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MathGroup Archive 1995

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  • To: mathgroup at
  • Subject: [mg553] New Book - MATHEMATICA FOR PHYSICS
  • From: steve at (Steve Christensen)
  • Date: Wed, 15 Mar 1995 16:02:57 -0600

Dear MathGroup:

I would like to call your attention to the following new Mathematica book
that was recently published,  MATHEMATICA FOR PHYSICS. This book is intended
for the advanced undergraduate and graduate  physics student taking core
courses in the physics curriculum. This text was developed with many years
of student input. 

Although this book addresses physics problems, the introductory chapters are
relevant for most Mathematica users. (In particular, many queries answered
on the MathGroup are addressed in the first chapter.)  

The source code (without annotations) to selected  problems is available  on
the MathSource server (item 0206-862 or and 
search for Zimmerman) so you can "test drive" the problems. 

I have attached a description of the text, and the table of contents. 

Thank you,

Fredrick I. Olness:  olness at


WHAT:    A new book for doing Physics with Mathematica
AUTHORS: Robert L. Zimmerman (U. of Oregon)
         Fredrick I. Olness (Southern Methodist University)
         with a foreword by Stephen Wolfram
LENGTH:  436 pages

         ISBN 0-201-53796-6
         For ordering information, call 1-800-822-6339
         Addison-Wesley Publishing Company
         MathSource Number: 0206-862 

Communication with the authors:
Fredrick I. Olness:  olness at
Robert L. Zimmerman: bob at

TOPICS:  General Physics
         Classical Mechanics
         Quantum Mechanics
         Oscillation Systems (Linear and Non-Linear)
         Special Relativity
         General Relativity


This book is intended for the advanced undergraduate and graduate 
physics student taking core courses in the physics curriculum. 

We expect this text to be a supplement to the standard course
text.   The student would  use this book to get ideas on how to
use  Mathematica to solve the problems assigned by the

Since we cover the canonical problems from the core courses, the 
student can  practice with our solutions, and then modify our 
solutions to solve the particular problems assigned.  This should
help the student move up the Mathematica learning curve quickly.


Mathematica is a powerful mathematical software system
for students, researchers, and anyone seeking an
effective tool for mathematical analysis. Tools such
as Mathematica have begun to revolutionize the way
science is taught, and research performed.  Now there
is a book specifically for students and teachers of
physics who wish to use Mathematica to visualize and
display physics concepts and to generate numerical and
graphical solutions to physics problems.

 Mathematica for Physics chooses the canonical
problems from the physics curriculum, and solves these
problems using Mathematica. This book takes the reader
beyond the "textbook" solutions by challenging the
student to cross check the results using the wide
variety of Mathematica's analytical, numerical, and
graphical tools. Throughout the book, the complexity
of both the physics and Mathematica is systematically
extended to broaden the tools  the reader has at his
or her disposal, and to broaden the range of problems
that can be solved.

As such, this text is an appropriate supplement for
any of the core advanced undergraduate and graduate
physics courses.

Highlights include:

Provides Mathematica solutions for the canonical
problems in the physics curriculum.

Covers essential problems in: Mechanics,
Electrodynamics, Quantum Mechanics, Special and
General Relativity, Cosmology,  Elementary Circuits,
Oscillating Systems.

Uses the power of Mathematica to go beyond "textbook"
solutions and bring the problems alive with
animations, and other graphical tools.

Emphasizes the graphical capability of Mathematica to
develop the reader's intuition and visualization in
problem solving.

Introduces the reader to the aspects of Mathematica
that are particularly useful for physics.

Brief Table of contents 

Chpt 1: Getting Started   
Chpt 2: General Physics 
Chpt 3: Oscillating Systems
Chpt 4: Lagrangians and Hamiltonians  
Chpt 5: Electrostatics  
Chpt 6: Quantum Mechanics 
Chpt 7: Relativity and Cosmology 


With Mathematica, the entire approach to problem solving can be 
drastically changed. We give some brief examples. 

DOUBLE PENDULUM: This is a topic that is generally treated as an 
"advanced" topic.  With Mathematica, the solution is relatively 
straightforward.  Once the solutions is obtained, the textbooks
try to  describe (in words) the general properties of the system,
and the  normal modes.  (In particular, the property that the
energy is  transferred back and forth between the two segments of
the pendulum.)  With the animation capability of Mathematica, we
do not need to lead  the student to these conclusions, but we can
point them in the general  direction, and let them discover these
results on their own by varying  the amplitudes of the separate
normal modes. 

