MATHEMATICAL PHYSICS

Transcription

1 Advanced Methods of MATHEMATICAL PHYSICS R.S. Kaushal D. Parashar Alpha Science International Ltd.

2 Contents Preface Abbreviations, Notations and Symbols vii xi 1. General Introduction 1 2. Theory of Finite Groups A Brief Review of Set Theory 6 Cartesian products 6 Mappings 7 Binary compositions 7 Composition tables Abstract Groups 8 Preliminaries 8 Cyclic groups 10 Permutation group (S,,) 12 Groups of symmetry 14 Conjugate elements and classes 18 Subgroups 19 Center of a group 20 Cosets 20 Normal subgroups 22 Factor groups Homomorphisms 24 Automorphisms 26 Inner and outer automorphisms 27 Group of automorphisms 27 Direct product of groups 28 Semidirect product of groups Group Representations 30 Invariant subscpaces 32 Reducible representations 32 Irreducible representations 34 Unitary representations 36 Schur's lemmas 37 The orthogonality theorem 39 Characters of a representation Introduction to Continous Groups 42 Lie groups 43 Lie groups of transformations 44 Infinite continous groups 45 Generators of a Lie group 45

3 xiv Contents- 2.6 Applications to Physical Problems 48 Permutation group (S,,) 48 Unitary group (SU(»)) 5/ Symmetry group of a square Summary and Further Reading 59 Problems Rudiments of Topology and Differential Geometry Preliminaries 63 Denumerable and countable sets 64 Lower and upper bounds 66 Neighbourhoods, open and closed sets 66 Continuity 68 Limit points 69 Bolzano-Weierstrass theorem 70 Isolated, dense and perfect sets Metric Spaces 72 Euclidean space 73 Hilbert space 74 Distance between sets 74 Open and closed spheres 75 Equivalence of metric spaces Topological Spaces 76 Definition 76 Union and intersection of topologies 78 Limit points 79 Closure of a set 79 Interior, exterior and boundary of a set 80 Base for a topology 82 Housdorff spaces 82 Relative topologies Compactness 83 Some definitions 83 Heine-Borel theorem Connectedness 86 Separated sets 86 Connected sets 86 Connected spaces Homotopy 88 Homotopic paths 88 Simply connected spaces 90 The fundamental group Essentials of Differential Geometry 94 Some basic concepts and definitions 94 Differentiable manifolds 96 Diffeomorphism 98 Vector fields 98 Differential forms Summary and Further Reading 103 Problems 104

4 Contents xv 4. Integral Equations, Sturm-Liouville Theory and Green's Functions Terminology and Definitions 107 Fredholm integral equations 108 Volterra integral equations 109 Differentiation of a function under the integral sign 110 Relation between differential and integral equations Solution of Integral Equations 113 The Liouville-Neumann series method 113 The Fredholm method 124 The Hilbert-Schmidt theory 134 A3 Sturm-Liouville Theory 142 Adjoint differential sytstem 143 The Sturm-Liouville problems: Eigenvalues and eigenfunctions The Green's Functions 148 Determination of G(x, t) 149 Connection with inhomogeneous Sturm-Liouville equation Applictions to Physics Problems" 153 The influence function 154 The Abel's integral equation Summary and Further Reading 158 Problems Stochastic Processes and Stochastic Differential Equations Random Variable and Distribution Function of Random Variables 163 Some basic definitions and results 163 Multidimensional distribution functions 168 Functions of random variables 171 The Stieltjes integral Numerical Characterstics of Random Variables: Moments of the distribution function 176 Mathematical expectation 176 Variance 178 Covariance and covariance matrix 182 Characteristic function of random variables Stochastic Processes: Markov process 756 Classification of stochastic processes 187 Markov processes: Fokker-Planck equation 191 General theory of continuous (Markov) processes Stochastic differential equations: an introduction Applications to physical problems Summary and further reading 205 Problems Methods of Nonlinear Dynamics I: Phase Portraits A Brief Survey of Nonlinear Operators and Differential Equations 213

