Cover -- Contents -- Preface -- Acknowledgments -- Introduction -- Chapter 1 Arbitrary Measures of the Physical World -- 1.1 Similarity -- 1.2 Dimensional Similarity -- 1.3 Physical Equations and the 'Pi' Theorem -- 1.4 Applications of the Pi Theorem -- 1.4.1 Plane Pendulum -- 1.4.2 Pipe Flow of a Fluid -- 1.4.3 Steady Motion of a Rigid Object in Viscous 'Fluid' -- 1.4.4 Diffusion and Self-Similarity -- 1.4.5 Ship Wave Drag -- 1.4.6 Adiabatic Gas Flow -- 1.4.7 Time-Dependent Adiabatic Flow -- 1.4.8 Point Explosion in a Gaseous Medium -- 1.4.9 Applications in Fundamental Physics -- 1.4.10 Drag on a Flexible Object in Steady Motion -- 1.4.11 Dimensional Analysis of Mammals -- 1.4.12 Trees -- References -- Chapter 2 Lie Groups and Scaling Symmetry -- 2.1 The Rescaling Group -- 2.1.1 Rescaling Physical Objects -- 2.1.2 Reconciliation with the Buckingham Pi Theorem -- 2.1.3 Rescaling and Self-Similarity as a Lie Algebra -- 2.1.4 Practical Lie Self-Similarity -- 2.2 Familiar Physical Examples -- 2.2.1 Line Vortex Diffusion: Reprise -- 2.2.2 Burgers' Equation -- 2.3 Less Familiar Examples -- 2.3.1 Self-Gravitating Collisionless Particles: The Boltzmann-Poisson Problem -- References -- Chapter 3 Poincaré Group Plus Rescaling Group -- 3.1 Galilean Space-Time -- 3.2 Minkowski Space-Time -- 3.2.1 Self-Similar Lorentz Boost -- 3.2.2 Self-Similar Boost/Rotation -- 3.3 Kinematic General Relativity -- References -- Chapter 4 Instructive Classic Problems -- 4.1 Introduction -- 4.2 Ideal Fluid Flow Past a Wedge: Self-Similarity of the 'Second Kind' -- 4.3 Boundary Layer on a Flat Plate: the Blasius Problem -- 4.4 Adiabatic Self-Similarity in the Diffusion Equation -- 4.5 Waves in a Uniformly Rotating Fluid -- References -- Chapter 5 Variations on Lie Self-Similarity.

5.1 Variations on the Boltzmann-Poisson System -- 5.1.1 Infinite Self-Gravitating Collisionless Spheres -- 5.1.2 Finite Self-Gravitating Collisionless Spheres -- 5.1.3 Other Approaches to Finite Spheres -- 5.2 Hydrodynamic Examples -- 5.2.1 General Navier-Stokes Theory -- 5.2.2 Modified Couette Flow -- 5.2.3 Flow at Large Scale inside a Laminar Wake -- 5.3 Axi-Symmetric Ideal Magnetohydrodynamics -- 5.3.1 Incomplete Self-Similarity as Separable Multi-variable Self-Similarity -- 5.3.2 Isothermal Collapse -- References -- Chapter 6 Explorations -- 6.1 Anisotropic Self-Similarity -- 6.1.1 Anisotropic Similarity -- 6.2 Mathematical Variations -- 6.3 Periodicity and Similarity -- 6.3.1 Log Periodicity and Self-Similarity: Diffusion Equation -- References -- Chapter 7 Renormalization Group and Noether Invariants -- 7.1 Hybrid Lie Self-Similarity/Renormalization Group -- 7.1.1 Renormalizing More Complicated Equations -- 7.1.2 Schrödinger: Adiabatic and Fractal -- 7.1.3 Noether Invariants and Self-Similarity -- References -- Chapter 8 Scaling in Hydrodynamical Turbulence -- 8.1 General Introduction -- 8.2 Homogeneous, Isotropic, Decaying Turbulence -- 8.2.1 Third-Order Correlation Negligible -- 8.2.2 Renormalization and Homogeneous, Isotropic, Turbulence -- 8.3 Dimensional Phenomenology of Stationary Turbulence -- 8.4 Structure in 2D Turbulence -- 8.4.1 Similarity of Time-Dependent 2D Vortical Fluid Flow -- 8.4.2 Similarity in Physically Steady, Inviscid Vortical Fluid Flow -- References -- Epilogue -- Appendix: Examples from the literature -- Index -- EULA.