E&M BOUNDARY VALUE PROBLEMS:  For the beginning student, it is
easy to  become overwhelmed by boundary value problems. With the
power of  Mathematica, it is easy to show how straightforward
these solutions  are--especially with the help of the different
coordinate systems  built into Mathematica.  When the student
finishes the problem with  pen and paper, they have only a set of
formulas that may mean very  little to the student.  With
Mathematica, we encourage the student to  plot the final solution
so that they can verify visually if the  boundary conditions are
satisfied.  This techniques encourages the  student to think
about the solution, and not simply grind out the  math. 

HYDROGEN ATOM: In the standard solution of the hydrogen atom,
the  student is completely lost in the mathematics. Mathematica is
able to  recognize that the solution of the radial equation is a
Laguerre  polynomial, assemble the constants to form the
principal quantum  number, and plot the solutions. The student
then has the energy and  the curiosity to numerically investigate
the behavior of the  wavefunctions, and consider the disastrous
consequences of choosing a  non-integral value for the principal
quantum number. 


Robert Zimmerman is a Professor of Physics and  research 
associate in the Institute of Theoretical Science at the
University of Oregon. He has written papers on  Mathematical
Physics, Elementary Particles, Astrophysics, Cosmology, and
General Relativity.  He has taught graduate courses in
Mathematical Physics, Theoretical Mechanics, Electrodynamics,
Quantum Mechanics, General Relativity and Cosmology. He received
his Ph.D. from the University of Washington.

 Fredrick Olness is an Assistant Professor of Physics at Southern
Methodist University in Dallas Texas.  His research is in
Theoretical High Energy Physics and he studies the  Quantum
Chromodynamic (QCD) theory of the strong interaction.  He
received his Ph.D. from the University of Wisconsin, received an 
SSC Fellowship in 1993,  and is a member of the CTEQ
collaboration--a novel collaboration of  theorists and


 Foreword v 
 Introduction vii 
 Troubleshooting xi 
CHAPTER 1 Getting Started  1 
1.1 Introduction    1 
1.1.1 Computers as a Tool    1 
1.1.2 Suggestions on Approaching the Exercises    2 
1.2 Arithmetic and Algebra    2 
1.2.1 Arithmetic and Notation    2 
1.2.2 Algebra    3 
1.2.3 Mapping Expressions    3 
1.2.4 Rules    4 
1.2.5 Conjugation    5 
1.2.6 User-defined Complex Conjugate Rule    5 
1.2.7 Algebraic Equations    5 
1.2.8 Threading Expressions    7 
1.3 Functions and Procedures    7 
1.3.1 User-defined Functions    7 
1.3.2 Discontinuous Functions    8 
1.3.3 Nonanalytic Functions    8 
1.3.4 Rules    9 
1.3.5 Procedures    10 
1.4 Miscellaneous    10 
1.4.1 Packages    10 
1.4.2 Contexts    11 
1.4.3 Protecting Commands    12 
1.5 Calculus    12 
1.5.1 Integration    12 
1.5.2 Analytic Solutions of Differential Equations    13 
1.5.3 Changing Variables and Pure Functions    13 
1.5.4 Numerical Solutions of Differential Equations    14 
1.6 Graphics    15 
1.6.1 Animated Plots    15 
1.6.2 Vector Field Plots    16 
1.6.3 Shadowing    16 
1.6.4 Three-dimensional Graphics    18 
1.6.5 Space Curve    18 
1.7 Exercises    18 

