Classical Theory of the Electron" Addison-Wesley, Reading,

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276. Kiehn, R. M. (2004) Plasmas and non-equilibrium ics, "Non-Equilibrium Systems and Irreversible Processes (http://www.lulu.com/kiehn).

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Equations", Translations of Mathematical Monographs, 113, AMS Rhode Island. 300. Zimin, V. (1991), "Hierarchical Models of Turbulence" in: Non-linear Dynamics of Structures, edited by R.Z. Sagdeev, U. Frisch, F. Hussain, S. S. Moiseev and N. S. Erokhin, World Scientific Press, Singapore, 273-288.

10.1

Acknowledgments and Index

I must thank David Radabaugh, at one time a student, then a colleague, and always a friend. Not only did he supply the first photos in 1986 of the Falaco Solitons as topological defects in a fluid, but also he spent many painstaking hours editing, suggesting changes, and checking cross references, and all the other things that I hate to do. Thanks, David.

Index Adiabatic Processes and Transversality , 69 associative vectors, 58 Characteristic vectors, 62 Covariant derivative, 77 Irreversible, 65 Irreversible , 80 Irreversible Torsion, 73 Local, 64 Adjoint Current, 144 Applied Topology, 26 Axioms (of non-equilibrium thermodynamics), 28 Bernoulli Invariants, 85 Thermodynamic Potentials, 86 Bifurcations, 220 in Projective spaces, 114 Tertiary, 226, 282 Brand Invariants, 210 Bulk viscosity Navier-Stokes, 176 topological torsion, 73 turbulence, 177 cosmological expansion, 75 dissipation, 95 entropy production, 36 Caratheodory, 13, 113, 315, 436, 447 Cardano envelope, 217 Cartan Connections, 449 Cartan’s Calculus, 418 Closure, 436 Exterior Algebra, 419 Exterior differential forms, 418 Exterior differntial, 423

Interior Product, 429 Lie Differential, 430 Pullbacks, 431 Topological features, 434 Cartan’s Magic Formula, 14, 50 Cartan’s Topological Structure, 301 Applications to Continuous Topological Evolution, 317 Connected and Disconnected Topologies, 315 Differential Form language, 311 Limit Points and the exterior derivative, 309 Point Set example, 303 Closure and Continuity, 436 Continous Process definition, 38 Continuity, 325 Uniform, 319 Continuous Evolutionary Processes, 42 Conformal , 87 Continuous Process Classes Associated class, 58 Characteristic Class, 62 Extremal class, 58 Topological Torsion Class, 61 Continuous Topological Change, 12, 14, 16, 25, 28, 31, 34, 42, 44, 45, 56, 63, 151, 251, 293, 301, 331, 340, 342, 350, 418 Continuous Topological Evolution, 323 Cartan’s Magic Formula, 49 Cartan’s magic Formula, 327 Deformation Invariants, 331 Evolutionary Invariants, 330 The Lie differential, 328 511

512

Continuous Topological Evolution Symplectic to Contact structures, 293 Curvature Invariants, 209 Curvatures, 467 DeRham Categories, 338 Derivative vs Differential, 50 Differential Covariant Differential, 76 Exterior Differential, 423 Lie Differential, 77 Distributions, 473 Dynamical Systems examples ABC Henon Flow, 233 Belsouov-Zhabotinsky, 241 Chiral Separation, 240 Eulerian Rotator, 231 Lorenz system, 235 Magnetic Dynamo, 237 Moffat STF flow, 234 Rossler Attractor, 238 The Brusselator, 239 Edge of Regression, 140 Envelopes, 137, 215 Cardano function, 491 Cayley-Hamilton cubic polynomial, 486 Cayley-Hamilton quartic polynomial, 137 General theory, 481 Mixed Phases, 140 Topological Torsion, 478 Universal Gibbs function swallowtail, 139 Equilibrium Submanifolds, 88 Equivalence Principle, 455 Euler Index, 348 Evolutionary invariants deformation invariants, 38 Exterior Algebra, 419 Exterior Differential, 423

INDEX

Exterior differential forms, 16, 418 Scalars and Densities, 359 Extremal Vectors (2n+1), 43 Falaco Solitons, 257 Cosmology , 279 Landau Ginsburg theory, 275 First Law of Thermodynamics Cartan’s magic formula, 50 First law of thermodynamics, 28 Frame Fields Implicit, 474 Line Bundles, 462 Parametric, 475 Projective, 477 Frobenius Integrability, 13, 17, 58, 62, 89, 315, 316, 349, 409, 436 Non-uniqueness and Topological Torsion, 36, 447 Geometry and Physics, 409 Affine (transitive) geometry, 413 Conformal geometry, 415 Intransitive geometry, 414 Projective geometry, 410 Gibbs Swallowtail, 138 Global Conservation Laws, 343 Heat inexact 1-form Q, 50 Homogeneous p-forms, 353 Braids and closed 3-forms, 385 Gauss Integrals (2-forms), 379 Gauss Link integral, 382 Scrolls, 384 Tangential Developables, 383 Hopf Bifurcation, 131 Integral Invariants, absolute, 39 Integral Invariants, relative, 40 Interior Product, 429 Internal Energy EM interaction energy , 51 Internal Energy U, 50 Irreversibility (Logical)

