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• Contents
  Volume I
  Preface vii
  Introduction 1
  1.1 Basics 1
  1.1.1 Structure of Mathematica 2
  1.1.2 Interactive Use of Mathematica 4
  1.1.3 Symbolic Calculations 6
  1.1.4 Numerical Calculations 11
  1.1.5 Graphics 13
  1.1.6 Programming 23
  2 Classical Mechanics 31
  2.1 Introduction 31
  2.2 Mathematical Tools 35
  2.2.1 Introduction 35
  2.2.2 Coordinates 36
  2.2.3 Coordinate Transformations and Matrices 38
  2.2.4 Scalars 54
  2.2.5 Vectors 57
  2.2.6 Tensors 59
  2.2.7 Vector Products 64
  2.2.8 Derivatives 69
  2.2.9 Integrals 73
  2.2.10 Exercises 74
  2.3 Kinematics 76
  2.3.1 Introduction 76
  2.3.2 Velocity 77
  2.3.3 Acceleration 81
  2.3.4 Kinematic Examples 82
  2.3.5 Exercises 94
  2.4 Newtonian Mechanics 96
  2.4.1 Introduction 96
  2.4.2 Frame of Reference 98
  2.4.3 Time 100
  2.4.4 Mass 101
  2.4.5 Newton's Laws 103
  2.4.6 Forces in Nature 106
  2.4.7 Conservation Laws 111
  2.4.8 Application of Newton's Second Law 118
  2.4.9 Exercises 188
  2.4.10 Packages and Programs 188
  2.5 Central Forces 201
  2.5.1 Introduction 201
  2.5.2 Kepler's Laws 202
  2.5.3 Central Field Motion 208
  2.5.4 Two-Particle Collisons and Scattering 240
  2.5.5 Exercises 272
  2.5.6 Packages and Programs 273
  2.6 Calculus of Variations 274
  2.6.1 Introduction 274
  2.6.2 The Problem of Variations 276
  2.6.3 Euler's Equation 281
  2.6.4 Euler Operator 283
  2.6.5 Algorithm Used in the Calculus of Variations 284
  2.6.6 Euler Operator for q Dependent Variables 293
  2.6.7 Euler Operator for q + p Dimensions 296
  2.6.8 Variations with Constraints 300
  2.6.9 Exercises 303
  2.6.10 Packages and Programs 303
  2.7 Lagrange Dynamics 305
  2.7.1 Introduction 305
  2.7.2 Hamilton's Principle Hisorical Remarks 306
  2.7.3 Hamilton's Principle 313
  2.7.4 Symmetries and Conservation Laws 341
  2.7.5 Exercises 351
  2.7.6 Packages and Programs 351
  2.8 Hamiltonian Dynamics 354
  2.8.1 Introduction 354
  2.8.2 Legendre Transform 355
  2.8.3 Hamilton's Equation of Motion 362
  2.8.4 Hamilton's Equations and the Calculus of Variation 366
  2.8.5 Liouville's Theorem 373
  2.8.6 Poisson Brackets 377
  2.8.7 Manifolds and Classes 384
  2.8.8 Canonical Transformations 396
  2.8.9 Generating Functions 398
  2.8.10 Action Variables 403
  2.8.11 Exercises 419
  2.8.12 Packages and Programs 419
  2.9 Chaotic Systems 422
  2.9.1 Introduction 422
  2.9.2 Discrete Mappings and Hamiltonians 431
  2.9.3 Lyapunov Exponents 435
  2.9.4 Exercises 448
  2.10 Rigid Body 449
  2.10.1 Introduction 449
  2.10.2 The Inertia Tensor 450
  2.10.3 The Angular Momentum 453
  2.10.4 Principal Axes of lnertia 454
  2.10.5 Steiner's Theorem 460
  2.10.6 Euler's Equations of Motion 462
  2.10.7 Force-Free Motion of a Symmetrical Top 467
  2.10.8 Motion of a Symmetrical Top in a Force Field 471
  2.10.9 Exercises 481
  2.10.10 Packages and Programms 481
  3 Nonlinear Dynamics 485
  3.1 Introduction 485
  3.2 The Korteweg-de Vries Equation 488
  3.3 Solution of the Korteweg-de Vries Equation 492
  3.3.1 The Inverse Scattering Transform 492
  3.3.2 Soliton Solutions of the Korteweg-de Vries Equation 498
  3.4 Conservation Laws of the Korteweg-de Vries Equation 505
  3.4.1 Definition of Conservation Laws 506