CHAPTER 2 General Physics  23 
2.1 Introduction    23 
2.2 General Physics    23 
2.2.1 Newtonian Motion    23 
2.2.2 Electricity, Magnetism, and Circuits    24 
2.3   Commands    24 
2.3.1 Packages    24 
2.3.2 User-defined Procedures    25 
 Procedure to find a least-squares fit to a set of data  25 
  Example: Slope of a Straight Line  25 
  Example: An Exponential  25 
 Procedure to find the electric potential for point charges  26 
  Example: Dipole  26 
2.3.3 Protect User-defined Procedures    26 
2.4 Problems    26 
2.4.1 Projectile Motion in a Constant Gravitational Field    26 
 Problem 1: Escape Velocity  26 
 Problem 2: Projectile in a Uniform Gravitational Field  27 
 Problem 3: Projectile with Air Resistance  31 
 Problem 4: Rocket with Varying Mass  36 
2.4.2 Projectile Motion in Rotating Reference Frames    42 
 Problem 1: Coriolis and Centrifugal Forces  42 
 Problem 2: Foucault Pendulum  47 
2.4.3 Electricity and Magnetism    52 
 Problem 1: Charged Disk  52 
  Problem 2: Uniformly Charged Sphere  54 
 Problem 3: Electric Dipole  60 
 Problem 4: Magnetic Vector Potential for a Linear Current  63 
2.4.4 Circuits    66 
 Problem 1: Series RC Circuit  66 
 Problem 2: Series RL Loop  68 
  Problem 3: RLC Loop  71 
2.4.5 Modern Physics    75 
 Problem 1: The Bohr Atom  75 
 Problem 2: Relativistic Collision  77 
2.5 Unsolved Problems    79 

CHAPTER 3  Oscillating Systems  83 
3.1 Introduction    83 
3.1.1 Oscillations    83 
 Potentials  83 
 Phase Planes  83 
 Small Oscillations and Normal Modes  84 
3.2   Commands    84 
3.2.1 Packages    84 
3.2.2 User-defined Procedures    85 
 Series expansion for second-order equation  85 
  Example 1: Second order solution  85 
  Example 2: How the  diffSeriesOne  routine works  86 
 Phase plot for one-dimensional system  87 
  Example: Phase plots for harmonic motion  87 
 Time behavior of phase plot for a one-dimensional system  88 
  Example: Time-evolved phase plots for harmonic motion  88 
  doublePlot : Phase plots and time evolution for a one-dimensional system  89 
  Example: Phase plots and time evolution for harmonic motion  89 
 Fourier spectrum of a one-dimensional oscillating system  89 
  Example: Fast Fourier transform  90 
 Eigenvalues and eigenvectors for small oscillating systems  90 
  Example: Two coupled particles  91 
 Animation for linear motion  93 
  Example: Linear harmonic oscillator  93 
3.2.3 Protect User-defined Procedures    94 
3.3 Problems    94 
3.3.1 Linear Oscillations    94 
 Problem 1: Analysis of Linear Oscillator  94 
 Problem 2: Solution of Linear Oscillator  97 
 Problem 3: Damped Linear Oscillator  99 
 Problem 4: Damped Harmonic Oscillator and Driving Forces  104 
3.3.2 Nonlinear Oscillations    109 
 Problem 1: Duffing's Oscillator Equation  109 
 Problem 2: Forced Duffing Oscillator for Double-well Potential  114 
 Problem 3: van der Pole Oscillator and Limiting Cycles  118 
 Problem 4: Motion of a Damped, Forced Nonlinear Pendulum  121 
3.3.3 Small Oscillations    124 
 Problem 1: Two Coupled Harmonic Oscillators  124 
 Problem 2: Three Coupled Harmonic Oscillators  130 
 Problem 3: Double Pendulum  135 
3.4 Unsolved Problems    139 

