The first eleven chapters in this volume, 18 through 28, contain material
that was developed in the third year of the three-year mathematical physics
sequence at the University of Colorado. The central concepts are groups,
manifolds, and differential geometry. I wish to thank Professors Wesley
Brittin and Russel Dubisch for extensive discussions of this material, and
I wish to thank Professor Wolf Beiglbock for advice and suggestions on the
overall plan and on the material on group representations.
The material in the last three chapters, related broadly to recent work in
differentiable dynamical systems, has been discussed in special courses on
hydrodynamic stability and seminars on mathematical physics. That
material is somewhat less well organized than the older subjects, but has
been included because it contains various concepts of great potential value
in physical science.
Author(s): Robert D. Richtmyer
Series: Texts and Monographs in Physics
Publisher: Springer-Verlag
Year: 1982
Language: English
Pages: 331
Tags: Математика;Математическая физика;
Preface to Volume II
Elementary Group Theory
18.l The group axioms; examples
18.2 Elementary consequences of the axioms; further definitions 3
18.3 Isomorphism 5
18.4 Permutation groups 6
18.5 Homomorphisms; normal subgroups 8
18.6 eosets 10
18.7 Factor groups 10
18.8 The Law of Homomorphism 1J
18.9 The structure of cyclic groups II
18.10 Translations, inner automorphisms 12
18.l1 The subgroups of /1'4 13
18.l2 Generators and relations; free groups IS
18.13 Multiply periodic functions and crystals 16
18.l4 The space and point groups 17
18.15 Direct and semidirect products of groups; symmorphic space
groups 20
Continuous Groups
19.1 Orthogonal and rotation groups 25
19.2 The rotation group SO(3); Euler's theorem 27
19.3 Unitary groups 28
19.4 The Lorentz groups 29
19.5 Group manifolds 34
19.6 Intrinsic coordinates in the manifold of the rotation group 35
19.7 The homomorphism of SU(2) onto SO(3) 37
19.8 The homomorphism of SL(2, q onto the proper Lorentz
group ~p 38
19.9 Simplicity of the rotation and Lorentz groups 38
20 Group Representations I: Rotations and Spherical Harmonics 40
20.1 Finite-dimensional representations of a group 41
20.2 Vector and tensor transformation laws 41
20.3 Other group representations in physics 44
20.4 Infinite-dimensional representations 45
20.5 A simple case: SO(2) 46
20.6 Representations of matrix groups on Xoo 47
20.7 Homogeneous spaces 48
20.8 Regular representations 49
20.9 Representations of the rotation group SO(3) 50
20.10 Tesseral harmonics; Legendre functions 53
20.11 Associated Legendre functions 55
20.12 Matrices of the irreducible representations of SO(3); the
Euler angles 57
20.13 The addition theorem for tesseral harmonics 59
20.14 Completeness of the tesseral harmonics 60
Group Representations II: General; Rigid Motions;
Bessel Functions
21.1 Equivalence; unitary representations 62
21.2 The reduction of representations 63
21.3 Schur's Lemma and its corollaries 65
21.4 Compact and noncompact groups 66
21.5 Invariant integration; Haar measure 67
21.6 Complete system ofrepresentations of a compact group 71
21.7 Homogeneous spaces as configuration spaces in physics 72
21.8 M 2 and related groups 73
21.9 Representations of M 2 73
21.10 Some irreducible representations 74
21.11 Bessel functions 75
21.12 Matrices of the representations 76
21.13 Characters 77
Group Representations and Quantum Mechanics
22.1 Representations in quantum mechanics 80
22.2 Rotations of the axes 81
22.3 Ray representations 82
22.4 A finite-dimensional case 83
22.5 Local representations 83
22.6 Origin of the two-valued representations 84
22.7 Representations of SU(2) and SL(2, IC) 85
22.8 Irreducible representations of SU(2) 87
22.9 The characters of SU(2) 89
22.10 Functions of z and z 89
22.11 The finite-dimensional representations of SL(2, IC) 90
22.12 The irreducible invariant subspaces of xro for SL(2, IC) 92
22.13 Spinors 93
Elementary Theory of Manifolds
23.1 Examples of manifolds; method of identification 96
23.2 Coordinate systems or charts; compatibility; smoothness 98
23.3 Induced topology 101
23.4 Definition of manifold; Hausdorff separation axiom 101
23.5 Curves and functions in a manifold 103
23.6 Connectedness; components of a manifold 104
23.7 Global topology; homotopic curves; fundamental group 105
23.8 Mechanical linkages: Cartesian products 111
Covering Manifolds
24.1 Definition and examples 114
24.2 Principles of lifting 117
24.3 Universal covering manifold 119
24.4 Comments on the construction of mathematical models 121
24.5 Construction of the universal covering 123
24.6 Manifolds covered by a given manifold 125
Lie Groups
25.1 Definitions and statement of objectives 130
25.2 Theexpansions ofm(" .) andI(" .) 132
25.3 The Lie algebra of a Lie group 133
25.4 Abstract Lie algebras 135
