Scale invariance

www.pdfgrip.com www.pdfgrip.com Richard N Henriksen Scale Invariance www.pdfgrip.com Related Titles Wolf, J.P Logan, J The Scaled Boundary Finite Element Method Applied Mathematics, Fourth Edition 2003 Print ISBN: 978-0-471-48682-4 Adobe PDF ISBN: 978-0-470-86148-6 Fourth Edition 2013 Print ISBN: 978-1-118-47580-5, also available in digital formats Krantz, W.B Scaling Analysis in Modeling Transport and Reaction Processes A Systematic Approach to Model Building and the Art of Approximation 2007 Print ISBN: 978-0-471-77261-3, also available in digital formats Abry, P., Goncalves, P., Vehel, J (eds.) Scaling, Fractals and Wavelets 2008 Print ISBN: 978-1-848-21072-1, also available in digital formats www.pdfgrip.com Richard N Henriksen Scale Invariance Self-Similarity of the Physical World www.pdfgrip.com Author Richard N Henriksen Queen’s University Department of Physics Engineering Physics and Astronomy Kingston, ON Canada All books published by Wiley-VCH are carefully produced Nevertheless, authors, editors, and publisher not warrant the information contained in these books, including this book, to be free of errors Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at © 2015 Wiley-VCH Verlag GmbH & Co KGaA, Boschstr 12, 69469 Weinheim, Germany All rights reserved (including those of translation into other languages) No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers Registered names, trademarks, etc used in this book, even when not specifically marked as such, are not to be considered unprotected by law Print ISBN: 978-3-527-41335-5 ePDF ISBN: 978-3-527-68733-6 ePub ISBN: 978-3-527-68735-0 Mobi ISBN: 978-3-527-68736-7 oBook ISBN: 978-3-527-68734-3 Typesetting Laserwords Private Limited, Chennai, India Printing and Binding Markono Print Media Pte Ltd., Singapore Printed on acid-free paper www.pdfgrip.com V To colleagues past, Judith, my parents and my children www.pdfgrip.com www.pdfgrip.com VII Contents Preface XI Acknowledgments XIII Introduction XV 1.1 1.2 1.3 1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.4.5 1.4.6 1.4.7 1.4.8 1.4.9 1.4.10 1.4.11 1.4.12 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.2 2.2.1 2.2.2 2.3 Arbitrary Measures of the Physical World Similarity Dimensional Similarity Physical Equations and the ‘Pi’ Theorem Applications of the Pi Theorem 10 Plane Pendulum 11 Pipe Flow of a Fluid 16 Steady Motion of a Rigid Object in Viscous ‘Fluid’ Diffusion and Self-Similarity 20 Ship Wave Drag 26 Adiabatic Gas Flow 28 Time-Dependent Adiabatic Flow 30 Point Explosion in a Gaseous Medium 33 Applications in Fundamental Physics 35 Drag on a Flexible Object in Steady Motion 41 Dimensional Analysis of Mammals 42 Trees 47 References 51 18 Lie Groups and Scaling Symmetry 53 The Rescaling Group 53 Rescaling Physical Objects 55 Reconciliation with the Buckingham Pi Theorem 59 Rescaling and Self-Similarity as a Lie Algebra 60 Practical Lie Self-Similarity 63 Familiar Physical Examples 68 Line Vortex Diffusion: Reprise 69 Burgers’ Equation 71 Less Familiar Examples 77 www.pdfgrip.com VIII Contents 2.3.1 Self-Gravitating Collisionless Particles: The Boltzmann-Poisson Problem 77 References 84 Poincaré Group Plus Rescaling Group 3.1 3.2 3.2.1 3.2.2 3.3 Galilean Space-Time 87 Minkowski Space-Time 96 Self-Similar Lorentz Boost 96 Self-Similar Boost/Rotation 102 Kinematic General Relativity 108 References 119 Instructive Classic Problems 121 4.1 4.2 Introduction 121 Ideal Fluid Flow Past a Wedge: Self-Similarity of the ‘Second Kind’ 121 Boundary Layer on a Flat Plate: the Blasius Problem 126 Adiabatic Self-Similarity in the Diffusion Equation 133 Waves in a Uniformly Rotating Fluid 140 References 146 4.3 4.4 4.5 5.1 5.1.1 5.1.2 5.1.3 5.2 5.2.1 5.2.2 5.2.3 5.3 5.3.1 5.3.2 6.1 6.1.1 6.2 6.3 6.3.1 87 Variations on Lie Self-Similarity 147 Variations on the Boltzmann–Poisson System 147 Infinite Self-Gravitating Collisionless Spheres 147 Finite Self-Gravitating Collisionless Spheres 155 Other