Computer Algebra Recipes for Mathematical Physics

401 Pages·2005·3.845 MB·English

by  Richard H. Enns

This mathematical physics contribution
to the Computer Algebra Recipes series
is dedicated to my wife Karen,
who lights my path through life.
Richard H. Enns
Computer Algebra Recipes
for Mathematical Physics
Birkha¨user
Boston • Basel • Berlin
RichardH.Enns
SimonFraserUniversity
DepartmentofPhysics
Burnaby,B.C.V5A1S6
Canada
AMS Subject Classifications (2000): 15A90, 30-XX, 33-XX, 34-XX, 35-XX, 35Qxx, 40-XX,
42-XX,44-XX,49-XX,65-XX,68-XX,70-XX,97U50
ISBN0-8176-3223-9 Printedonacid-freepaper.
(cid:1)c2005Birkha¨userBoston
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Preface
Thisbookisaself-containedguidetoproblem-solvingandexplorationinmath-
ematicalphysicsusingthepowerfulMaple9.5computeralgebrasystem(CAS).
WithaCASonecannotonlycrunchnumbersandplotresults,butalsocarryout
the symbolic manipulations which form the backbone of mathematical physics.
The heart of this text consists of over 230 useful and stimulating “classic”
computer algebra worksheets or recipes, which are systematically organized to
cover the major topics presented in the standard Mathematical Physics course
offeredtothirdorfourthyearundergraduatephysicsandengineeringstudents.
The emphasis here is on applications, with only a brief summary of the un-
derlying theoretical ideas being presented. The aim is to show how computer
algebra can not only implement the methods of mathematical physics quickly,
accurately, and efficiently, but can be used to explore more complex examples
which are tedious or difficult or even impossible to implement by hand.
The recipes are grouped into three sections, the introductory Appetizers
dealing with linear ordinary differential equations (ODEs), series, vectors, and
matrices. ThemoreadvancedEntreescoverlinearpartialdifferentialequations
(PDEs), scalar and vector fields, complex variables, integral transforms, and
calculus of variations. Finally, in the Desserts the emphasis is on presenting
some analytic, graphical, and numerical techniques for solving nonlinear ODEs
andPDEs. ThenumericalmethodsarealsoappliedtolinearODEsandPDEs.
NopriorknowledgeofMapleisassumedinthistext, therelevantcommand
structures being introduced on a need-to-know basis. The recipes are thor-
oughly annotated and, on numerous occasions, presented in a “story” format
orinahistoricalcontext. Eachrecipetakesthereaderfromtheanalyticformu-
lationorstatementofarepresentativetypeofmathematicalphysicsproblemto
itsanalyticornumericalsolutionandtoagraphicalvisualizationoftheanswer,
where relevant. The graphical representations vary from static 2-dimensional
pictures, to contour and vector field plots, to 3-dimensional graphs that can
be rotated, to animations in time. For your convenience, all 230 recipes are
included on the accompanying CD.
The range of mathematical physics problems that can be solved with the
enclosedrecipesisonlylimitedbyyourimagination. Byalteringtheparameter
values,orinitialconditions,orequationstructure,thousandsofotherproblems
can be easily generated and solved. “What if?” questions become answerable.
This should prove extremely useful to instructor and student alike.
