一階偏微分方程及其在物理中的應(yīng)用PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER AND THEIR APPLICATIONS TO PHYSICS

內(nèi)容概要

This book is about the theory and applications of Partial Differential Equations of First Order (PDEFO). Many interesting topics in physics such as constant motion of dynamical systems, renormalization theory, Lagrange transformation, ray trajectories, and Hamilton–Jacobi theory are or can be formulated in terms of partial differential equations of first order. In this book, the author illustrates the utility of the powerful method of PDEFO in physics, and also shows how PDEFO are useful for solving practical problems in different branches of science. The book focuses mainly on the applications of PDEFO, and the mathematical formalism is treated carefully but without diverging from the main objective of the book.

書籍目錄

CHAPTER I. Geometric Concepts and Generalities  1. Surfaces and Curves in Three Dimensions  2. Methods of Solution of dx/P = dy/Q = dz/R  3. Orthogonal Trajectories of a System of Curves on a Surface ...  4. Pfaffian Differential Equation in Ra  5. Newton's Mechanics, Lagrangians, Hamiltonian, Hamilton-Jacobi Equation, and Liouville's Theorem  6. ReferencesCHAPTER II. Partial Differential Equations of First Order  1. Classification  2. Linear PDEFO for Functions Defined in  R2  3. Quasi-linear PDEFO for Functions Defined in  C R2  4. Quasi-linear PDEFO for Functions Defined in    5. ReferencesCHAPTER III. Physical Applications I  1. Mechanics  2. Angular Momentum in Quantum Mechanics  3. Heat Propagation between Two Superconducting Cables  4. Classical Statistical Mechanics in Equilibrium  5. Renormalization Group's Equations  6. Particle Multiplicity Distribution in High Energy Physics  7. Hamiltonian Perturbation Approach in Accelerator Physics  8. Perturbation Approach for the One-dimensional Constant of Motion  9. Constant of Motion for a Relativistic Particle under Periodic Perturbation  10. ReferencesCHAPTER IV. Nonlinear Partial Differential Equations of First Order  1. Non-linear PDEFO for Functions Defined in  C R2  2. Non-linear PDEFO for Functions Defined in  C Rn  3. ReferencesCHAPTER V. Physical Applications II  1. Motion of a Classical Particle    a) Hamilton-Jacobi equation for a one-dimensional Harmonic oscillator    b) The Lagrangian obtained directly from the Hamiltonian    c) Relativistic particle moving in a Coulomb field    d) Motion of a test particle in a Schwarchild's space    e) Interaction of a periodic gravitational wave with a test particle  2. Trajectory of a Ray of Light    a) Solution of the Eikonal equation for a refraction index depending on z    b) Solution of the Eikonal equation for a refraction index radially depending  3. ReferencesCHAPTER VI. Characteristic Surfaces of Linear Partial Differential Equation of Second Order  1. Characteristic Surfaces of a Linear PDESO Defined in   2. Characteristic Surfaces of a Linear PDESO Defined in   3. ReferencesINDEX

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