Principles of Advanced Mathematical Physics : Volume II

R. D. Richtmyer

ISBN 10: 3642510787 / ISBN 13: 9783642510786

Editore: Springer Berlin Heidelberg, 2012

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Druck auf Anfrage Neuware - Printed after ordering - Inhaltsangabe18 Elementary Group Theory.- 18.1 The group axioms; examples.- 18.2 Elementary consequences of the axioms; further definitions.- 18.3 Isomorphism.- 18.4 Permutation groups.- 18.5 Homomorphisms; normal subgroups.- 18.6 Cosets.- 18.7 Factor groups.- 18.8 The Law of Homomorphism.- 18.9 The structure of cyclic groups.- 18.10 Translations, inner automorphisms.- 18.11 The subgroups of 4.- 18.12 Generators and relations; free groups.- 18.13 Multiply periodic functions and crystals.- 18.14 The space and point groups.- 18.15 Direct and semidirect products of groups; symmorphic space groups.- 19 Continuous Groups.- 19.1 Orthogonal and rotation groups.- 19.2 The rotation group SO(3); Euler's theorem.- 19.3 Unitary groups.- 19.4 The Lorentz groups.- 19.5 Group manifolds.- 19.6 Intrinsic coordinates in the manifold of the rotation group.- 19.7 The homomorphism of SU(2) onto SO(3).- 19.8 The homomorphism of SL(2, ) onto the proper Lorentz group p. 19.9 Simplicity of the rotation and Lorentz groups. 20 Group Representations I: Rotations and Spherical Harmonics. 20.1 Finitedimensional representations of a group. 20.2 Vector and tensor transformation laws. 20.3 Other group representations in physics. 20.4 Infinitedimensional representations. 20.5 A simple case: SO(2). 20.6 Representations of matrix groups on X . 20.7 Homogeneous spaces. 20.8 Regular representations. 20.9 Representations of the rotation group SO(3). 20.10 Tesseral harmonics; Legendre functions. 20.11 Associated Legendre functions. 20.12 Matrices of the irreducible representations of SO(3); the Euler angles. 20.13 The addition theorem for tesseral harmonics. 20.14 Completeness of the tesseral harmonics. 21 Group Representations II: General; Rigid Motions; Bessel Functions. 21.1 Equivalence; unitary representations. 21.2 The reduction of representations. 21.3 Schur's Lemma and its corollaries. 21.4 Compact and noncompact groups. 21.5 Invariant integration; Haar measure. 21.6 Complete system of representations of a compact group. 21.7 Homogeneous spaces as configuration spaces in physics. 21.8 M2 and related groups. 21.9 Representations of M2. 21.10 Some irreducible representations. 21.11 Bessel functions. 21.12 Matrices of the representations. 21.13 Characters. 22 Group Representations and Quantum Mechanics. 22.1 Representations in quantum mechanics. 22.2 Rotations of the axes. 22.3 Ray representations. 22.4 A finitedimensional case. 22.5 Local representations. 22.6 Origin of the twovalued representations. 22.7 Representations of SU(2) and SL(2, ). 22.8 Irreducible representations of SU(2). 22.9 The characters of SU(2). 22.10 Functions of z and z . 22.11 The finitedimensional representations of SL(2, ). 22.12 The irreducible invariant subspaces of X for SL(2, ). 22.13 Spinors. 23 Elementary Theory of Manifolds. 23.1 Examples of manifolds; method of identification. 23.2 Coordinate systems or charts; compatibility; smoothness. 23.3 Induced topology. 23.4 Definition of manifold; Hausdorff separation axiom. 23.5 Curves and functions in a manifold. 23.6 Connectedness; components of a manifold. 23.7 Global topology; homotopic curves; fundamental group. 23.8 Mechanical linkages: Cartesian products. 24 Covering Manifolds. 24.1 Definition and examples. 24.2 Principles of lifting. 24.3 Universal covering manifold. 24.4 Comments on the construction of mathematical models. 24.5 Construction of the universal covering. 24.6 Manifolds covered by a given manifold. 25 Lie Groups. 25.1 Definitions and statement of objectives. 25.2 The expansions of m( , ) and l( , ). 25.3 The Lie algebra of a Lie group. 25.4 Abstract Lie algebras. 25.5 The Lie algebras of linear groups. 25.6 The exponential mapping; logarithmic coordinates. 25.7 An auxiliary lemma on inner automorphisms; the mappings Ad . 25.8 Auxiliary lemmas on formal derivatives. 25.9 An auxiliary lemma on the differentiation of exponentials. 25.10 The Campbe.