INDEX

Arrow of time, 46 Irreversibility (thermodynamic), 62 Irreversible Processes The sliding bowling ball example, 94 Kinematic Perfection, 49 Lie Differential, 430 Limit cycles, 71, 243 Lorentz Force, 169, 175 Maps, Basis Frames, 447 Maxwell-Faraday system, 166 Minimal surfaces, 134 Conjugate pairs, 135 Gibbs swallowtail, 136 Minimal Surfaces Fractal minimal surfaces, 136 Mole number, 14 Negative Pressure, 223 Non-Canonical Momentum, 80 Non-Equilibrium Systems, 146 Pair and Impair differential forms, 399 Period Integrals, 41 Pfaff Topological Dimension Applications, 52 Change of Pfaff Dimension, 56 Definition, 34 Pfaff sequence, 35 Phase (topological), 13, 15 Coherent Structures, 38, 40, 56, 171, 210, 246, 279, 316, 331, 357, 408 Phase Transitions, 254 Physical Systems, 53 Connected topology PD less than 3, 54 Disconnected topology PD more than 2, 54 Non-equilibrium, 54 Point Set Topology, 438

513

Boundary, 445 Closed and Open sets, 439 Closure, 443 Continuity, 443 Exterior, 445 Interior, 444 Limit Points, 441 Pre-geometry, 37 Processes and the Pfaff dimension of W Chaos PD 3, 341 Extremal PD=0 , 66 Helmholtz - Sympectic PD=1 , 66 Stokes PD=2 or 3 , 70 Pullbacks examples, 431 functional substitution, 401 Retrodiction vs Prediction, 403 Similarity Invariants, 207 Singularities Pfaff dimension 3, 143 Spinors eigendirections of 2-forms, 37 Hopf maps, 131 Spinors and Work, 59 thermodynamic processes, 153 Structural Equations (Cartan), 465 Tertiary Bifurcations, 282 Falaco Minimal Surface, 291 Hysteresis Hopf, 287 Pitchfork Hopf , 283 Saddle Node Hopf, 285 Transcritical Hopf, 289 Thermodynamic Phase function Binodal Line, 130 Cayley-Hamilton Polynomial, 119 Critical Isotherm, 129 Higgs potential, 127 Intensive and Extensive variables, 109, 114 Similarity Invariants, 122

514

Spinodal Line, 130 Thermodynamic Processes, 58 Thermodynamic Systems, 52 Tisza, 13 Top Pfaffian, 79, 80, 93 Topological Fluctuations Artifacts of Spinors, 60 Cartan-Hilbert 1-form, 78 Fluctuations in position and velocity, 89 Pressure and Temperature, 81 Second order and first order ODE’s, 97 Topological perspective, 21, 27 Topological Physics, 415 Topological properties, 32 Topological quantization, 41 Topological Spin, 145 Topological Thermodynamics, 49, 151 Contact Manifolds Pfaff Dimension 3, 152 Dynamical Systems, 195 Helmholtz processes, 189 Hydrodynamic systems, 174 Isovectors, 191 Lagrangian systems, 181 Mechanical systems, 177 Navier-Stokes systems, 175 Pfaff dimension 4 systems, 183 Sol geometry, 186 Symplectic Manifolds Pfaff Dimension 4, 156 The Master Equation, 190 The van der Waals gas as a dynamiical system, 202 Topological Torsion, 62, 64, 73, 88, 96 chaos, 385 Disconnected Topology, 316 Electromagnetism, 170 Envelopes, 493 evolution of, 350 extended to n > 4, 80

INDEX

Hydrodynamics, 176 Isovectors, 191 Torsion bursting, 253 Torsion Current, 347 Torsion Quanta, 172 Topological Torsion Current, 346 Topological Torsion example, 72 Topological Torsion Vector, 44 Transversality, 64 Turbulence, 46, 173, 317, 319, 356 Turbulent State, 63, 76 Van der Waals gas, 103 Binodal Line, 130 Critical Point, 104 Edge of regression = Spinodal Line, 207 Gibbs dual surface, 107 Spinodal Line, 130 Universal P-rho-T hypersurface, 106 Work inexact 1-form W, 50

Symbols

10.2

515

Symbols

The notation used in this monograph is:

L . L(V ) . H . H . . K . A . F . . G . . J . W . Q . PTD PD

Lagrange function. − Lie differential with respect to V . − Hamiltonan function (typically H = pV − L). − The 3-form of Topological Torsion, AˆF (not necessarily equal to Frenet Torsion, or Affine Torsion). − The 4-form of Topological Parity. − The 1-form of system Action (can be path dependent). − The exact 2-form of field intensities F = dA (coefficients are covariant tensors wrt diffeomorphisms). − The non-exact 2-form of field quantities (coefficients are contravariant tensor densities wrt diffeomophisms). − The exact 3-form of charge current densities. − The 1-form of Work (can be non-exact and path dependent). − The 1-form of Heat (can be non-exact and path dependent). − Pfaff Topological Dimension.