Approaches to Finite Spheres 159 Hydrodynamic Examples 164 General Navier–Stokes Theory 164 Modified Couette Flow 166 Flow at Large Scale inside a Laminar Wake 170 Axi-Symmetric Ideal Magnetohydrodynamics 178 Incomplete Self-Similarity as Separable Multi-variable Self-Similarity 182 Isothermal Collapse 185 References 187 Explorations 189 Anisotropic Self-Similarity 189 Anisotropic Similarity 192 Mathematical Variations 193 Periodicity and Similarity 198 Log Periodicity and Self-Similarity: Diffusion Equation 203 References 207 www.pdfgrip.com References 20 Anselmet, F., Gagne, Y., Hopfinger, E.J and Antonia, R.A (1984) J Fluid Mech., 140, 63 21 Kraichnan, R.H and Montgomery, D (1980) Rep Prog Phys., 43, 547 22 Ditlevsen, P.D (2012) Phys Fluids, 24, 105109 23 Goldstein, H., Poole, C and Safko, J (2002) Classical Mechanics, 3rd edn, Addison Wesley (Pearson), San Fracncisco, CA 24 Henriksen, R.N and Turner, B.E (1984) Astrophys J., 287, 200 25 Henriksen, R.N (1986) Astrophys J., 310, 189 26 Cagliotti, E., Lions, P.L., Marchioro, C and Pulverenti, M (1992) Commun Math Phys., 143, 501 265 www.pdfgrip.com www.pdfgrip.com 267 Epilogue The chapter on turbulence concludes the discussion in this book We have been concerned throughout to assess the practical power of the notion of Scale Invariance This was done mostly by examining particular real problems where such ‘scale freedom’ promotes understanding The study of pure Scale invariance is efficiently conducted in the context of Lie group theory The formulation is uncomplicated and can replace many ‘ad hoc’ approaches found in the literature (see e.g Appendix) The formal treatment of the Lie theory following [1] led us to a more general symmetry group than that of pure rescaling This expanded group contained the Poincaré group, the Rescaling group and a ‘Boost’ or Galilean transformation This allowed us to go beyond the more typical applications of Scale Invariance Thus, for example, the study of the group invariants under a Boost led to the Lorentz transformations The application to space-time implied kinematic SelfSimilar symmetry and a solution for the ‘missing’ Self-Similar solutions in General Relativity In the course of our practical adventures, we introduced the notion of ‘asymmetric’ Self-Similarity and the use of ‘limiting scales’ The latter method implies that one could ‘coarse grain’ spatially and temporally by letting scales be arbitrarily large (while maintaining the Similarity class) or ‘fine grain’ by taking the opposite limit In the context of temporal scaling, these limits are known as the ‘slowtime’–‘fast-time’ limits This procedure led to novel approximations to physical scaling systems Our formal considerations caused us to reflect on how to find the constants associated with Self-Similarity as asymptotic behaviour We were able to make contact with the Renormalization group approach to Scale Invariance and to suggest some simplification due to our formalism We were also able to identify the presence of ‘discrete’ Scale Invariance in our formalism, which was related to imaginary or complex scaling A vitally important property of our approach to Similar and Self-Similar symmetries is that it can apply to arbitrary systems These may be physical, biological, chemical or even financial and perhaps political The first chapter of this book was dedicated to a survey of many of these potential applications It was done in Scale Invariance: Self-Similarity of the Physical World, First Edition Richard N Henriksen © 2015 Wiley-VCH Verlag GmbH & Co KGaA Published 2015 by Wiley-VCH Verlag GmbH & Co KGaA www.pdfgrip.com 268 Epilogue the familiar language of Buckingham theory, but subsequently this language was subsumed under the Lie formalism The details of these claims are to be found in the chapters of this book I believe that a creative appreciation of the methods used here may lead the reader to greater accomplishment in many subjects Reference Carter, B and Henriksen, R.N (1991) J