® Mathematica for Theoretical Physics ® Mathematica for Theoretical Physics Classical Mechanics and Nonlinear Dynamics Second Edition Gerd Baumann CD-ROMIncluded GerdBaumann DepartmentofMathematics GermanUniversityinCairoGUC NewCairoCity MainEntranceofAlTagamoaAlKhames Egypt [email protected] This is a translated, expanded, and updated version of the original German version of the work “Mathematica® in der Theoretischen Physik,” published by Springer-Verlag Heidelberg,1993©. LibraryofCongressCataloging-in-PublicationData Baumann,Gerd. [MathematicaindertheoretischenPhysik.English] Mathematicafortheoreticalphysics/byGerdBaumann.—2nded. p.cm. Includesbibliographicalreferencesandindex. Contents:1. Classicalmechanicsandnonlineardynamics—2. Electrodynamics,quantum mechanics,generalrelativity,andfractals. ISBN0-387-01674-0 1. Mathematicalphysics—Dataprocessing. 2. Mathematica(Computerfile) I. Title. QC20.7.E4B3813 2004 530′.285′53—dc22 2004046861 ISBN-10:0-387-01674-0 e-ISBN0-387-25113-8 Printedonacid-freepaper. ISBN-13:978-0387-01674-0 ©2005SpringerScience+BusinessMedia,Inc. All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermissionofthepublisher(SpringerScience+BusinessMedia,Inc.,233SpringStreet,New York,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis. Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,com- putersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisfor- bidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if theyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornot theyaresubjecttoproprietaryrights. Mathematica,MathLink,andMathSourceareregisteredtrademarksofWolframResearch,Inc. PrintedintheUnitedStatesofAmerica. (HAM) 9 8 7 6 5 4 3 2 1 springeronline.com To Carin, for her love, support, and encuragement. Preface As physicists, mathematicians or engineers, we are all involved with mathematical calculations in our everyday work. Most of the laborious, complicated, and time-consuming calculations have to be done over and over again if we want to check the validity of our assumptions and derive new phenomena from changing models. Even in the age of computers, we often use paper and pencil to do our calculations. However, computer programs like Mathematica have revolutionized our working methods. Mathematica not only supports popular numerical calculations but also enables us to do exact analytical calculations by computer. Once we know the analytical representations of physical phenomena, we are able to use Mathematica to create graphical representations of these relations. Days of calculations by hand have shrunk to minutes by using Mathematica. Results can be verified within a few seconds, a task that took hours if not days in the past. The present text uses Mathematica as a tool to discuss and to solve examples from physics. The intention of this book is to demonstrate the usefulness of Mathematica in everyday applications. We will not give a complete description of its syntax but demonstrate by examples the use of its language. In particular, we show how this modern tool is used to solve classical problems. viii Preface This second edition of Mathematica in Theoretical Physics seeks to prevent the objectives and emphasis of the previous edition. It is extended to include a full course in classical mechanics, new examples in quantum mechanics, and measurement methods for fractals. In addition, there is an extension of the fractal's chapter by a fractional calculus. The additional material and examples enlarged the text so much that we decided to divide the book in two volumes. The first volume covers classical mechanics and nonlinear dynamics. The second volume starts with electrodynamics, adds quantum mechanics and general relativity, and ends with fractals. Because of the inclusion of new materials, it was necessary to restructure the text. The main differences are concerned with the chapter on nonlinear dynamics. This chapter discusses mainly classical field theory and, thus, it was appropriate to locate it in line with the classical mechanics chapter. The text contains a large number of examples that are solvable using Mathematica. The defined functions and packages are available on CD accompanying each of the two volumes. The names of the files on the CD carry the names of their respective chapters. Chapter 1 comments on the basic properties of Mathematica using examples from different fields of physics. Chapter 2 demonstrates the use of Mathematica in a step-by-step procedure applied to mechanical problems. Chapter 2 contains a one-term lecture in mechanics. It starts with the basic definitions, goes on with Newton's mechanics, discusses the Lagrange and Hamilton representation of mechanics, and ends with the rigid body motion. We show how Mathematica is used to simplify our work and to support and derive solutions for specific problems. In Chapter 3, we examine nonlinear phenomena of the Korteweg–de Vries equation. We demonstrate that Mathematica is an appropriate tool to derive numerical and analytical solutions even for nonlinear equations of motion. The second volume starts with Chapter 4, discussing problems of electrostatics and the motion of ions in an electromagnetic field. We further introduce Mathematica functions that are closely related to the theoretical considerations of the selected problems. In Chapter 5, we discuss problems of quantum mechanics. We examine the dynamics of a free particle by the example of the time-dependent Schrödinger equation and study one-dimensional eigenvalue problems using the analytic and Preface ix numeric capabilities of Mathematica. Problems of general relativity are discussed in Chapter 6. Most standard books on Einstein's theory discuss the phenomena of general relativity by using approximations. With Mathematica, general relativity effects like the shift of the perihelion can be tracked with precision. Finally, the last chapter, Chapter 7, uses computer algebra to represent fractals and gives an introduction to the spatial renormalization theory. In addition, we present the basics of fractional calculus approaching fractals from the analytic side. This approach is supported by a package, FractionalCalculus, which is not included in this project. The package is available by request from the author. Exercises with which Mathematica can be used for modified applications. Chapters 2–7 include at the end some exercises allowing the reader to carry out his own experiments with the book. Acknowledgments Since the first printing of this text, many people made valuable contributions and gave excellent input. Because the number of responses are so numerous, I give my thanks to all who contributed by remarks and enhancements to the text. Concerning the historical pictures used in the text, I acknowledge the support of the http://www-gapdcs.st-and.ac.uk/~history/ webserver of the University of St Andrews, Scotland. My special thanks go to Norbert Südland, who made the package FractionalCalculus available for this text. I'm also indebted to Hans Kölsch and Virginia Lipscy, Springer-Verlag New York Physics editorial. Finally, the author deeply appreciates the understanding and support of his wife, Carin, and daughter, Andrea, during the preparation of the book. Ulm, Winter 2004 Gerd Baumann Contents Volume I 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 xii Contents 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 qDependent 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