Contents
Preface v
INTRODUCTION 1
A. Computer Algebra Systems . . . . . . . . . . . . . . . . . . . . . . 1
B. Computer Algebra Recipes . . . . . . . . . . . . . . . . . . . . . . . 2
C. Maple Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
D. Introductory Recipes . . . . . . . . . . . . . . . . . . . . . . . . . . 4
D.1 A Dangerous Ride?. . . . . . . . . . . . . . . . . . . . . . . . . 4
D.2 The Patrol Route of Bertie Bumblebee . . . . . . . . . . . . . . 7
E. How to Use this Text . . . . . . . . . . . . . . . . . . . . . . . . . . 10
I THE APPETIZERS 11
1 Linear ODEs of Physics 13
1.1 Linear ODEs with Constant Coefficients . . . . . . . . . . . . . . 13
1.1.1 Dazzling Dsolve Debuts . . . . . . . . . . . . . . . . . . . 14
1.1.2 The Tale of the Turbulent Tail . . . . . . . . . . . . . . . 18
1.1.3 This Bar Doesn’t Serve Drinks . . . . . . . . . . . . . . . 21
1.1.4 Shake, Rattle, and Roll . . . . . . . . . . . . . . . . . . . 25
1.1.5 “Resonances”, A Recipe by I. M. Curious . . . . . . . . . 27
1.1.6 Mr. Dirac’s Famous Function . . . . . . . . . . . . . . . . 32
1.2 Linear ODEs with Variable Coefficients . . . . . . . . . . . . . . 36
1.2.1 Introducing the Sturm–Liouville Family . . . . . . . . . . 36
1.2.2 Onset of Bending in a Vertical Antenna . . . . . . . . . . 42
1.2.3 The Quantum Oscillator . . . . . . . . . . . . . . . . . . . 44
1.2.4 Going Green, the Mathematician’s Way . . . . . . . . . . 48
1.2.5 In Search of a More Stable Existence . . . . . . . . . . . . 51
1.3 Supplementary Recipes . . . . . . . . . . . . . . . . . . . . . . . 54
01-S01 Newton’s Law of Cooling . . . . . . . . . . . . . . . . . . 54
01-S02 Charging a Capacitor . . . . . . . . . . . . . . . . . . . . 55
01-S03 Radioactive Chain . . . . . . . . . . . . . . . . . . . . . . 55
viii CONTENTS
01-S04 Newton and Stokes Join Forces . . . . . . . . . . . . . . . 55
01-S05 Exploring the RLC Series Circuit . . . . . . . . . . . . . . 55
01-S06 The Whirling Bar Revisited . . . . . . . . . . . . . . . . . 56
01-S07 Driven Coupled Oscillators . . . . . . . . . . . . . . . . . 56
01-S08 Some Properties of the Delta Function . . . . . . . . . . . 56
01-S09 A Green Function . . . . . . . . . . . . . . . . . . . . . . 56
01-S10 A Potpourri of General Solutions . . . . . . . . . . . . . . 57
01-S11 Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . 57
01-S12 The Growing Pendulum . . . . . . . . . . . . . . . . . . . 57
01-S13 Another Green Function . . . . . . . . . . . . . . . . . . . 58
01-S14 Going Green, Once Again . . . . . . . . . . . . . . . . . . 58
2 Applications of Series 59
2.1 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.1.1 Polynomial Approximations . . . . . . . . . . . . . . . . . 59
2.1.2 Finite Difference Approximations . . . . . . . . . . . . . . 61
2.2 Series Solutions of LODEs . . . . . . . . . . . . . . . . . . . . . . 65
2.2.1 Jennifer Renews an Old Acquaintance . . . . . . . . . . . 65
2.2.2 Another Old Acquaintance . . . . . . . . . . . . . . . . . 68
2.3 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.3.1 Madeiran Levadas and the Gibb’s Phenomenon . . . . . . 74
2.3.2 Sine or Cosine Series? . . . . . . . . . . . . . . . . . . . . 76
2.3.3 How Sweet This Is! . . . . . . . . . . . . . . . . . . . . . . 78
2.4 Summing Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.4.1 I. M. Curious Sums a Series . . . . . . . . . . . . . . . . . 80