Math Phys., 32 (10), 2580 www.pdfgrip.com 269 Appendix: Examples from the Literature After a great deal of hesitation, I have added this appendix It is meant to illustrate the widespread application of the symmetry discussed in this book, but it cannot be a complete summary of even the astrophysical literature I hope that analysis of the cited research will show how the results might have been achieved, or even extended, by the methods of this book If one types ‘Self Similar Solutions’ as an object in the SAO/NASA/ADS search form under astronomy, one finds 579 abstracts (including arXived abstracts) between 01/1960 and 12/2005 The latter is my adopted cut-off date and I take 1960 as a reasonable date for the beginning of the modern era in astrophysics If one adds physics to the search criteria, the number returned is 1115 Moreover, the count is sensitive to the description employed Under ‘Scale Invariance’, the number jumps to 1604 If one only wants to track any reference to ‘Self Similar Solutions’, Google announces 12,200,000 hits! The question that presents itself for the appendix is simply what to include? This is the reason for my long hesitation Any selection will inevitably be somewhat arbitrary, since I am not attempting any sort of review Ultimately, I have chosen those research works that are either known to me directly or which illustrate an unusual use of Self-Similarity The selection is thus influenced both by my ignorance and by my judgement The latter is necessarily based on a rather superficial reading, given the magnitude of the task Prolific authors are represented only by a sampling of their work I have chosen the cut-off date so as to assure some objectivity, granted by the passage of time A reader who seeks a more comprehensive exposure to the wealth of applications and the overwhelming ingenuity employed by authors need only conduct his or her own search of the literature 01/1960 to 12/1980 1) Zel’dovich, Ya.B and Raizer, Yu.P (1966) Physics of Shock Waves and High Temperature Hydrodynamic Phenomena, vol I, Academic Press, New York 2) Penston, M.V (1969) Dynamics of self-gravitating gaseous spheres – III Analytical results in the free-fall of isothermal cases Mon Not R Astron Soc., 144 (4), 425 Scale Invariance: Self-Similarity of the Physical World, First Edition Richard N Henriksen © 2015 Wiley-VCH Verlag GmbH & Co KGaA Published 2015 by Wiley-VCH Verlag GmbH & Co KGaA www.pdfgrip.com 270 Appendix: Examples from the Literature 3) Barenblatt, G.I and Zel’dovich, Ya.B (1972) Self-similar solutions as intermediate asymptotics Annu Rev Fluid Mech., 4, 285 4) Shu, F.H (1977) Self-similar collapse of isothermal spheres and star formation Astrophys J., 214, 488 5) Bicknell, G.V and Henriksen, R.N (1978) Self-similar growth of primordial black holes – II, general sound speed Astrophys J., 225, 237 6) Lacombe, C (1979) Self-similar solutions for the distribution function of particles accelerated by Alfvén waves Astron Astrophys., 71, 169 7) Kirkland, K.B and Sonnerup, B.U.Ö (1979) Self-similar resistive decay of a current sheet in a compressible plasma J Plasma Phys., 22, 289 8) Lonngren, K.E and Axford, R.A (1980) On the self-similar solution for the distribution function of particles accelerated by Alfvén waves Astron Astrophys., 81, 363 9) Lynden-Bell, D and Eggleton, P.P (1980) On the consequences of the gravothermal catastrophe Mon Not R Astron Soc., 191, 483 01/1981 to 12/1990 1) Chevalier, R (1982) Self-similar solutions for the interaction of stellar ejecta with an external medium Astrophys J., 258, 790 2) Sedov, L.I (1982) Similarity and Dimensional Methods in Mechanics MIR Publishers, English Translation, Moscow 3) Inagaki, S and Lynden-Bell, D (1983) Self-similar solutions for post-collapse evolution of globular clusters Mon Not R Astron Soc., 205, 91 4) Hamilton, A.J.S and Sarazin, C.L (1984) A new similarity solution for reverse shocks in supernovae remnants Astrophys J., 281, 682 5) Whitworth, A