5 xvi Contents- 6.2 Solution of nonlinear Differential Equations: Existence and Uniqueness Theorems Critical Point Analysis of Differential Equations 223 Generalization to the case of n variables 223 Definitions: linear systems 224 Definitions (continued): A two dimensional linear system 225 Further remarks on linear systems Nonlinear Systems in the Plane 238 Linearization at a critical point 239 Volterra-Lotka system 241 General remarks 244 Problems Methods of Nonlinear Dynamics II: Stability and Bifurcation Stability of Critical Points and Liapunov Functions 249 Stability for non-autonomous systems 249 Stability for autonomous systems 253 Liapunov functions 256 Stability and linear aproximation in two dimensions Limit Cycle Index of a Critical Point and Bendixson Criterion for Periodic Solutions 265 Index of a critical point 268 Bendixson's criterion for periodic solutions Bifurcation and Structural Stability 273 Phenomenon of bifurcation 273 One-dimensional bifurcation 277 Hopf bifurcation 281 Structural stability 283 Chaos and stange attractor Applications to Physical Problems 257 Conservative systems 287 Hamiltonian systems Summary and Further Reading 297 Problems Some Nonlinear Differential Equations and their Solutions Van der Pol Equation 300 Lienard systems and Vab der pol equation 300 Dependence of the solution on the parameter e 303 Large parameter behaviour of the solution Solitary-Wave Solutions of Nonlinear Differential Equations 310 A brief introduction 310 Korteweg-de Vries (KdV) equation and its solutions 313 Some remarks about KdV and modified KdV equations 325

6 8.3 Solitary-Wave Solution of Nonlinear Schrodinger Equation Application to Physical Problems 329 Applications of Van der Pol equation 329 Applictions of KdV and NLS equations Summary and Further Reading 334 Problems Contents xvii 9. Some Nonlinear Integral Equations and their Solutions Inverse Scattering Transform Method 339 A brief introduction 339 Two typical examples of u(x) 342 Inverse scattering problem 346 Connection with the KdV equation Backlund Transformation and the Solution of KdV Equation The Lax Pair Method Hirota's Method of Bilinear Derivatives Painleve' Property and Painleve' Transcendents Kadomstev-Petviashvili (KP) Equation and its Solutions Solution of Some Nonlinear Integral Equations 368 Existence theorems for NL integral equations 368 Some representative nonlinear integral equations Summary and Further Reading 376 Problems Exact Solution of Some Nonlinear Differential Equations Riccati Equation 378 Riccati equation and the linear differential equation of second order 379 Solution of the original Riccati equation 381 Further remarks on Riccati equation Exact Solution of Some Other NLODEs Nonlinear Diffusion Equations 357 Case when F(Q = C 388 Case when F(Q = exp (Q Exact Solution of Some Other NLPDEs Applications to Physical Problems 396 Classical mechanics 396 Quantum mechanics 396 Astrophysics Concluding Discussion 399 Problems 400

7 xviii Contents- 11. Symmetries of Differential Equations Symmetry Groups of Differential Equations: An Introduction 402 Symmetries of algebraic equations 402 Groups and differential equations 404 Some basic results and definitions Extended Transformations or Prolongations Extended Infinitesimal Transformations 425 Case of one dependent and one independent variables 426 Case of one dependent and p independent variables 426" 11.4 Invariance of an Ordinary Differential Equation Invariance of a Partial Differential Equation Symmetry Groups and Conservation Laws Noether's Theorem and Lie-Backlund Symmetries Summary and Further Reading 440 Problems Normal Modes in Nonlinear Dynamical Systems Normal Modes of Linear Systems: A Brief Survey Normal Modes of Nonlinear Systems: A Simple Generalization Types of Mode Interactions: A Group Theoretic Approach 454 An overview 454 Various modes and their interactions Bushes of Modes for a Dynamical System 459 Interactions between modes of an N-particle nonlinear system 454 Bush of dynamical variables Dynamical Bush Equations: Normal Forms Theory 463 An introduction to normal form theory 463 Indicator of a resonance 465 Indicator of resonance for Hamiltonian systems Concluding Discussion 469 Appendix: Some Numerical Aspects of Nonlinear Dynamical Systems Vis-a-Vis Chaos 470 Al. Introduction 470 A2. Phase Flow and Maps 471 A3. Characterization of the Chaotic Motion 478 A4. Routes to Chaos 486 A5. Fractals and Hausdorff Dimension 493 A6. Application to Physical problems 495 A7. Summary and Further Reading 497 Bibliography 499 Index 503