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Dati bibliografici

• Titolo: Principles of Advanced Mathematical Physics : Volume II
• Editore: Springer Berlin Heidelberg
• Data di pubblicazione: 2012
• Lingua: Inglese
• ISBN 10: 3642510787
• ISBN 13: 9783642510786
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 235 L x 155 A x 19 P (millimetri) 235 millimetri di larghezza per 155 millimetri di altezza per 19 millimetri di profondità
• Peso: 511 grammi

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Contenuti

18 Elementary Group Theory.- 18.1 The group axioms; examples.- 18.2 Elementary consequences of the axioms; further definitions.- 18.3 Isomorphism.- 18.4 Permutation groups.- 18.5 Homomorphisms; normal subgroups.- 18.6 Cosets.- 18.7 Factor groups.- 18.8 The Law of Homomorphism.- 18.9 The structure of cyclic groups.- 18.10 Translations, inner automorphisms.- 18.11 The subgroups of ?4.- 18.12 Generators and relations; free groups.- 18.13 Multiply periodic functions and crystals.- 18.14 The space and point groups.- 18.15 Direct and semidirect products of groups; symmorphic space groups.- 19 Continuous Groups.- 19.1 Orthogonal and rotation groups.- 19.2 The rotation group SO(3); Euler’s theorem.- 19.3 Unitary groups.- 19.4 The Lorentz groups.- 19.5 Group manifolds.- 19.6 Intrinsic coordinates in the manifold of the rotation group.- 19.7 The homomorphism of SU(2) onto SO(3).- 19.8 The homomorphism of SL(2, ?) onto the proper Lorentz group ? p.- 19.9 Simplicity of the rotation and Lorentz groups.- 20 Group Representations I: Rotations and Spherical Harmonics.- 20.1 Finite-dimensional representations of a group.- 20.2 Vector and tensor transformation laws.- 20.3 Other group representations in physics.- 20.4 Infinite-dimensional representations.- 20.5 A simple case: SO(2).- 20.6 Representations of matrix groups on X?.- 20.7 Homogeneous spaces.- 20.8 Regular representations.- 20.9 Representations of the rotation group SO(3).- 20.10 Tesseral harmonics; Legendre functions.- 20.11 Associated Legendre functions.- 20.12 Matrices of the irreducible representations of SO(3); the Euler angles.- 20.13 The addition theorem for tesseral harmonics.- 20.14 Completeness of the tesseral harmonics.- 21 Group Representations II: General; Rigid Motions; Bessel Functions.- 21.1 Equivalence; unitary representations.- 21.2 The reduction of representations.- 21.3 Schur’s Lemma and its corollaries.- 21.4 Compact and noncompact groups.- 21.5 Invariant integration; Haar measure.- 21.6 Complete system of representations of a compact group.- 21.7 Homogeneous spaces as configuration spaces in physics.- 21.8 M2 and related groups.- 21.9 Representations of M2.- 21.10 Some irreducible representations.- 21.11 Bessel functions.- 21.12 Matrices of the representations.- 21.13 Characters.- 22 Group Representations and Quantum Mechanics.- 22.1 Representations in quantum mechanics.- 22.2 Rotations of the axes.- 22.3 Ray representations.- 22.4 A finite-dimensional case.- 22.5 Local representations.- 22.6 Origin of the two-valued representations.- 22.7 Representations of SU(2) and SL(2, ?).- 22.8 Irreducible representations of SU(2).- 22.9 The characters of SU(2).- 22.10 Functions of z and z?.- 22.11 The finite-dimensional representations of SL(2, ?).- 22.12 The irreducible invariant subspaces of X? for SL(2, ?).- 22.13 Spinors.- 23 Elementary Theory of Manifolds.- 23.1 Examples of manifolds; method of identification.- 23.2 Coordinate systems or charts; compatibility; smoothness.- 23.3 Induced topology.- 23.4 Definition of manifold; Hausdorff separation axiom.- 23.5 Curves and functions in a manifold.- 23.6 Connectedness; components of a manifold.- 23.7 Global topology; homotopic curves; fundamental group.- 23.8 Mechanical linkages: Cartesian products.- 24 Covering Manifolds.- 24.1 Definition and examples.- 24.2 Principles of lifting.- 24.3 Universal covering manifold.- 24.4 Comments on the construction of mathematical models.- 24.5 Construction of the universal covering.- 24.6 Manifolds covered by a given manifold.- 25 Lie Groups.- 25.1 Definitions and statement of objectives.- 25.2 The expansions of m(·, ·) and l(·, ·).- 25.3 The Lie algebra of a Lie group.- 25.4 Abstract Lie algebras.- 25.5 The Lie algebras of linear groups.- 25.6 The exponential mapping; logarithmic coordinates.- 25.7 An auxiliary lemma on inner automorphisms; the mappings Ad?.- 25.8 Auxiliary lemmas on formal derivatives.