2.4.2 Spiegel’s Series Problem . . . . . . . . . . . . . . . . . . . 82
2.5 Supplementary Recipes . . . . . . . . . . . . . . . . . . . . . . . 84
02-S01 Euler and Bernoulli Numbers . . . . . . . . . . . . . . . . 84
02-S02 Ms. Curious Approximates an Integral . . . . . . . . . . . 84
02-S03 More Finite Difference Approximations . . . . . . . . . . 84
02-S04 Series Solution . . . . . . . . . . . . . . . . . . . . . . . . 85
02-S05 Chebyshev Polynomials Revisited . . . . . . . . . . . . . . 85
02-S06 A Fourier Series . . . . . . . . . . . . . . . . . . . . . . . 85
02-S07 Fourier Sine Series . . . . . . . . . . . . . . . . . . . . . . 85
02-S08 Fourier Cosine Series . . . . . . . . . . . . . . . . . . . . . 85
02-S09 Legendre Series . . . . . . . . . . . . . . . . . . . . . . . . 86
02-S10 Directly Evaluating Series Sums . . . . . . . . . . . . . . 86
02-S11 Another Cosine Series . . . . . . . . . . . . . . . . . . . . 86
02-S12 The Complex Series Trick Again . . . . . . . . . . . . . . 86
3 Vectors and Matrices 87
3.1 Vectors: Cartesian Coordinates . . . . . . . . . . . . . . . . . . . 87
3.1.1 Bobby Blowfly . . . . . . . . . . . . . . . . . . . . . . . . 88
3.1.2 Hiking in the Southern Chilkotin . . . . . . . . . . . . . . 90
3.1.3 Establishing These Identities is Easy . . . . . . . . . . . . 94
CONTENTS ix
3.1.4 This Task is Not a Chore . . . . . . . . . . . . . . . . . . 95
3.2 Vectors: Curvilinear Coordinates . . . . . . . . . . . . . . . . . . 98
3.2.1 From Scale Factors to Vector Operators . . . . . . . . . . 98
3.2.2 Vector Operators the Easy Way. . . . . . . . . . . . . . . 100
3.2.3 These Operators Do Not Have an Identity Crisis . . . . . 102
3.2.4 Is This Vector Field Conservative? . . . . . . . . . . . . . 103
3.2.5 The Divergence Theorem . . . . . . . . . . . . . . . . . . 105
3.3 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.3.1 Some Matrix Basics . . . . . . . . . . . . . . . . . . . . . 109
3.3.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . 111
3.3.3 Diagonalizing a Matrix. . . . . . . . . . . . . . . . . . . . 115
3.3.4 Orthogonal and Unitary Matrices . . . . . . . . . . . . . . 117
3.3.5 Introducing the Euler Angles . . . . . . . . . . . . . . . . 119
3.4 Supplementary Recipes . . . . . . . . . . . . . . . . . . . . . . . 122
03-S01 Bobby Blowfly Seeks a Warmer Clime . . . . . . . . . . . 122
03-S02 Jennifer’s Vector Assignment . . . . . . . . . . . . . . . . 122
03-S03 Another Vector Operator Identity . . . . . . . . . . . . . 123
03-S04 Another Maple Approach . . . . . . . . . . . . . . . . . . 123
03-S05 Conservative or Non-conservative? . . . . . . . . . . . . . 123
03-S06 Basic Matrix Operations . . . . . . . . . . . . . . . . . . . 123
03-S07 The Cayley–Hamilton Theorem . . . . . . . . . . . . . . . 123
03-S08 Simultaneous Diagonalization . . . . . . . . . . . . . . . . 124
03-S09 Orthonormal Vectors . . . . . . . . . . . . . . . . . . . . . 124
03-S10 Stokes’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 124
03-S11 Solving Linear Equation Systems . . . . . . . . . . . . . . 124
II THE ENTREES 125
4 Linear PDEs of Physics 127
4.1 Three Cheers for the String . . . . . . . . . . . . . . . . . . . . . 128
4.1.1 Jennifer Finds the General Solution . . . . . . . . . . . . 128
4.1.2 Daniel Separates Strings: I Separate Variables . . . . . . 130
4.1.3 Daniel Strikes Again: Mr. Fourier Reappears . . . . . . . 133