  3.4.2 Derivation of Conservation Laws 508
  3.5 Numerical Solution of the Korteweg-de Vries Equation 511
  3.6 Exercises 515
  3.7 Packagesand Programs 516
  3.7.1 Solution of the KdV Equation 516
  3.7.2 Conservation Laws for the KdV Equation 517
  3.7.3 Numerical Solution of the KdV Equation 518
  References 521
  Index 529
  Volume II
  Preface vii
  4 Electrodynamics 545
  4.1 Introduction 545
  4.2 Potential and Electric Field of Discrete Charge Distributions 548
  4.3 Boundary Problem of Electrostatics 555
  4.4 Two Ions in the Penning Trap 566
  4.4.1 The Center of Mass Motion 569
  4.4.2 Relative Motion of the Ions 572
  4.5 Exercises 577
  4.6 Packagesand Programs 578
  4.6.1 Point Charges 578
  4.6.2 Boundary Problem 581
  4.6.3 Penning Trap 582
  5 Quantum Mechanics 587
  5.1 Introduction 587
  5.2 The Schrodinger Equation 590
  5.3 0ne-Dimensional Potential 595
  5.4 The Harmonic Oscillator 609
  5.5 Anharmonic Oscillator 619
  5.6 Motion in the Central Force Field 631
  5.7 Second Virial Coefficient and Its Quantum Corrections 642
  5.7.1 The SVC and Its Relation to Thermodynamic Properties 644
  5.7.2 Calculation of the Classical SVC Bc.(T) for the (2 n - n) -Potential 646
  5.7.3 Quantum Mechanical Corrections Bq,(T) and Bq2(T) of the SVC 655
  5.7.4 Shape Dependence of the Boyle Temperature 680
  5.7.5 The High-Temperature Partition Function for Diatomic Molecules 684
  5.8 Exercises 687
  5.9 Packages and Programs 688
  5.9.1 Quantum Well 688
  5.9.2 Harmonic Oscillator 693
  5.9.3 Anharmonic Oscillator 695
  5.9.4 Central Field 698
  6 General Relativity 703
  6.1 Introduction 703
  6.2 The Orbits in General Relativity 707
  6.2.1 Quasielliptic Orbits 713
  6.2.2 Asymptotic Circles 719
  6.3 Light Bending in the Gravitational Field 720
  6.4 Einstein's Field Equations (Vacuum Case) 725
  6.4.1 Examples for Metric Tensors 727
  6.4.2 The Christoffel Symbols 731
  6.4.3 The Riemann Tensor 731
  6.4.4 Einstein's Field Equations 733
  6.4.5 The Cartesian Space 734
  6.4.6 Cartesian Space in Cylindrical Coordinates 736
  6.4.7 Euclidean Space in Polar Coordinates 737
  6.5 The Schwarzschild Solution 739
  6.5.1 The Schwarzschild Metric in Eddington-Finkelstein Form 739
  6.5.2 Dingle's Metric 742
  6.5.3 Schwarzschild Metric in Kruskal Coordinates 748
  6.6 The Reissner-Nordstrom Solution for a Charged Mass Point 752
  6.7 Exercises 759
  6.8 Packages and Programs 761
  6.8.1 Euler Lagrange Equations 761
  6.8.2 Perihelion Shift 762
  6.8.3 Light Bending 767
  7 Fractals 773
  7.1 Introduction 773
  7.2 Measuringa Borderline 776
  7.2.1 Box Counting 781
  7.3 The Koch Curve 790
  7.4 Multifractals 795
  7.4.1 Multifractals with Common Scaling Factor 798
  7.5 The Renormlization Group 801
  7.6 Fractional Calculus 809
  7.6.1 Historical Remarks on Fractional Calculus 810
  7.6.2 The Riemann-Liouville Calculus 813
  7.6.3 Mellin Transforms 830
  7.6.4 Fractional Differential Equations 856
  7.7 Exercises 883
  7.8 Packagesand Programs 883
  7.8.1 Tree Generation 883
  7.8.2 Koch Curves 886
  7.8.3 Multifactals 892
  7.8.4 Renormalization 895
  7.8.5 FractionaI Calculus 897
  Appendix 899
  A.1 Program Installation 899
  A.2 Glossary of Files and Functions 900
  A.3 Mathematica Functions 910
  References 923
  Index 931