Partial differential equations of first order and their applications to physics / Gustavo Lópes Velázguez.

Format:

Book

Author/Creator:

López Velázquez, Gustavo.

Language:

English

Subjects (All):

Differential equations, Partial.

Physics.

Physical Description:

xi, 188 pages ; 24 cm

Edition:

Second edition.

Place of Publication:

Singapore ; Hackensack, NJ : World Scientific, [2012]

Summary:

This book tries to point out the mathematical importance of the Partial Differential Equations of First Order (PDEFO) in Physics and Applied Sciences. The intention is to provide mathematicians with a wide view of the applications of this branch in physics, and to give physicists and applied scientists a powerful tool for solving some problems appearing in Classical Mechanics, Quantum Mechanics, Optics, and General Relativity. This book is intended for senior or first year graduate students in mathematics, physics, or engineering curricula.

This book tries unique in the sense that it covers the applications of PDEFO in several branches of applied mathematics, and fills the theoretical gap between the formal mathematical presentation of the theory and the pure applied tool to" physical problems that are contained in other books.

Improvements made in this second edition include corrected typographical errors; rewritten text to improve the flow and enrich the material; added exercises in all chapters; new applications in Chapters 1, 2, and 5 and expanded examples. Book jacket.

Contents:

1 Geometric Concepts and Generalities 1

1.1 Surfaces and Curves in Three Dimensions 1

1.2 Parallelism of Vector Fields 9

1.3 Methods of Solution of dx/P = dy/Q = dz/R 11

1.4 Orthogonal Trajectories of a System of Curves on a Surface 15

1.5 Pfaffian Differentia! Equation in R³ 17

1.6 Newton's Mechanics, Lagrangians, Hamiltonian, Hamilton-Jacobi Equation, and Liouville's Theorem 19

1.6.1 Equivalent Hamiltonians 27

1.7 Problems 30

1.8 References 32

2 Partial Differential Equations of First Order 33

2.1 Classification 33

2.2 Linear PDEFO for Functions Defined in Ω ⊂ R² 35

2.3 Quasi-Linear PDEFO for Functions Defined in Ω ⊂ R² 39

2.4 Quasi-Linear PDEFO for Functions Defined in Ω ⊂ Rⁿ 45

2.5 Problems 49

2.6 References 52

3 Physical Applications I 53

3.1 Mechanics 53

3.2 Angular Momentum in Quantum Mechanics 60

3.3 Heat Propagation between Two Superconducting Cables 67

3.4 Classical Statistical Mechanics in Equilibrium 71

3.5 Renormalization Group's Equations 73

3.6 Particle Multiplicity Distribution in High Energy Physics 76

3.7 Hamiltonian Perturbation Approach in Accelerator Physics 79

3.8 Perturbation Approach for the One-Dimensional Constant of Motion 84

3.9 Constant of Motion for a Relativistic Particle under Periodic Perturbation 86

3.10 Uniqueness of Constant of Motion 87

3.11 Vlasov Equation and Bunched Beam Instabilities 91

3.11.1 Potential Well Distortion 92

3.11.2 Longitudinal Modes 94

3.11.3 Transverse Modes 96

3.12 Interaction Plasma-Electromagnetic Field 100

3.12.1 Diluted Plasma 100

3.12.2 Diluted Linear, Dispersed and Nonlocal Plasma 105

3.12.3 Non Dilute Plasma 107

3.13 Decoherence in a Quantum System 109

3.14 Problems 119

3.15 References 122

4 Nonlinear Partial Differential Equations of First Order 127

4.1 Non-Linear PDEFO for Functions Defined in R² 127

4.2 Non-Linear PDEFO for Functions Defined in Rⁿ 143

4.3 Problems 149

4.4 References 151

5 Physical Applications II 153

5.1 Motion of a Classical Particle 153

5.1.1 Hamilton-Jacobi Equation for a One-Dimensional Harmonic Oscillator 153

5.1.2 The Lagrangian Obtained Directly from Hamiltonian 155

5.1.3 Relativistic Particle Moving in a Coulomb Field 157

5.1.4 Motion of a Test Particle in a Schwarchild's Space 160

5.1.5 Interaction of a Periodic Gravitational Wave with a Test Particle 163

5.2 Trajectory of a Ray of Light 167

5.2.1 Solution of the Eikonal Equation for a Refraction Index Depending on z 168

5.2.2 Solution of the Eikonal Equation for a Refraction Index Radially Depending 169

5.2.3 Eikonal Equation for a Refraction Index Radially Depending in Sphere 170

5.3 Equivalent Hamiltonians 170

5.4 Problems 174

5.5 References 176

6 Characteristic Surfaces of Linear Partial Differential Equation of Second Order 177

6.1 Characteristic Surfaces of a Linear PDESO Defined in Rⁿ 177

6.2 Characteristic Surfaces of a Linear PDESO Defined in R² 180

6.3 Problems 184

6.4 References 185.

Notes:

Includes index.

ISBN:

9814390372

9789814390378

OCLC:

788213090

Publisher Number:

99947894788