and Summers, D (1985) Self-similar condensations of spherically symmetric, self-gravitating, isothermal gas clouds Mon Not R Astron Soc., 214, 6) Fillmore, J.A and Goldreich, P (1984) Self-similar collapse in an expanding universe Astrophys J., 281, 7) Bertschinger, E (1985) Self-similar secondary infall and accretion in an Einstein-deSitter universe Astrophys J Suppl Ser., 58, 39 8) Lynden-Bell, D and Lemos, J.P.S (1988) On Penston’s self-similar solution for cold collapse Mon Not R Astron Soc., 233, 197 9) Blottiau, P., Chieze, J.P and Bouquet, S (1988) An asymptotic self-similar solution for gravitational collapse Astron Astrophys., 207, 24 10) Suto, Y and Silk, J (1988) Self-similar dynamics of polytropic gaseous spheres Astrophys J., 326, 527 11) Ostriker, J.P and McKee, C.F (1988) Astrophysical blast waves Rev Mod Phys., 60, 12) Lemos, J.P.S and Lynden-Bell, D (1989) A general class of self-similar solutions for a cold fluid – negligible gravity Mon Not R Astron Soc., 240, 303 13) Lemos, J.P.S and Lynden-Bell, D (1989) A general class of self-similar solutions for a cold fluid – with gravity Mon Not R Astron Soc., 240, 317 www.pdfgrip.com Appendix: Examples from the Literature 01/1991 to 12/2000 1) Ryden, B.S (1991) Compression of dark halos by baryon infall – self-similar solutions Astrophys J., 370, 15 2) Boily, C and Lynden-Bell, D (1993) Self-similar collapse of flat systems Mon Not R Astron Soc., 264, 1003 3) Foglizzo, T and Henriksen, R.N (1993) General relativistic collapse of homothetic ideal gas spheres and planes Phys Rev D, 48, 4645 4) Lynden-Bell, D and Boily, C (1994) Self-similar solutions up to flashpoint in highly wound magnetostatics Mon Not R Astron Soc., 267, 146 5) Pen, U.-L (1994) A general class of self-similar self-gravitating fluids Astrophys J., 429, 759 6) Evans, C.R and Coleman, J.S (1994) Critical Phenomena and self-similarity in the gravitational collapse of radiation fluid Phys Rev Lett., 72, 1782 7) Henriksen, R.N and Valls-Gabaud, D (1994) ‘Cored apple’ bipolarity – a global instability to convection in radial accretion Mon Not R Astron Soc., 266, 681 8) Traschen, J (1994) Discrete self-similarity and critical point behaviour in fluctuations about extremal black holes Phys Rev D, 50, 7144 9) Narayan, R and Yi, I (1995) Advection dominated accretion: self-similarity and bipolar outflows Astrophys J., 444, 231 10) Henriksen, R.N and Widrow, L.M (1997) Self-similar relaxation of selfgravitating collisionless particles Phys Rev Lett., 78, 3426 11) Carr, B.J and Coley, A.A (1999) Self-similarity in general relativity Classical Quantum Gravity, 16, 31 12) Henriksen, R.N and Widrow, L.M (1999) Relaxing and virializing a dark matter halo Mon Not R Astron Soc., 302, 321 13) Frolov, A.V (2000) Continuous self-similarity breaking in critical collapse Phys Rev D, 61, 4006 14) Lai, D (2000) Global nonradial instabilities of dynamically collapsing gas spheres Astrophys J., 540, 94 15) Carr, B.J and Coley, A.A (2000) Complete classification of sphericallysymmetric self-similar perfect fluid solutions Phys Rev D, 62, 4023 01/2001 to 12/2005 1) Pittard, J.M., Dyson, J.E and Hartquist, T.W (2001) Self-similar evolution of wind-blown bubbles with mass loading by conductive evaporation Astron Astrophys., 367, 1000 2) Martin-Garcia, J.M and Gundlach, C (2002) Self-similar spherically symmetric solutions of the massless Einstein-Vlasov system Phys Rev D, 65, 4026 3) Chavanis, P.