Theoretical Physics. Josef Honerkamp Hartmann Romer. A Classical Approach

Transcription

1 Theoretical Physics

2 Josef Honerkamp Hartmann Romer Theoretical Physics A Classical Approach Translated by H. Pollack With 141 Figures and 39 Problems Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

3 Professor Dr. Josef Honerkamp Professor Dr. Hartmann Romer Albert-Ludwigs-Universitat, Fakultat fur Physik, Hennann-Herder-StraBe 3, Freiburg, Gennany Translator: Howard Pollack 715 South Washington Street, Bloomington, IN 47401, USA Title of the original German edition: Klassische Theoretische Physik, 3. Auflage (Springer-Lebrbuch) ISBN-13 : Springer-Verlag Berlin Heidelberg 1986, 1989 and 1993 ISBN-13 : e-isbn-13 : DOl: / Library of Congress Cataloging in-publication Data. Honerkamp, J. [Klassische Theoretische Physik. English] Theoretical physics: a classical approach / Josef Honerkamp, Hartmann Romer; translated by H. Pollack. p. cm. Translation of: Klassische Theoretische Physik. Includes bibliographical references and index. ISBN 13: (New York: alk. paper). I. Mathematical physics. I. Romer, H. (Hartmann) II. Title. QC20.H '.1'51-dc This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only uoder the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1993 Softcover reprint of the hardcover 1st edition 1993 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Production editor: A. Kubler Typesetting: Macmillan India Ltd., India 56/ Printed on acid-free paper

4 Preface This introduction to classical theoretical physics emerged from a course for students in the third and fourth semester, which the authors have given several times at the University of Freiburg (Germany). The goal of the course is to give the student a comprehensive and coherent overview of the principal areas of classical theoretical physics. In line with this goal, the content, the terminology, and the mathematical techniques of theoretical physics are all presented along with applications, to serve as a solid foundation for further courses in the basic areas of experimental and theoretical physics. In conceiving the course, the authors had four interdependent goals in mind: the presentation of a consistent overview, even at this elementary level the establishment of a well-balanced interactive relationship between physical content and mathematical methods a demonstration of the important applications of physics, and an acquisition of the most important mathematical techniques needed to solve specific problems. In relation to the first point, it was necessary to limit the amount of material treated. This introductory course was not intended to preempt a later, primarily theoretical, course. On the other hand, we aimed for a certain completeness in the presentation of the basic principles and concepts of classical theoretical physics, which would serve as a lasting basis for later work. Emphasis was placed on presenting the material clearly and coherently, in the form of a clearly thought out (but not formalistic) introduction to the fundamental concepts and methods. To achieve clarity, the presentations, with few exceptions, go from the general to the particular. The conceptual framework is prepared first and is not developed much further in the examples. Nevertheless, after the structural fundamentals have been clearly explained, the carefully chosen examples play an essential role in each section of this course. Using these examples, the material which we have explained earlier becomes concrete and is demonstrated in a meaningful way. In addition, we have provided a number of summaries, reviews of earlier material, and tastes of what is to come. This places the subject matter in a larger context, and anticipates further developments, all of which helps to provide a broad perspective on the entire material.

5 VI Preface We also demonstrate, in many cases, how particular mathematical concepts and structures appear in different physical fields and contexts with different physical interpretations, for example in our treatment of the elementary results of linear algebra. We deliberately present mathematical concepts in a familiar manner, as they might be introduced in lectures in analysis and linear algebra. In this context, they are already familiar to the student, and this should help the student to recognize them in a physical context. Thus, mathematical knowledge is utilized. We have found that knowledge and understanding of these areas in physics as well as in mathematics have profited from this method. We cannot talk of an appropriate interaction between physics and mathematics if physics is seen only as an example of the realization of mathematical structures, or if conceptual exactness is confused with formalistic pedantry. Much is done to combat such a misunderstanding which often arises among students, particularly among talented ones. Physical and mathematical arguments are often developed in parallel, and carefully held apart from each other. Wherever possible, the physical origins of mathematical assumptions are revealed. Thus, it is not only from lack of space that mathematical proofs are often avowedly incomplete, or even omitted; rather, this corresponds to our intention. For example, the theory of distributions is developed as far as possible within the conceptual framework of linear algebra, ignoring mathematical subtleties. Here, again, the many examples we use are significant. We use not only dry, highly idealized systems, chosen for their easy treatment, such as the simple pendulum, but rather the manifold of physical phenomena, including examples from applied branches of physics like geophysics and physical chemistry. We discuss the examples as completely as possible, with particular emphasis on the physical interpretations of the results obtained. Thus, the connection is made between the physical situation, the mathematical formulation and discussion, and the intuitive physical results. It is here that the close relationship between mathematical deduction and intuitive interpretation, which is the essence of theoretical physics, emerges clearly. These thoroughly discussed examples also serve the last primary goal: they demonstrate the value of mathematical-technical dexterity in the solution of problems. This technical and methodological knowledge represents, so to speak, the tools of the trade. Familiarity with these techniques does not come from just listening to the lectures or reading, or even following the individual steps in the argument, it must also proceed from individual practice. It is essential for progress towards mastery that the student learns to use the equations, to find possible methods of solution, to go through the calculations in a problem, to interpret a result in its physical meaning, and to examine its plausibility himself or herself. This is naturally the purpose of the exercises which always accompany an introductory theoretical course. For reasons of space, we have given only a small