CHAPTER 4 Lagrangians and Hamiltonians  143 
4.1 Introduction    143 
4.1.1 Lagrange's Equations    143 
 Generalized Coordinates and Constraints  143 
 Lagrangian  144 
 Nonholonomic Constraints and Lagrangian Multipliers  144 
4.1.2 Hamilton's and Hamilton-Jacobi Equations    144 
 Hamilton's Equations  144 
 Hamilton-Jacobi Technique  145 
4.1.3   Commands    146 
 Packages  146 
 User-defined Rules  147 
  HyperbolicToComplex  and  ComplexToHyperbolic   147 
  Example:  HyperbolicToComplex  and  ComplexToHyperbolic   147 
 User-defined Procedures  147 
 Finding Lagrange's equations  147 
  Example: Lagrange's equation for a particle in a potential V[x]  148 
 Finding the canonical momentum, Hamiltonian, and equations of motion  148 
  Example 1: Hamilton's equations of motion in one dimension  149 
  Example 2: Hamilton's equations of motion in two dimensions  149 
  Example 3: How  Hamilton  works  150 
 Finding the canonical momentum, Hamilton's principal function, and Hamilton-Jacobi equations  151 
  Example 1: One-dimensional particle in a potential V[x]  151 
  Example 2: Two-dimensional particle in a potential V[x]  152 
  Example 3: How  HamiltonJacobi  works  152 
 Series expansion solution for second-order equation  153 
 Series expansion solution for two first-order equations  153 
  Example 1: Expansion of harmonic oscillator  154 
  Example 2: How  firstDiffSeries  works  154 
 First-order perturbation solution  155 
  Example 1: Perturbed harmonic oscillator  156 
  Example 2: Details of  firstOrderPert   156 
4.1.4 Protect User-defined Procedures    158 
4.2 Problems    158 
4.2.1 Lagrangian Problems    158 
 Problem 1: Atwood Machine  158 
 Problem 2: Bead Sliding on a Rotating Wire  161 
 Problem 3: Bead on a Rotating Hoop  165 
 Problem 4: Hoop Rolling on an Incline  171 
 Problem 5: Sphere Rolling on a Fixed Sphere  174 
 Problem 6: Mass Falling Through a Hole in a Table  178 
4.2.2 Orbiting Bodies    183 
 Problem 1: Equivalent One-body Problem  183 
 Problem 2: Kepler Problem  188 
 Problem 3: Precessing Ellipse and Generalized Kepler Problem  192 
 Problem 4: Numerical Solution for Orbits with Central Forces  195 
 Problem 5: Quadripole Potential and Perturbative Solutions  199 
4.2.3 Hamilton and Hamilton-Jacobi Problems    206 
  Problem 1: Harmonic Oscillator and Hamilton's Equations  206 
 Problem 2: Hamilton's Equations in Cylindrical and Spherical Coordinates  210 
 Problem 3: Spherical Pendulum and Hamilton's Equations  213 
 Problem 4: Harmonic Oscillator and Hamilton-Jacobi Equations  218 
 Problem 5: Kepler's Problem and Hamilton-Jacobi Equations  221 
4.3 Unsolved Problems    225 

CHAPTER 5 Electrostatics  229 
5.1 Introduction    229 
5.1.1 Electric Field and Potential    229 
 Electric field    229 
 Electrostatic potential    229 
5.1.2 Laplace's Equation    230 
 Cartesian coordinates  230 
 Cylindrical coordinates  230 
 Spherical coordinates  230 
5.1.3   Commands    231 
 Packages  231 
 User-defined procedures  231 
 Operator:  TrigToY   231 
  Example  232 
 Operator:  TrigToP   232 
  Example  233 
  Monopole   233 
  Example  233 
  PotentialExpansion   234 
  Example: Asymptotic potential of two-point charges  234 
  VEPlot   234 
  Example: Equipotential surface and electric field of two-point charges  235 
  MultipoleSH   236 
  Example: Potential of non-axially symmetric charge density  236 
  MultipoleP   237 
  Example: Potential of an axially symmetric charge distribution  238 
5.1.4 Protect User-defined Procedures    238 
5.2 Problems    238 
5.2.1 Point Charges, Multipoles, and Image Problems    238 
 Problem 1: Superposition of point charges  238 
 Problem 2: Point charges and grounded plane  242 
 Problem 3: Point charges and grounded sphere  245 
 Problem 4: Line charge and grounded plane  249 
 Problem 5: Multipole expansion of a charge distribution  252 
5.2.2 Cartesian and Cylindrical Coordinates    258 
 Problem 1 : Separation of variables in Cartesian and cylindrical coordinates  258 
 Problem 2 : Potential and a rectangular groove  261 
 Problem 3: Rectangular conduit  265 
 Problem 4: Potential inside a rectangular box with five sides at zero potential  269 
 Problem 5: Conducting cylinder with a potential on the surface  274 
5.2.3 Legendre Polynomials and Spherical Harmonics    278 
  Problem 1: A charged ring  278 
 Problem 2: Grounded sphere in an electric field  285 
 Problem 3: Sphere with an axially symmetric charge distribution  288 
 Problem 4: Sphere with a given axially symmetric potential  292 
 Problem 5: Sphere with upper hemisphere  V  and lower hemisphere - .75pt  V  295 
5.3 Unsolved Problems    299 