-H., Rosier, C and Sire, C (2002) Thermodynamics of selfgraviatating systems Phys Rev E, 66, 6105 4) Harada, T and Maeda, H (2004) Stability criterion for self-similar solutions with a scalar field amd with a stiff fluid in general relativity Classical Quantum Gravity, 21, 371 271 www.pdfgrip.com 272 Appendix: Examples from the Literature 5) Szell, A., Merritt, D and Kevrekidis, I.G (2005) Core collapse via coarse dynamic renormalization Phys Rev Lett., 95, 1902 6) Maeda, H and Harada, T (2005) Kinematic self-similar solutions in general relativity, Horizons in World Physics, vol 249, Nova Science, New York, p 123 7) Gallay, T and Wayne, C.E (2005) Global stability of vortex solutions of the two-dimensional Navier-Stokes equation Commun Math Phys., 255, 97 8) Hamilton, A.J.S and Pollack, S.E (2005) Inside charged black holes I – Baryons Phys Rev D, 71, 4031 www.pdfgrip.com 273 Index a A-SMHD, see axi-symmetric ideal magnetohydrodynamics (A-SMHD) – Buckingham theory 221 adiabatic and fractal variation – free particle self-similar solution 222 – harmonic oscillator 221 – Lie transformed equation 220 – oscillating function phase 222 – physical significance 219 – Planck’s constant 220 – Schrödinger equation 221 anisotropic self-similarity – Blasius boundary layer 190, 192 – Cartesian invariant coordinates 190 – diagonal matrices 191, 192 – Lie motion 192 – Lie parameter condition 190 – matrix transformation 191 – physical continuum 190 – physical stress 192 – rotational symmetry 191 – strain tensor 189 – Taylor series 191 – temporal scaling and spatial scaling 189 – turbulence 193 axi-symmetric ideal magnetohydrodynamics (A-SMHD) – boundary conditions 181 – description 178 – dynamical pressure 179 – electric field quantities 180 – gravitational potential 180 – induction and mass conservation equations 183 – integral constant 184 – isothermal collapse 185, 186 – Newton’s constant constrains 178 – – – – – – – – – – – – physical invariants 179 physical quantities 185 Pi quantities 182 Poisson equation 184 self-consistency 184 self-gravity 182, 183 similarity class 185 spatial incompressibility 182 spherical polar coordinates 181 steady state condition 181 temperature 179 velocity and magnetic field 182 b biological trees – branching generation 50 – fractal geometry 48 – generation rule 47 – geometric similarity 47 – Leonardo’s observation of structure 48 Blasius problem – analytic and numeric solutions 132 – incident velocity 127 – kinematic viscosity 127 – numerical solution 130 – parameter 127, 128 – physical units 131 – plate surface 128 – plate thickness, width and length 126 – room temperature 131 – spatial anisotropy 126 – transverse velocity 130 – velocity measurement 128, 129 Boltzmann-Poisson system – adiabatic variation 159 – arbitrary constant 149 – boundary conditions 153, 154 – density integral 153 Scale Invariance: Self-Similarity of the Physical World, First Edition Richard N Henriksen © 2015 Wiley-VCH Verlag GmbH & Co KGaA Published 2015 by Wiley-VCH Verlag GmbH & Co KGaA www.pdfgrip.com 274 Index Boltzmann-Poisson system (contd.) – dimensional covectors 148 – finite polytropes 150 – global constraints 149 – homothetic class 149 – Lie parameter 156 – N-body systems 156 – non-singular isothermal sphere 152 – phase-space mass density 147 – physical density 154 – polytropic index 150 – renormalization 154 – Schuster polytrope 153, 154 – self-similar density profile 150 – singular isothermal sphere 151 – small scale behaviour 158 – steady state 157, 158 – transition shell potential 158 – velocity space, isotropy 147 – zeroth order 155 – Buckingham theory Burgers equation – arbitrary constant 76 – asymptotic structure 76 – Hopf–Cole transformation 72 – Lambert function 74 – multi-variable 76 – Navier–Stokes equation 71 – self-similar motion 72 – velocity function symmetry 73 c collisionless Boltzmann–Poisson N body systems 146 Couette flow – angular momentum 167 – angular velocity 167 – anisotropic self-similarity 169 – azimuthal equation 169 – Bessel functions 168 – circular velocity vanishes 168 – cylindrical coordinates 166 – diffusing line vortex 168 – pipe flow 169 – power law viscosity 169 – simple waves 170 – spatial and temporal scaling 167 – transformation, Self-Similar variables 166 – variable viscosity 170 – zero radial velocity 166 d decaying turbulence, see homogeneous and isotropic turbulence diffusion and self-similarity – diffusion equation 22, 26 – dimensional analysis 25 – intermediate asymptotic 23 – invariant profile 22 – kinematic