4.1.4 The 3-Piece String . . . . . . . . . . . . . . . . . . . . . . 135
4.1.5 Encore? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.2 Beyond the String . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.2.1 Heaviside’s Telegraph Equation . . . . . . . . . . . . . . . 141
4.2.2 Spiegel’s Diffusion Problems . . . . . . . . . . . . . . . . . 143
4.2.3 Introducing Laplace’s Equation . . . . . . . . . . . . . . . 146
4.2.4 Grandpa’s “Trampoline”. . . . . . . . . . . . . . . . . . . 148
4.2.5 Irma Insect’s Isotherm . . . . . . . . . . . . . . . . . . . . 150
4.2.6 Daniel Hits Middle C . . . . . . . . . . . . . . . . . . . . 152
4.2.7 A Poisson Recipe . . . . . . . . . . . . . . . . . . . . . . . 155
4.3 Beyond Cartesian Coordinates . . . . . . . . . . . . . . . . . . . 158
x CONTENTS
4.3.1 Is It Separable? . . . . . . . . . . . . . . . . . . . . . . . . 158
4.3.2 A Shell Problem, Not a Shell Game . . . . . . . . . . . . 162
4.3.3 The Little Drummer Boy . . . . . . . . . . . . . . . . . . 166
4.3.4 The Cannon Ball . . . . . . . . . . . . . . . . . . . . . . . 169
4.3.5 Variation on a Split-sphere Potential . . . . . . . . . . . . 171
4.3.6 Another Poisson Recipe . . . . . . . . . . . . . . . . . . . 174
4.4 Supplementary Recipes . . . . . . . . . . . . . . . . . . . . . . . 178
04-S01 General Solutions. . . . . . . . . . . . . . . . . . . . . . . 178
04-S02 Balalaika Blues . . . . . . . . . . . . . . . . . . . . . . . . 179
04-S03 Damped Oscillations . . . . . . . . . . . . . . . . . . . . . 179
04-S04 Kids Will Be Kids . . . . . . . . . . . . . . . . . . . . . . 179
04-S05 Energy of a Vibrating String . . . . . . . . . . . . . . . . 179
04-S06 Vibrations of a Tapered String . . . . . . . . . . . . . . . 180
04-S07 Green Function for Forced Vibrations . . . . . . . . . . . 180
04-S08 Plane-wave Propagation in a 5-Piece String . . . . . . . . 180
04-S09 Transverse Vibrations of a Whirling String . . . . . . . . 180
04-S10 Newton Would Think That This Recipe Is Cool. . . . . . 181
04-S11 Locomotive on a Bridge . . . . . . . . . . . . . . . . . . . 181
04-S12 The Temperature Switch . . . . . . . . . . . . . . . . . . 181
04-S13 Telegraph Equation Revisited . . . . . . . . . . . . . . . . 181
04-S14 Another “Trampoline” Example . . . . . . . . . . . . . . 182
04-S15 An Electrostatic Poisson Problem . . . . . . . . . . . . . 182
04-S16 SHE Does Not Want to Separate . . . . . . . . . . . . . . 182
04-S17 WE Can Separate . . . . . . . . . . . . . . . . . . . . . . 182
04-S18 The Stark Effect . . . . . . . . . . . . . . . . . . . . . . . 183
04-S19 Annular Temperature Distribution . . . . . . . . . . . . . 183
04-S20 Split-boundary Temperature Problem . . . . . . . . . . . 183
04-S21 Fluid Flow Around a Sphere . . . . . . . . . . . . . . . . 183
04-S22 Sound of Music? . . . . . . . . . . . . . . . . . . . . . . . 183
5 Complex Variables 185
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.1.1 Jennifer Tests Basics . . . . . . . . . . . . . . . . . . . . . 186
5.1.2 The Stream Function . . . . . . . . . . . . . . . . . . . . 188
5.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.2.1 Jennifer Tests Cauchy’s Theorem . . . . . . . . . . . . . . 191
5.2.2 Cauchy’s Residue Theorem . . . . . . . . . . . . . . . . . 193
5.3 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.3.1 Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . . 196
5.3.2 Poles on the Contour. . . . . . . . . . . . . . . . . . . . . 198