CHAPTER 6 Quantum Mechanics  303 
6.1 Introduction    303 
6.1.1 Foundations of Quantum Mechanics    303 
 Historical beginnings  303 
 Time-independent quantum mechanics  304 
6.1.2   Commands    304 
 Packages  304 
 User-defined procedures and rules  304 
 Change of variables procedure  304 
  Example: Change of variable  305 
 Series expansion solution for second-order equation  305 
  HyperbolicToComplex  and  ComplexToHyperbolic   306 
 Complex conjugate rule  306 
  Example: Complex exponential  306 
 User-defined solutions of differential equations  307 
 Solution: Hermite polynomials  307 
  Example: Hermite solution  307 
 Solution: Legendre polynomials  307 
  Example: Legendre solution  307 
 User-defined one-dimensional wave properties  308 
 One-dimensional wave function with constant potential  308 
6.1.3 User-defined Three-dimensional Quantum Equations    308 
 Operator:  schrodinger   308 
  Example: Spherical potential and spherical wave function  308 
 Operator:  hamiltonian   309 
  Example: Harmonic oscillator  309 
 Operator:  flux   309 
  Example: Plane wave flux  309 
6.1.4 Protect User-defined Operators    310 
6.2 Problems    310 
6.2.1 One-dimensional Schrodinger's Equation    310 
 Problem 1: Particle bound in an infinite potential well  310 
 Problem 2: Particle bound in a finite potential well  314 
 Problem 3: Particle hitting a finite step potential  322 
 Problem 4: Particle propagating towards a rectangular potential  328 
 Problem 5: The one-dimensional harmonic oscillator  336 
6.2.2 Three-dimensional Schrodinger's Equation    341 
 Problem 1: Three-dimensional harmonic oscillator in Cartesian coordinates  341 
 Problem 2: Schrodinger's equation for spherically symmetric potentials  344 
 Problem 3: Particle in an infinite, spherical box  349 
 Problem 4: Particle with negative energy in a finite, spherical box  353 
 Problem 5: The hydrogen atom in spherical coordinates  359 
 Problem 6: Separation in cylindrical and paraboloidal coordinates  364 
6.3 Unsolved Problems    369 

CHAPTER 7 Relativity and Cosmology  371 
7.1 Introduction    371 
7.1.1 Special Relativity    371 
 The two basic postulates of special relativity  372 
 Lorentz transformations  372 
 Covariant equations and tensors  373 
 Cartesian coordinates and ``flat'' spacetime  373 
7.1.2 General Relativity and Cosmology    374 
 Spacetime metric  374 
 Field equations  374 
 Free-falling test particles and light trajectories  374 
 Robertson-Walker cosmology  375 
7.1.3   Commands    375 
 Packages  375 
 User-defined metric, boost, and velocity parameters  375 
 Metric  375 
 Rule for relativistic velocity parameters  376 
 Boost along the x-axis  376 
 User-defined geometric procedures  376 
 Christoffel symbols  376 
  Example: Christoffel symbols for a pseudo-Euclidean metric  377 
 Curvature tensor  377 
  Example: Curvature tensor for pseudo-Euclidean metric  378 
 Ricci tensor  378 
  Example: Ricci tensor for Go del metric  378 
 Killing's equations  379 
  Example: Killing vector equations in pseudo-Euclidean space  379 
 Einstein tensor  380 
  Example: Einstein tensor for wave metric  380 
 Geodesic equations  380 
  Example: Geodesics for a pseudo-Euclidean metric  381 
 User-defined metrics and Christoffel symbols  381 
 Schwarzschild metric  381 
 Kerr metric  382 
 Protect user-defined operators  384 
7.2 Problems    384 
7.2.1 Special Relativity Problems    384 
 Problem 1: Decay of a particle  384 
 Problem 2: Two-particle collision  385 
 Problem 3: Compton scattering  386 
 Problem 4: Moving mirror and generalized Snell's law  389 
 Problem 5: One-dimensional motion of a relativistic particle with constant acceleration  392 
 Problem 6: Two-dimensional motion of a relativistic particle in a uniform electric field  395 
7.2.2 General Relativity and Cosmology    398 
 Problem 1: Schwarzschild solution in null coordinates  398 
 Problem 2: The horizons and surfaces of infinite redshift  400 
 Problem 3: Killing vectors and constants of motion  402 
 Problem 4: Potential analysis for timelike geodesics  405 
 Problem 5: Time it takes to fall into a black hole  408 
 Problem 6: Circular geodesics for the Schwarzschild metric  414 
 Problem 7: Field equations for Robertson-Walker cosmology  416 
 Problem 8: Zero-pressure cosmological models  421 
 Problem 9: The expansion and age of the standard model  424 
7.3 Unsolved Problems    430 
 Index 433 

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