viscosity 21 – multivariable self-similar symmetry 24 – Oseen vortex 25 – rescaled systems 26 – time dependent systems 20 diffusion equation adiabatic self-similarity – arbitrary components 135 – boundary conditions 133, 135 – Einstein tensor 134 – heat-conducting bar 133 – invariant length 134 – invariants and Lie path parameter 134 – physical units 136 – similarity subclasses 136, 137 – symmetric envelope curves 139 – temperature rescaling 134 – time and space 134 – Whittaker functions 137, 138 dimensional similarity – economic theory – solar masses – space measure f fundamental physics – Buckingham theorem 35 – electrostatic force 36 – free particle wave function 39 – gravitational bending, light rays 39 – linear partial differential equation 39 – logarithmic variables 37 – Newtonian particle, gravity 40 – scaled wave function 37 – spherical mass distribution 39 – wave vectors/momenta 37 g Galilean/Newtonian classical continuum system 63, 64 Galilean space-time – arbitrary coordinates 87 – Buckingham theory 90 – characteristic (Lie path) equations 90 – matrix/vector notation 91 – maximum Lie group symmetry 87 – Minkowski space 92 – planar spiral arms 96 – polar coordinates 96 – rescaling group invariants 94 www.pdfgrip.com Index – rotation and translation/boost subgroups 92 – rotation effects 89 – self-similar space-times 87 – space-time coordinates 89 – spiral structure 95 Galilean velocity transformations 87 Galileo’s law of falling bodies 14 h Hamilton-Jacobi method 227 homogeneous and isotropic turbulence – ad hoc approach 235 – description 232 – dissipation rate decays 235 – fluctuation velocities 232 – Kolmogorov four-fifths law and Loitsiansky integral 233 – Lie transformed equation 234 – mean flow velocity 233 – renormalization 240–242 – self preservation hypothesis 234 – third order correlation, see third order correlation – time dependence 236 – velocity correlation tensors 232 – viscosity, renormalization 235 – von Karman-Howarth equation 233, 234 Hopf–Cole transformation 217 hybrid Lie self-similarity – ‘anti-chaotic’ behaviour 209 – application 213 – asymptotic power law 209 – asymptotic self-similar class 219 – diffusion equation 210 – dimensional coherence, equation 210 – Dirac delta functions 211 – direct numerical integration of equation 212 – Fourier transform 212 – Kummer function 213 – linear initial value problem (IVP) 210 – numerical iteration 215 – physical dependence 213 – renormalization group methods 211 – symmetric solution 213 – Whittaker functions 217 hydrodynamical turbulence 229 k kinematic general relativity (GR) – Bianchi equation 117 – conservation laws 109 – covariant metric tensor 108 – dimensional coherence and rule 112 – Einstein field equations 109, 111, 114, 116 – Galilean space-time 110 – invariant ‘shape’ 113 – Lie derivatives 112 – Lie group parameter 112 – physical metrics 114 – projection tensor 110 – relativistic demonstration 116 – rescaling algebra components 111 – Riemannian space-time 110 – Schwarzschild metric 118 – self-similar motion 113 – spherical symmetry 111 – steady space-time 114 – zero pressure particles 115 l Lambert function 74 Laminar wake – arbitrary function 174, 176 – decomposition 172 – description 171 – envelope approximation, first order 177 – first order function 176 – fundamental equations 172 – global approximation 177 – irrotational components 174 – Laplace equation 174 – Lie parameter and transformations 175 – net velocity 171 – Oseen equation 172 – perturbation 175 – renormalization 176 – rescaling algebra vector 171 – rotational and irrotational velocities 173 – stream lines 175 – three dimensional flow 170 – transverse scale 171 – velocity potential 174 – velocity potential and pressure 172 – zero divergence condition 173 Lie algebra – Buckingham theory 65–67 – classical dimensional analysis 67 – Galilean/Newtonian physics 63, 64 – Galilean space-time coordinates 68 – multi-variable self-similar symmetry 65 – rescaling and self-similarity 