25.5 The Lie algebras of linear groups 135
25.6 The exponential mapping; logarithmic coordinates 136
96
114
129
25.7 An auxiliary lemma on inner automorphisms; the mappings Ad p 139
25.8 Auxiliary lemmas on formal derivatives 141
25.9 An auxiliary lemma on the differentiation of exponentials 143
25.10 The Campbell-Baker-Hausdorf (CBH) formula 144
25.11 Translation of charts; compatibility; G as an analytic manifold 146
25.12 Lie algebra homomorphisms 149
25.13 Lie group homomorphisms 151
25.14 Law of homomorphism for Lie groups 155
25.15 Direct and semidirect sums of Lie algebras 160
25.16 Classification of the simple complex Lie algebras 162
25.17 Models of the simple complex Lie algebras 167
25.18 Note on Lie groups and Lie algebras in physics 170
Appendix to Chapter 25-Two nonlinear Lie groups 171
Metric and Geodesics on a Manifold
26.1 Scalar and vector fields on a manifold 175
26.2 Tensor fields 180
26.3 Metric in Euclidean space 182
26.4 Riemannian and pseudo-Riemannian manifolds 183
26.5 Raising and lowering of indices 185
26.6 Geodesics in a Riemannian manifold 186
26.7 Geodesics in a pseudo-Riamannian manifold 9Ji 190
26.8 Geodesics; the initial-value problem; the Lipschitz condition 190
26.9 The integral equation; Picard iterations 192
26.10 Geodesics; the two-point problem 193
26.11 Continuation of geodesics 194
26.12 Affinely connected manifolds 195
26.13 Riemannian and pseudo-Riemannian covering manifolds 197
Riemannian, Pseudo-Riemannian, and Affinely
Connected Manifolds
27.1 Topology and metric 199
27.2 Geodesic or Riemannian coordinates 199
27.3 Normal coordinates in Riemannian and pseudo-Riemannian
manifolds 202
27.4 Geometric concepts; principle of equivalence 203
27.5 Covariant differentiation 206
27.6 Absolute differentiation along a curve 208
27.7 Parallel transport 209
27.8 Orientability 210
27.9 The Riemann tensor, general; Laplacian and d'Alembertian 211
27.10 The Riemann tensor in a Riemannian or pseudo-Riemannian
manifold 214
27.11 The Riemann tensor and the intrinsic curvature of a manifold 216
27.12 Flatness and the vanishing of the Riemann tensor 218
27.13 Eisenhart's analysis of the Stackel systems 221
The Extension of Einstein Manifolds
28.1 Special relativity 223
28.2 The Einstein gravitational field equations 224
28.3 The Schwarzschild charts 227
28.4 The Finkelstein extensions of the Schwarzschild charts 231
28.5 The Kruskal extension 233
28.6 Maximal extensions; geodesic completeness 235
28.7 Other extensions of the Schwarzschild manifolds 235
28.8 The Kerr manifolds 237
28.9 The Cauchy problem 240
28.10 Concluding remarks 243
Bifurcations in Hydrodynamic Stability Problems
29.1 The classical problems of hydrodynamic stability 244
29.2 Examples of bifurcations in hydrodynamics 245
29.3 The Navier-Stokes equations 247
29.4 Hilbert space formulation 248
29.5 The initial-value problem; the semiflow in,5 248
29.6 The normal modes 249
29.7 Reduction to a finite-dimensional dynamical system 250
29.8 Bifurcation to a new steady state 254
29.9 Bifurcation to a periodic orbit 255
29.10 Bifurcation from a periodic orbit to an invariant torus 257
29.11 Subharmonic bifurcation 261
Appendix to Chapter 29-Computational details for the invariant torus 261
Invariant Manifolds in the Taylor Problem
30.1 Survey of the Taylor problem to 1968 263
30.2 Calculation of invariant manifolds 265
30.3 Cylindrical coordinates 268
30.4 The Hilbert space 270
30.5 Separation of variables in cylindrical coordinates 27l
30.6 Results to date for the Taylor problem 272
Appendix to Chapter 30-The matrices in Eagles' formulation 274
263
31
The Early Onset ofTurbulence
276
31.1 The Landau~Hopfmodel 276
31.2 The Hopf example 278
31.3 The Ruelle~ Takens model 279
31.4 The w-limit set of a motion 280
31.5 Attractors 282
31.6 The power spectrum for motions in [Rn 283
31.7 Almost periodic and aperiodic motions 284
31.8 Lyapounov stability 285
31.9 The Lorenz system; the bifurcations 286
31.10 The Lorenz attractor; general description 288
31.11 The Lorenz attractor; aperiodic motions 290
31.12 Statistics of the mapping! and 9 293
31.13 The Lorenz attractor; detailed structure I 294
31.14 The symbols [i,j] of Williams 297
31.15 Prehistories 299
31.16 The Lorenz attractor; detailed structure II 300
31.17 Existence of I-cells in F 301
31.18 Bifurcation to a strange attractor 302
31.19 The Feigenbaum model 303
Appendix to Chapter 3I (Parts A~H)-Generic properties of systems: 304
31.A Spaces of systems 304
31.B Absence of Lebesgue measure in a Hilbert space 304
31.C Generic properties of systems 305
31.D Strongly generic; physical interpretation 305
31.E Peixoto's theorem 306
Other examples of generic and nongeneric properties 306
Lack of correspondence between genericity and Lebesgue measure 308
Probability and physics 308
References 313
Index 317