- 25.9 An auxiliary lemma on the differentiation of exponentials.- 25.10 The Campbell-Baker-Hausdorf (CBH) formula.- 25.11 Translation of charts; compatibility; G as an analytic manifold.- 25.12 Lie algebra homomorphisms.- 25.13 Lie group homomorphisms.- 25.14 Law of homomorphism for Lie groups.- 25.15 Direct and semidirect sums of Lie algebras.- 25.16 Classification of the simple complex Lie algebras.- 25.17 Models of the simple complex Lie algebras.- 25.18 Note on Lie groups and Lie algebras in physics.- Appendix to Chapter 25—Two nonlinear Lie groups.- 26 Metric and Geodesics on a Manifold.- 26.1 Scalar and vector fields on a manifold.- 26.2 Tensor fields.- 26.3 Metric in Euclidean space.- 26.4 Riemannian and pseudo-Riemannian manifolds.- 26.5 Raising and lowering of indices.- 26.6 Geodesies in a Riemannian manifold.- 26.7 Geodesies in a pseudo-Riamannian manifold M.- 26.8 Geodesies; the initial-value problem; the Lipschitz condition.- 26.9 The integral equation; Picard iterations.- 26.10 Geodesies; the two-point problem.- 26.11 Continuation of geodesies.- 26.12 Affmely connected manifolds.- 26.13 Riemannian and pseudo-Riemannian covering manifolds.- 27 Riemannian, Pseudo-Riemannian, and Affinely Connected Manifolds.- 27.1 Topology and metric.- 27.2 Geodesic or Riemannian coordinates.- 27.3 Normal coordinates in Riemannian and pseudo-Riemannian manifolds.- 27.4 Geometric concepts; principle of equivalence.- 27.5 Covariant differentiation.- 27.6 Absolute differentiation along a curve.- 27.7 Parallel transport.- 27.8 Orientability.- 27.9 The Riemann tensor, general; Laplacian and d’Alembertian.- 27.10 The Riemann tensor in a Riemannian or pseudo-Riemannian manifold.- 27.11 The Riemann tensor and the intrinsic curvature of a manifold.- 27.12 Flatness and the vanishing of the Riemann tensor.- 27.13 Eisenhart’s analysis of the Stäckel systems.- 28 The Extension of Einstein Manifolds.- 28.1 Special relativity.- 28.2 The Einstein gravitational field equations.- 28.3 The Schwarzschild charts.- 28.4 The Finkelstein extensions of the Schwarzschild charts.- 28.5 The Kruskal extension.- 28.6 Maximal extensions; geodesic completeness.- 28.7 Other extensions of the Schwarzschild manifolds.- 28.8 The Kerr manifolds.- 28.9 The Cauchy problem.- 28.10 Concluding remarks.- 29 Bifurcations in Hydrodynamic Stability Problems.- 29.1 The classical problems of hydrodynamic stability.- 29.2 Examples of bifurcations in hydrodynamics.- 29.3 The Navier-Stokes equations.- 29.4 Hilbert space formulation.- 29.5 The initial-value problem; the semiflow in ?.- 29.6 The normal modes.- 29.7 Reduction to a finite-dimensional dynamical system.- 29.8 Bifurcation to a new steady state.- 29.9 Bifurcation to a periodic orbit.- 29.10 Bifurcation from a periodic orbit to an invariant torus.- 29.11 Subharmonic bifurcation.- Appendix to Chapter 29—Computational details for the invariant torus.- 30 Invariant Manifolds in the Taylor Problem.- 30.1 Survey of the Taylor problem to 1968.- 30.2 Calculation of invariant manifolds.- 30.3 Cylindrical coordinates.- 30.4 The Hilbert space.- 30.5 Separation of variables in cylindrical coordinates.- 30.6 Results to date for the Taylor problem.- Appendix to Chapter 30—The matrices in Eagles’ formulation.- 31 The Early Onset of Turbulence.- 31.1 The Landau-Hopf model.- 31.2 The Hopf example.- 31.3 The Ruelle-Takens model.- 31.4 The co-limit set of a motion.- 31.5 Attractors.- 31.6 The power spectrum for motions in ?n.- 31.7 Almost periodic and aperiodic motions.- 31.8 Lyapounov stability.- 31.9 The Lorenz system; the bifurcations.- 31.10 The Lorenz attractor; general description.- 31.11 The Lorenz attractor; aperiodic motions.- 31.12 Statistics of the mapping f and g.- 31.13 The Lorenz attractor; detailed structure I.- 31.14 The symbols [i, j] of Williams.- 31.15 Prehistories.- 31.16 The Lorenz attractor; detailed structure II.- 31.17 Existence of 1-cells in F.- 31.18 Bifurcation to a strange attractor.- 31.19 The Feigenbaum model.- Appendix to Chapter 31 (Parts A-H)—Generic properties of systems:.- 31.A Spaces of systems.- 31.B Absence of Lebesgue measure in a Hilbert space.- 31.C Generic properties of systems.- 31.D Strongly generic; physical interpretation.- 31.E Peixoto’s theorem.- 31.F Other examples of generic and nongeneric properties.- 31.G Lack of correspondence between genericity and Lebesgue measure 308 31.H Probability and physics.- References.