Mathematica for Theoretical Physics: Electrodynamics, Quantum Mechanics, General Relativity, and Fractals, Vol. 2 Book

	Classical Mechanics and Nonlinear Dynamics
	Preface	vii
1	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 Historical 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 Inertia	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	Packages and 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
	Electrodynamics, Quantum Mechanics, General Relativity, and Fractals
	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	Packages and 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	One-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 B[subscript c](T) for the (2n-n)-Potential	646
5.7.3	Quantum Mechanical Corrections B[subscript q1](T) and B[subscript q2](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	QuantumWell	688
5.9.2	HarmonicOscillator	693
5.9.3	AnharmonicOscillator	695
5.9.4	CentralField	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	EulerLagrange Equations	761
6.8.2	PerihelionShift	762
6.8.3	LightBending	767
7	Fractals	773
7.1	Introduction	773
7.2	Measuring a 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	Packages and 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	Fractional 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