60–62 – self-similarity generator 63 – space-time coordinates 68 275 www.pdfgrip.com 276 Index Lie self-similarity – Boltzmann–Poisson system, see Boltzmann-Poisson system – Couette flow, see Couette flow – Navier-Stokes equations 164, 165 line vortex diffusion 69–71 log periodicity – boundary conditions 204 – electric and magnetic fields 207 – Fourier cosine transform 206 – invariant constant 204 – Kummer function 205 – Stokes 203 – wave equation 205 – Whittaker functions 204 m mammals, dimensional analysis – free energy 43 – human resting metabolic rate 44 – internal/microscopic timescale 43 – linear behaviour 46 – living structure 47 – lungs and blood distribution 45 – MacLaurin series 44 – mechanical explanation, metabolism 43 – motor nervous system 46 – Young’s modulus 50 mathematical variations – arbitrary function 197 – calibration 195 – canonical coordinates 193 – dimensional covector component 194 – exponential factors 196 – general relativity 194 – harmonic function 195 – Lie group variables 196 – Liouville equation 193, 194, 197, 198 – space-time 195 – transformations 194 Minkowski space-time – boost/rotation, see self-similar boost/rotation – Lorentz boost, see self-similar Lorentz boost Multiplicative procedure, Buckingham Theorem n Navier–Stokes equations – boundary conditions 165 – Cartesian coordinates 164 – creeping flow 165 – kinematic viscosity 164 – Lie parameter 164 – velocity gradient tensor 165 Newton’s second law Noether invariants and self-similarity – conservative Lagrangian system 227 – dimensional constraint 226 – discrete symmetry 228 – Lagrangian mechanics 223 – Lie self-similar technique 228 – momentum and energy conservation 223 – planar Kepler problem 228 – point space-time coordinate transformations 223, 224 – radial equation, motion 229 – self-similar rescaling theorem 225 – spatial rescaling component 225 Noether/renormalization group theory 119 p periodicity – amplitude 201 – complex scaling 202, 203 – diffusion equation, log periodicity, see log periodicity – Floquet theorem 201, 202 – linear harmonic oscillator 199, 200 – Mathieu equation 201, 202 – oscillatory behaviour 198 – phase mixing 201 – power law solutions 200 – rescaling symmetry 199 – similarity class 200 – Whittaker functions 203 Photon 40 physical equations and ‘Pi’ theorem – application – description – hydrodynamics – informal dimensional analysis – numerical quantities – thermodynamic contexts physical objects rescaling – definition 55 – dimensional covectors 58 – homothetic’ system 58 – mechanical/thermodynamic systems 55 – numerical components 55 – orbits, solar system 58 – relativistic quantum mechanics 59 – rescaling group action 56 – space and time 57 – uncertainty principle 59 physical systems and structures – scaling, displacement and rotation www.pdfgrip.com Index – shape invariance Pi theorem – adiabatic gas flow 28–30 – diffusion and self-similarity see diffusion and self-similarity – flexible object, steady motion 41, 42 – MacLaurin series 10 – pipe flow, fluid 16–18 – plane pendulum see plane pendulum – Rescaling group invariants 10 – ship wave drag 26, 27 – steady motion, rigid object in viscous fluid 18–20 – time dependent, adiabatic flow of a gas 30–32 Pi theorem approach – arbitrary function 162 – finite, collisionless and spheres 160 – Lie formalism 159 – mean field and density 161 – power law behaviour 160 – second order expansion 163 – velocity distribution 163 – velocity renormalization 161 – zeroth order density 162 plane pendulum – continuous and invertible functions 13 – Coriolis force 11 – definition 11 – Foucault precession 14 – MacLaurin expansion 16 – quantity, dimensional analysis 13 – rescaling invariance 14 point explosion, gaseous medium 33–35 Pythagorean theorem – invariant distribution function (DF) 79 – Lie parameter 78 – non-linear oscillation 81 – phase-mixing’ relaxation 80, 81 – physical units 81 – Poisson equation in invariants 82 – radial orbit instability 84 – self-similar motion 77 