5.3.3 An Angular Integral . . . . . . . . . . . . . . . . . . . . . 201
5.3.4 A Branch Cut. . . . . . . . . . . . . . . . . . . . . . . . . 202
5.4 Laurent Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5.4.1 Ms. Curious Meets Mr. Laurent . . . . . . . . . . . . . . 205
5.4.2 Converge or Diverge? . . . . . . . . . . . . . . . . . . . . 207
CONTENTS xi
5.5 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . 208
5.5.1 Field Around a Semi-infinite Plate . . . . . . . . . . . . . 209
5.5.2 A Clever Transformation . . . . . . . . . . . . . . . . . . 212
5.5.3 Schwarz–Christoffel Transformation . . . . . . . . . . . . 216
5.6 Supplementary Recipes . . . . . . . . . . . . . . . . . . . . . . . 218
05-S01 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
05-S02 Fluid Flow Around a Cylinder . . . . . . . . . . . . . . . 218
05-S03 Constructing f(z) . . . . . . . . . . . . . . . . . . . . . . 219
05-S04 Analytic or Non-analytic? . . . . . . . . . . . . . . . . . . 219
05-S05 A Contour Integral . . . . . . . . . . . . . . . . . . . . . . 219
05-S06 A Higher-order Pole . . . . . . . . . . . . . . . . . . . . . 219
05-S07 Another Angular Integral . . . . . . . . . . . . . . . . . . 219
05-S08 A Removable Singularity . . . . . . . . . . . . . . . . . . 219
05-S09 Another Contour Integral . . . . . . . . . . . . . . . . . . 219
05-S10 Fluid Flow & Electric Field Around a Plate . . . . . . . . 219
05-S11 Another Branch Cut . . . . . . . . . . . . . . . . . . . . . 220
05-S12 Laurent Expansion . . . . . . . . . . . . . . . . . . . . . . 220
05-S13 Capacitor Edge Effects. . . . . . . . . . . . . . . . . . . . 220
6 Integral Transforms 221
6.1 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.1.1 Some Fourier Transform Shapes . . . . . . . . . . . . . . 223
6.1.2 A Northern Weenie Roast . . . . . . . . . . . . . . . . . . 225
6.1.3 Turn Off the Boob Tube and Concentrate . . . . . . . . . 227
6.1.4 Diffusive Heat Flow . . . . . . . . . . . . . . . . . . . . . 230
6.1.5 Deja Vu . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
6.2 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 234
6.2.1 Jennifer Consults Mr. Spiegel . . . . . . . . . . . . . . . . 235
6.2.2 Jennifer’s Heat Diffusion Problem . . . . . . . . . . . . . 236
6.2.3 Daniel Strikes Yet Again: Mr. Laplace Appears . . . . . . 238
6.2.4 Infinite-medium Green’s Function . . . . . . . . . . . . . 239
6.2.5 Our Field of Dreams . . . . . . . . . . . . . . . . . . . . . 241
6.3 Bromwich Integral and Contour Integration . . . . . . . . . . . . 243
6.3.1 Spiegel’s Transform Problem Revisited . . . . . . . . . . . 245
6.3.2 Ms. Curious’s Branch Point . . . . . . . . . . . . . . . . . 246
6.3.3 Cooling That Weenie Rod . . . . . . . . . . . . . . . . . . 249
6.4 Other Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
6.4.1 Meet the Hankel Transform . . . . . . . . . . . . . . . . . 253
6.5 Supplementary Recipes . . . . . . . . . . . . . . . . . . . . . . . 255
06-S01 Verifying the Convolution Theorem. . . . . . . . . . . . . 255
06-S02 Bandwidth Theorem . . . . . . . . . . . . . . . . . . . . . 255
06-S03 Solving an Integral Equation . . . . . . . . . . . . . . . . 255
06-S04 Verifying Parseval’s Theorem . . . . . . . . . . . . . . . . 255
06-S05 Heat Diffusion in a Copper Rod . . . . . . . . . . . . . . 255
06-S06 Solving Another Integral Equation . . . . . . . . . . . . . 255