Scale Invariance Self-Similarity of the Physical World

Acknowledgments XIII

Introduction XV

1 Arbitrary Measures of the PhysicalWorld 1

1.1 Similarity 1

1.2 Dimensional Similarity 3

1.3 Physical Equations and the ‘Pi’ Theorem 6

1.4 Applications of the PiTheorem 10

1.4.1 Plane Pendulum 11

1.4.2 Pipe Flow of a Fluid 16

1.4.3 Steady Motion of a Rigid Object in Viscous ‘Fluid’ 18

1.4.4 Diffusion and Self-Similarity 20

1.4.5 ShipWave Drag 26

1.4.6 Adiabatic Gas Flow 28

1.4.7 Time-Dependent Adiabatic Flow 30

1.4.8 Point Explosion in a Gaseous Medium 33

1.4.9 Applications in Fundamental Physics 35

1.4.10 Drag on a Flexible Object in Steady Motion 41

1.4.11 Dimensional Analysis of Mammals 42

1.4.12 Trees 47

References 51

2 Lie Groups and Scaling Symmetry 53

2.1 The Rescaling Group 53

2.1.1 Rescaling Physical Objects 55

2.1.2 Reconciliation with the Buckingham PiTheorem 59

2.1.3 Rescaling and Self-Similarity as a Lie Algebra 60

2.1.4 Practical Lie Self-Similarity 63

2.2 Familiar Physical Examples 68

2.2.1 Line Vortex Diffusion: Reprise 69

2.2.2 Burgers’ Equation 71

2.3 Less Familiar Examples 77

2.3.1 Self-Gravitating Collisionless Particles: The Boltzmann-Poisson Problem 77

References 84

3 Poincaré Group Plus Rescaling Group 87

3.1 Galilean Space-Time 87

3.2 Minkowski Space-Time 96

3.2.1 Self-Similar Lorentz Boost 96

3.2.2 Self-Similar Boost/Rotation 102

3.3 Kinematic General Relativity 108

References 119

4 Instructive Classic Problems 121

4.1 Introduction 121

4.2 Ideal Fluid Flow Past aWedge: Self-Similarity of the ‘Second Kind’ 121

4.3 Boundary Layer on a Flat Plate: the Blasius Problem 126

4.4 Adiabatic Self-Similarity in the Diffusion Equation 133

4.5 Waves in a Uniformly Rotating Fluid 140

References 146

5 Variations on Lie Self-Similarity 147

5.1 Variations on the Boltzmann–Poisson System 147

5.1.1 Infinite Self-Gravitating Collisionless Spheres 147

5.1.2 Finite Self-Gravitating Collisionless Spheres 155

5.1.3 Other Approaches to Finite Spheres 159

5.2 Hydrodynamic Examples 164

5.2.1 General Navier–StokesTheory 164

5.2.2 Modified Couette Flow 166

5.2.3 Flow at Large Scale inside a LaminarWake 170

5.3 Axi-Symmetric Ideal Magnetohydrodynamics 178

5.3.1 Incomplete Self-Similarity as Separable Multi-variable Self-Similarity 182

5.3.2 Isothermal Collapse 185

References 187

6 Explorations 189

6.1 Anisotropic Self-Similarity 189

6.1.1 Anisotropic Similarity 192

6.2 Mathematical Variations 193

6.3 Periodicity and Similarity 198

6.3.1 Log Periodicity and Self-Similarity: Diffusion Equation 203

References 207

7 Renormalization Group and Noether Invariants 209

7.1 Hybrid Lie Self-Similarity/Renormalization Group 209

7.1.1 Renormalizing More Complicated Equations 216

7.1.2 Schrödinger: Adiabatic and Fractal 219

7.1.3 Noether Invariants and Self-Similarity 223

References 229

8 Scaling in Hydrodynamical Turbulence 231

8.1 General Introduction 231

8.2 Homogeneous, Isotropic, Decaying Turbulence 232

8.2.1 Third-Order Correlation Negligible 236

8.2.2 Renormalization and Homogeneous, Isotropic, Turbulence 240

8.3 Dimensional Phenomenology of Stationary Turbulence 242

8.4 Structure in 2D Turbulence 246

8.4.1 Similarity of Time-Dependent 2D Vortical Fluid Flow 248

8.4.2 Similarity in Physically Steady, Inviscid Vortical Fluid Flow 258

References 264

Epilogue 267

Appendix: Examples from the literature 269

Index 273

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