6 Preface VII collection of 39 worked-through homework problems. Comprehensive collections of such exercises already exist in great numbers. We should offer one word of explanation as to why this presentation of the fundamental principles of physics is limited to classical physics and thus leaves out modern, important, and "exciting" areas like relativity and quantum mechanics. First, in the opinions of the authors, the addition ofthis material would have made it impossible to present the course in two semesters - without simultaneously losing sight of the goal of active mastery of the basics as well as an overview of the entire subject matter. Furthermore, the classical fields of physics have the advantage that they work within the realm of phenomena more easily accessible to immediate intuitive observation. The interaction between formal deduction and intuitive interpretation, which is tremendously important in theoretical physics, is best practised within the framework of classical physics. Only with a greater sense of security can the student then progress into a realm where intuitive understanding is less forthcoming. We attempted to avoid unnecessary one-sidedness in the selection of material in the areas of classical physics represented. Thus, for example, statistical mechanics and thermodynamics, as well as the fundamentals of fluid mechanics, have received treatment here, based on their importance particularly for applied physics. As we have already stated, students should receive a sound foundation of knowledge as an initial preparation for further research in fields like quantum mechanics, relativity theory, fluid dynamics, analytical mechanics, irreversible thermodynamics, or the theory of dynamical systems. Finally, we wish to thank all those who have contributed to the publication of this book. In particular, we want to name Mrs. H. Kranz, Mrs. E. Rupp, Mrs. E. Ruf, and Mrs. W. Wanoth, who wrote out the long, difficult manuscript and never lost patience during the countless corrections. We thank Mrs. I. Weber and Mrs. B. Miiller for drawing the figures. We also express our gratitude to the participants in our course "Introduction to theoretical physics", in which this concept was first tested, for their many suggestions: also to those who took care of the accompanying exercises, above all Dr. H.C. Oettinger and Mr. R. Seitz, as well as Mr. P. Biller, Dr. H. Hess, Dr. M. Marcu, Mr. J. Miiller, Mr. G. Mutschler, and Dr. A. Saglio de Simonis. Mr. A. Geidel, Dr. H. Simonis, Mr. F.K. Schmatzer, and Mr. M. Zahringer gave us valuable assistance in proofreading. Freiburg, June 1993 J. Honerkamp. H. Romer

7 Contents 1. Introduction Newtonian Mechanics Space and Time in Classical Mechanics Newton's Laws A Few Important Force Laws The Energy of a Particle in a Force Field Line Integrals Work and Energy Several Interacting Particles Momentum and Momentum Conservation Angular Momentum The Two-Body Problem The Kepler Problem Scattering Relative Motion in the Scattering Process The Center of Mass System and the Laboratory System The Scattering Cross-Section The Virial Theorem Mechanical Similarity Some General Observations About the Many-Body Problem. : 81 Problems Lagrangian Methods in Classical Mechanics A Sketch of the Problem and Its Solution in the Case of a Pendulum The Lagrangian Method of the First Type The Lagrangian Method of the Second Type The Conservation of Energy in Motions Which are Limited by Constraints Non-holonomic Constraints Invariants and Conservation Laws The Hamiltonian Lagrange's Equations and Hamilton's Equations

8 X Contents Aside on the Further Development of Theoretical Mechanics and the Theory of Dynamical Systems The Hamiltonian Principle of Stationary Action Functionals and Functional Derivatives Hamilton's Principle Hamilton's Principle for Systems with Holonomic Constraints Problems Rigid Bodies The Kinematics of the Rigid Body The Inertia Tensor and the Kinetic Energy of a Rigid Body Definition and Elementary Properties of the Inertia Tensor Calculation of Inertia Tensors The Angular Momentum of a Rigid Body, Euler's Equations The Equations of Motion for the Eulerian Angles Problems Motion in a Noninertial System of Reference Fictitious Forces in Noninertial Systems Foucault's Pendulum Linear Oscillations Linear Approximations About a Point of Equilibrium A Few General Remarks About Linear Differential Equations Homogeneous Linear Systems with One Degree of Freedom and Constant Coefficients Homogeneous Linear Systems with n Degrees of Freedom and Constant Coefficients Normal Modes and Eigenfrequencies Examples of the Calculation of Normal Modes The Response of Linear Systems to External Forces External Oscillating Forces Superposition of External Harmonic Forces Periodic External Forces Arbitrary External Forces Problems Classical Statistical Mechanics Thermodynamic Systems and Distribution Functions Entropy Temperature, Pressure, and Chemical Potential Systems with Exchange of Energy Systems with an Exchange of Volume Systems with Exchanges of Energy and Particles