self-similar boost/rotation – coordinate transformations 102, 105–107 – electromagnetic wave equation 103, 108 – hyperbolic functions 105 – kinematical interaction 103 – Klein-Gordon equation 102 – Minkowski ‘metric’ invariant 102 – orbital angular momentum 108 – time dilation formula 104, 105, 107 self-similarity – algorithmic manner 141 – angular velocity, fluid 141 – Bessel function 144 – boundary conditions 123, 124 – cylindrical polar coordinates 121 – description 121 – flow velocity 122 – fluid, constant density 140 – integrated equation 142, 143 – invariant variables 142 – Laplace equation 125, 126 – Lie motion 142, 144 – perturbed velocity 140, 141 – pressure perturbation 144, 145 – renormalization 126 – rescalings, time and space 122 – separability 126 – Sobolev problem 144 – velocity potential 123–125 r reconciliation, Buckingham Pi theorem – wedge length 121 – automorphisms 59 self-similar Lorentz boost – Dirac equation 59 – Cartesian coordinates 97 – elementary plane pendulum 59 – Euclidian geometry 101 relativistic self-similarity 56 – Lie motion 97 rescaling symmetry – Minkowski metrics 97 – physical invariance 53 – planar complexification 100 – rescaling parameter 97 – self-similarity 54 – spherical trigonometry, Riemann sphere s 99 Scrödinger equation 219, 36 – transformed phasor 99 self-gravitating collisionless particles stationary turbulence – collisionless Boltzmann equation (CBE) 77 – angle dependence 243 – description 77 – anomalous scaling 243 – differential equations 80 – beta model 244 – Dirac delta function 78 – eddy velocity 244, 245 – distribution function (DF), particles 77 – energy dissipation and viscosity 245 277 www.pdfgrip.com 278 Index – exponential factor 245 – kinetic dissipation 246 – Kolmogorov prediction 243 – Lie group variables 245 – Lie path variable 242 – scale invariance 243 – third order correlations 246 – viscosity renormalization 244 strain tensor 189, 209 t third order correlation – magnitude estimation 236 – similarity class 238, 239 – time dependence 238 – turbulent dissipation rate 238 – viscous dissipation 238 – Whittaker function 237 turbulence – homogeneous and isotropic, see homogeneous and isotropic turbulence – Navier–Stokes equation 231 – rescaling symmetry 231 – structure functions 231 – two-dimension, see two-dimensional turbulence two-dimensional turbulence – axisymmetric solution 250 – azimuthal velocity 252 – Bessel function 255, 256 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – biharmonic condition 257 Cartesian coordinates 247, 252 coherent vortices 258 delta and epsilon matrices 248, 253, 255 Eulerian velocity 259, 260 fiducial units 252 fine-grained approximation 256 Galilean frame 254 invariant coordinates 247, 253 inverse transformation 249 Jacobian 253, 254, 258 Kelvin–Helmholtz instability 260 Lie group motion 258, 259 Lie path parameter 249 molecular clouds 255 Navier–Stokes equation 247, 256 passive coordinate transformations 248 Pauli matrix 253 periodic log solution 252 phi, angular dependence 263 physical quantities 251 polar invariants 249 rescaling, rotation and translation 255, 256 scaled vorticity 249–251 self-similar inviscid steady flow 259, 260 spatial displacement 258 spectral density 257 stream function 249 stream lines 260–262 transposed rotation matrix 248 www.pdfgrip.com WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA ... invariants are used as the description of the set of physical objects The set of invariant functions and the relation between them is the generalization of the typical Universal function of a Self-Similar... Measures of the Physical World Similarity Dimensional Similarity Physical Equations and the ‘Pi’ Theorem Applications of the Pi Theorem 10 Plane Pendulum 11 Pipe Flow of a Fluid 16 Steady Motion of. .. Measures of the Physical World each of the m quantities of independent Dimensions, in order to write the arguments of the function in (1.14) as we have done Equation (1.14) is the Buckingham Pi theorem