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Mathematica for Theoretical Physics Classical Mechanics and Nonl....pdf
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Preface
Contents
1 Introduction
1.1 Basics
1.1.1 Structure of Mathematica
1.1.2 Interactive Use of Mathematica
1.1.3 Symbolic Calculations
1.1.4 Numerical Calculations
1.1.5 Graphics
1.1.6 Programming
2 Classical Mechanics
2.1 Introduction
2.2 Mathematical Tools
2.2.1 Introduction
2.2.2 Coordinates
2.2.3 Coordinate Transformations and Matrices
2.2.4 Scalars
2.2.5 Vectors
2.2.6 Tensors
2.2.7 Vector Products
2.2.8 Derivatives
2.2.9 Integrals
2.210 Exercises
2.3 Kinematics
2.3.1 Introduction
2.3.2 Velocity
2.3.3 Acceleration
2.3.4 Kinematic Examples
2.3.5 Exercises
2.4 Newtonian Mechanics
2.4.1 Introduction
2.4.2 Frame of Reference
2.4.3 Time
2.4.4 Mass
2.4.5 Newton's Laws
2.4.6 Forces in Nature
2.4.6.1 The Fundamental Forces
2.4.7 Conservation Laws
2.4.7.1 Linear Momentum
2.4.7.2 Angular Momentum
2.4.7.3 Work and Energy
2.4.7.4 Constant Forces
2.4.8 Application of Newton's Second Law
2.4.8.1 Falling Particle
2.4.8.2 Harmonic Oscillator
2.4.8.3 The Phase Diagram
2.4.8.4 Damped Harmonic Oscillator
2.4.8.5 Driven Oscillations
2.4.8.6 Solution Procedures of Liner Differential Equations
2.4.8.7 Nonlinear Oscillation
2.4.8.8 Damped Driven Nonlinear Oscillator
2.4.9 Exercises
2.4.10 Packages and Programs
2.5 Central Forces
2.5.1 Introduction
2.5.2 Kepler's Laws
2.5.3 Central Field Motion
2.5.3.1 Equations of Motion
2.5.3.2 Orbits in a Central Force Field
2.5.3.3 Effective Potential
2.5.3.4 Planet Motions
2.5.4 Two-Particle Collisions and Scattering
2.5.4.1 Elastic Collisions
2.5.4.2 Scattering Cross Section
2.5.4.3 Rutherford Scattering
2.5.5 Exercises
2.5.6 Packages and Programs
2.6 Calculus of Variations
2.6.1 Introduction
2.6.2 The Problem of Variations
2.6.3 Euler’s Equation
2.6.4 Euler Operator
2.6.5 Algorithm Used in the Calculus of Variations
2.6.6 Euler Operator for q Dependent Variables
2.6.7 Euler Operator for q + p Dimensions
2.6.8 Variations with Constraints
2.6.9 Exercises
2.6.10 Packages and Programs
2.7 Lagrange Dynamics
2.7.1 Introduction
2.7.2 Hamilton's Principle Historical Remarks
2.7.3 Hamilton's Principle
2.7.3.1 Classes of Constraints
2.7.4 Symmetries and Conservation Laws
2.7.4.1 Conservation of Energy and Translation in Time
2.7.4.2 Conservation of Momentum
2.7.4.3 Conservation of Angular Momentum
2.7.5 Exercises
2.7.6 Packages and Programs
2.8 Hamiltonian Dynamics
2.8.1 Introduction
2.8.2 Legendre Transform
2.8.3 Hamilton's Equation of Motion
2.8.4 Hamilton's Equations and the Calculus of Variation
2.8.5 Liouville's Theorem
2.8.6 Poisson Brackets
2.8.7 Manifolds and Classes
2.8.7.1 A Two-Dimensional Poisson Manifold
2.8.7.2 A Four-Dimensional Poisson Manifold
2.8.7.3 Hamilton's Equations Derived from the Manifold
2.8.7.4 Hamilton's Equations Derived from the
Hamilton–Poisson Manifold
2.8.8 Canonical Transformations
2.8.9 Generating Functions
2.8.10 Action Variables
2.8.10.1 One-Dimensional Hamilton–Jacobi Equation
2.8.10.2 Action Angle Variables for one Dimension
2.8.10.3 Separation of Hamiltonians
2.8.11 Exercises
2.8.12 Packages and Programs
2.9 Chaotic Systems
2.9.1 Introduction
2.9.2 Discrete Mappings and Hamiltonians
2.9.3 Lyapunov Exponent
2.9.4 Exercises
2.10 Rigid Body
2.10.1 Introduction
2.10.2 The Inertia Tensor
2.10.3 The Angular Momentum
2.10.4 Principal Axes of Inertia
2.10.5 Steiner's Theorem
2.10.6 Euler's Equations of Motion
2.10.7 Force-Free Motion of a Symmetrical Top
2.10.8 Motion of a Symmetrical Top in a Force Field
2.10.9 Exercises
2.10.10 Packages and Programs
3
Nonlinear Dynamics
3.1 Introduction
3.2 The Korteweg–de Vries Equation
3.3 Solution of the Korteweg–de Vries Equation
3.3.1 The Inverse Scattering Transform
3.3.2 Soliton Solutions of the Korteweg–de Vries Equation
3.4 Conservation Laws of the Korteweg–de
Vries Equation
3.4.1 Definition of Conservation Laws
3.4.2 Derivation of Conservation Laws
3.5 Numerical Solution of the Korteweg–de
Vries Equation
3.6 Exercises
3.7 Packages and Programs
3.7.1 Solution of the KdV Equation
3.7.2 Conservation Laws for the KdV Equation
3.7.3 Numerical Solution of the KdV Equation
References
Volume I
Volume II
Index
Mathematica® for Theoretical Physics
Mathematica®
for Theoretical Physics
Classical Mechanics
and Nonlinear Dynamics
Second Edition
Gerd Baumann
CD-ROM Included
Gerd Baumann
Department of Mathematics
German University in Cairo GUC
New Cairo City
Main Entrance of Al Tagamoa Al Khames
Egypt
Gerd.Baumann@GUC.edu.eg
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 ©.
Library of Congress Cataloging-in-Publication Data
Baumann, Gerd.
[Mathematica in der theoretischen Physik. English]
Mathematica for theoretical physics / by Gerd Baumann.—2nd ed.
p. cm.
Includes bibliographical references and index.
Contents: 1. Classical mechanics and nonlinear dynamics — 2. Electrodynamics, quantum
mechanics, general relativity, and fractals.
ISBN 0-387-01674-0
1. Mathematical physics—Data processing.
QC20.7.E4B3813 2004
530′.285′53—dc22
2. Mathematica (Computer file)
I. Title.
2004046861
e-ISBN 0-387-25113-8
Printed on acid-free paper.
ISBN-10: 0-387-01674-0
ISBN-13: 978-0387-01674-0
© 2005 Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New
York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis.
Use in connection with any form of information storage and retrieval, electronic adaptation, com-
puter software, or by similar or dissimilar methodology now known or hereafter developed is for-
bidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if
they are not identified as such, is not to be taken as an expression of opinion as to whether or not
they are subject to proprietary rights.
Mathematica, MathLink, and Math Source are registered trademarks of Wolfram Research, Inc.
Printed in the United States of America.
(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
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