9 Contents XI 7.4 The Gibbs Equation and the Forms of Energy Exchange The Canonical Ensemble and the Free Energy Thermodynamic Potentials Material Constants Changes of State Reversible and Irreversible Processes Adiabatic and Non-adiabatic Processes The Joule-Thomson Process The Transformation of Heat into Work, the Carnot Efficiency The Laws of Thermodynamics The Phenomenological Basis of Thermodynamics Thermodynamics and Statistical Mechanics The First Law of Thermodynamics The Second and Third Laws The Thermal and Caloric Equations of State Equilibrium and Stability Conditions Equilibrium and Stability in Exchange Processes Equilibrium, Stability and Thermodynamic Potentials 296 Problems Applications of Thermodynamics Phase Transformations and Phase Diagrams The Latent Heat of Phase Transitions Solutions Henry's Law, Osmosis Henry's Law Osmosis Phase Transitions in Solutions Case (2): Miscibility in Only One Phase Case (3): Miscibility in Two Phases Problem Elements of Fluid Mechanics A Few Introductory Remarks About Fluid Mechanics The General Balance Equation Particular Balance Equations Entropy Production, Generalized Forces, and Fluids The Differential Equations of Fluid Mechanics A Few Elementary Applications of the Navier-Stokes Equations Problem The Most Important Linear Partial Differential Equations of Physics General Considerations Types of Linear Partial Differential Equations, the Formulation of Boundary and Initial Value Problems

10 XII Contents Initial Value Problems in 1RD Inhomogeneous Equations and Green's Functions Solutions of the Wave Equation Boundary Value Problems Initial Observations Examples of Boundary Value Problems The General Treatment of Boundary Value Problems The Helmholtz Equation in Spherical Coordinates, Spherical Harmonics, and Bessel Functions Separation of Variables The Angular Equations, Spherical Harmonics The Radial Equation, Bessel Functions Solutions of the Helmholtz Equation Supplementary Considerations Problems t 11. Electrostatics The Basic Equations of Electrostatics and Their First Consequences Coulomb's Law and the Electric Field Electrostatic Potential and the Poisson Equation Examples and Important Properties of Electrostatic Fields Boundary Value Problems in Electrostatics, Green's Functions Dirichlet and Neumann Green's Functions Supplementary Remarks on Boundary Value Problems in Electrostatics The Calculation of Green's Functions, the Method of Images The Calculation of Green's Functions, Expansion in Spherical Harmonics Localized Charge Distributions, the Multipole Expansion Electrostatic Potential Energy Problems Moving Charges, Magnetostatics The Biot-Savart Law, the Fundamental Equations of Magnetostatics Electric Current Density and Magnetic Fields The Vector Potential and Ampere's Law The SI-System of Units in Electrodynamics Localized Current Distributions The Magnetic Dipole Moment Force, Potential, and Torque in a Magnetic Field

11 Contents XIII 13. Time Dependent Electromagnetic Fields Maxwell's Equations Potentials and Gauge Transformations Electromagnetic Waves in a Vacuum, the Polarization of Transverse Waves Electromagnetic Waves, the Influence of Sources The Energy of the Electromagnetic Field Balance of Energy and the Poynting Vector The Energy Flux of the Radiation Field The Energy of the Electric Field The Energy of the Magnetic Field Self-Energy and Interaction Energy The Momentum of the Electromagnetic Field Elements of the Electrodynamics of Continuous Media The Macroscopic Maxwell Equations Microscopic and Macroscopic Fields The Average Charge Density and Electric Displacement The Average Current Density and the Magnetic Field Strength Electrostatic Fields in Continuous Media Magnetostatic Fields in Continuous Media Plane Waves in Matter, Wave Packets The Frequency Dependence of Susceptibility Wave Packets, Phase and Group Velocity Reflection and Refraction at Plane Boundary Surfaces Boundary Conditions, the Laws of Reflection and Refraction Fresnel's Equations Special Effects of Reflection and Refraction Appendices A. The r-function B. Conic Sections C. Tensors D. Fourier Series and Fourier Integrals D.1 Fourier Series D.2 Fourier Integrals and Fourier Transforms E. Distributions and Green's Functions E.1 Distributions E.2 Green's Functions F. Vector Analysis and Curvilinear Coordinates F.1 Vector Fields and Scalar Fields F.2 Line, Surface, and Volume Integrals

12 XIV Contents F.3 Stokes's Theorem F.4 Gauss's Theorem F.5 Applications of the Integral Theorems F.6 Curvilinear Coordinates Problems References Subject Index