Take care to distinguish the Lagrange function (L sans serif) from the Lie differential, L(V ) , and the Hamiltonian Function (H sans serif) and the 3-form of Topological Torsion, H.

516

10.3

INDEX

About the Cover Picture

Almost everyone who has been in a bowling alley has seen the evolutionary process described in the picture. A bowling ball is given linear momentum, and angular momentum (in this case backspin). When the ball touches the alley, the ball skids and slips as friction irreversibly causes the angular momentum and kinetic energy to decay. When the conditions of no-slip rolling are achieved, the irreversible decay stops, and, to first order, the further motion of the ball is without dissipation.

Emergence of a long lived state far from equilibirum This picture demonstrates the irreducible topological evolution from a symplectic state of Pfaff Topological dimension 2n+2 to a long lived contact state of Pfaff topological dimension 2n+1. The evolution is in the direction of the Topological Torsion vector. Note that the angular momentum changes sign. The Contact Structure supports an extremal (Hamiltonian) evolutionary process that is without irreversible decay. The topological change occurs when the ball stops skidding and then rolls without slipping. The Topological Torsion vector then has zero divergence and becomes an adiabatic characteristic vector field, with a Hamiltonian generator. This problem is considered in detail in Volume 2.

About the Author

10.4

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About the Author

Professor R. M. Kiehn, B.Sc. 1950, Ph.D. 1953 (Physics, Course VIII, MIT), started his career working (during the summers) at MIT, and then at the Argonne National Laboratory on the Navy’s nuclear powered submarine project. Argonne was near his parents home in the then small suburban community known as Elmhurst, Illinois. At Argonne, Dr. Kiehn was given the opportunity to do nuclear experiments using Fermi’s original reactor, CP1. The experience stimulated an interest in the development of nuclear energy. After receiving the Ph. D. degree as the Gulf Oil Fellow at MIT, Dr. Kiehn went to work at Los Alamos, with the goal of designing and building a plutonium powered fast breeder reactor, a reactor that would produce more fissionable fuel then it consumed. He was instrumental in the design and operation of LAMPRE, the Los Alamos Molten Plutonium Reactor Experiment. He also became involved with diagnostic experiments on nuclear explosions, both in Nevada on shot towers above ground, and in the Pacific from a flying laboratory built into a KC135 jet tanker. He is one of the diminishing number of people still alive who have witnessed atmospheric nuclear explosions. Dr. Kiehn has written patents that range from AC ionization chambers, plutonium breeder reactor power plants, to dual polarized ring lasers and down-hole oil exploration instruments. He is active, at present, in creating new devices and processes, from the nanometer world to the macroscopic world, which utilize the features of Non-Equilibrium Systems and Irreversible Processes, from the perspective of Continuous Topological Evolution. Dr. Kiehn left Los Alamos in 1963 to become a professor of physics at the University of Houston. He lived about 100 miles from Houston on his Pecan Orchard - Charolais Cattle ranch on the banks of the San Marcos river near San Antonio. As a pilot, he would commute to Houston, and his classroom responsibilities, in his Cessna 172 aircraft. He was known as the "flying professor". He is now retired, as an "emeritus" professor of physics, and lives in a small villa at the base of Mount Ventoux in the Provence region of southeastern France. He maintains an active scientific website at (http://www.cartan.pair.com).

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Other Volumes in the Series

Non-Equilibrium Systems and Irreversible Processes R. M. Kiehn Adventures in Applied Topology Series: Volume 1 Non-Equilibrium Thermodynamics Volume 2 Falaco Solitons, Cosmology, and the Arrow of Time Volume 3 Wakes, Coherent Structures and Turbulence Volume 4 Plasmas and non-equilibrium Electrodynamics Volume 5 Topological Torsion and Macroscopic Spinors Volume 6 Maple Magic and Differential Topology Volume 7 Selected Publications c Copyright CSDC INC. 2003-2006 °

Other Volumes in the Series

Future additions to this monograph will include Entropic constraints on the Action 1-form that lead to the Einstein equations Conformal transformations vs transitive Affine translations vs. intransitive rotations. Differential Forms and Cross Ratios

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