Physics with Maple: computer algebra for mathematical methods in physics

610 Pages·2006·4.33 MB·English

by  Frank Y. Wang

Physics with Maple™
The Computer Algebra Resource for Mathematical Methods
in Physics
Frank Y. Wang
WILEY-VCH Verlag GmbH&Co. KGaA
December11, 2005
Contents
Preface XI
GuideforUsers XVII
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIX
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 BasicAlgebraandSolvingEquations . . . . . . . . . . . . . . . . . . . . . 10
1.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 DifferentialEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.1 ExactSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.2 SpecialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.3 NumericalSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5 VectorsandMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 OscillatoryMotion 41
2.1 SimpleHarmonicOscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 DampedOscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.1 Overdamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.2 Underdamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.3 CriticalDamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3 SinusoidallyDrivenOscillation . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4 PhaseSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3 CalculusofVariations 71
3.1 Euler–LagrangeEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 MathematicalExamples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3 SymmetryProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4 PrincipleofLeastAction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5 SystemswithManyDegreesofFreedom . . . . . . . . . . . . . . . . . . . . 86
3.6 ForceofConstraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
PhysicswithMaple™:TheComputerAlgebraResourceforMathematicalMethodsinPhysics.FrankY.Wang
Copyright©2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:3-527-40640-9
VI Contents
4 IntegrationofEquationsofMotion 101
4.1 LinearizationofEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 DoublePendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3 Central-forceProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3.1 KeplerProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3.2 CorrectionTerms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.4 MotionofaSymmetricTop . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.5 NonlinearOscillationandChaos . . . . . . . . . . . . . . . . . . . . . . . . 126
4.6 SummaryofLagrangianMechanics . . . . . . . . . . . . . . . . . . . . . . 134
5 OrthogonalFunctionsandExpansions 139
5.1 FourierSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.2 FourierIntegrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.3 OrthogonalFunctionsinCompleteSets . . . . . . . . . . . . . . . . . . . . 146
5.4 LegendrePolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.4.1 GeneratingFunctionandRodriguesFormula . . . . . . . . . . . . . 151
5.5 BesselFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.6 SummaryofSpecialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . 163
6 Electrostatics 169
6.1 Coulomb’sLaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.2 CurvilinearCoordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.2.1 SphericalCoordinates . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.2.2 CylindricalCoordinates . . . . . . . . . . . . . . . . . . . . . . . . 178
6.3 DifferentialVectorCalculus . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.4 ElectricPotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.4.1 Cavendish’sApparatusfortheInverseSquareLaw . . . . . . . . . . 186
6.4.2 MultipoleExpansion . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.5 ElectricFieldandEquipotential . . . . . . . . . . . . . . . . . . . . . . . . 197
7 Boundary-valueProblems 205
7.1 TheoryofPotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.2 MethodofImages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.3 Complex-variableTechniques . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.4 LaplaceEquationinCartesianCoordinates. . . . . . . . . . . . . . . . . . . 214
7.5 LaplaceEquationinSphericalCoordinates. . . . . . . . . . . . . . . . . . . 218
7.6 LaplaceEquationinCylindricalCoordinates. . . . . . . . . . . . . . . . . . 223
7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
8 Magnetostatics 235
8.1 MagneticForces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
8.2 Biot–SavartLaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
8.3 VectorPotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.4 ForceandTorqueonMagneticDipoles. . . . . . . . . . . . . . . . . . . . . 250
8.5 SummaryofElectromagnetisminStaticConditions . . . . . . . . . . . . . . 253
Contents VII
9 ElectricCircuits 259
9.1 ResistorsinSeriesandinParallel. . . . . . . . . . . . . . . . . . . . . . . . 259
9.2 Kirchhoff’sRules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
9.3 Direct-currentCircuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.3.1 RC Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
9.3.2 RLCircuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
9.3.3 RLC Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
9.3.4 LissajousFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
9.4 Alternating-currentCircuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
9.4.1 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
9.4.2 Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
10 Waves 283
10.1 WaveEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
10.2 VibratingString . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
10.3 SinusoidalWavesinLinearCombinations . . . . . . . . . . . . . . . . . . . 290
10.3.1 ComplexNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
10.3.2 FourierIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
10.3.3 UncertaintyPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . 292
10.4 GaussianWavePacket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
10.5 Two-dimensionalCircularMembrane . . . . . . . . . . . . . . . . . . . . . 301
10.6 ElectromagneticWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
10.6.1 Electric-dipoleRadiation . . . . . . . . . . . . . . . . . . . . . . . . 306
10.6.2 SynchrotronRadiation . . . . . . . . . . . . . . . . . . . . . . . . . 312
11 PhysicalOptics 321
11.1 LightasanElectromagneticWave . . . . . . . . . . . . . . . . . . . . . . . 321
11.1.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
11.2 MathematicsofInterference . . . . . . . . . . . . . . . . . . . . . . . . . . 325
11.3 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
11.3.1 Double-slitInterference . . . . . . . . . . . . . . . . . . . . . . . . 329
11.3.2 Multiple-slitInterference . . . . . . . . . . . . . . . . . . . . . . . . 330
11.4 Diffraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
11.4.1 ResolutionofSingleSlitsandCircularApertures . . . . . . . . . . . 336
11.5 DiffractionGrating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
11.6 FourierTransformSpectrometry . . . . . . . . . . . . . . . . . . . . . . . . 343
11.7 FresnelDiffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
12 SpecialRelativity 353
12.1 LorentzTransformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
12.1.1 LengthContractionandTimeDilation . . . . . . . . . . . . . . . . . 357
12.1.2 AdditionofVelocity . . . . . . . . . . . . . . . . . . . . . . . . . . 361
12.1.3 DopplerShift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
12.2 RelativisticKinematicsandDynamics . . . . . . . . . . . . . . . . . . . . . 367
12.3 TransformationsofElectromagneticFields. . . . . . . . . . . . . . . . . . . 373
VIII Contents
13 QuantumPhenomena 379
13.1 BlackbodyRadiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
13.2 PhotoelectricandComptonEffects . . . . . . . . . . . . . . . . . . . . . . . 383
13.3 Wave–ParticleDuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
13.4 BohrModeloftheHydrogenAtom. . . . . . . . . . . . . . . . . . . . . . . 387
13.5 DielectricsandParamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . 390
14 SchrödingerEquationinOneDimension(I):UnboundStates 401
14.1 FormulationofQuantumMechanics . . . . . . . . . . . . . . . . . . . . . . 401
14.2 ZeroPotentialandPlaneWaves . . . . . . . . . . . . . . . . . . . . . . . . 403
14.3 StepPotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
14.3.1 StepPotential(E >V0) . . . . . . . . . . . . . . . . . . . . . . . . 404
14.3.2 StepPotential(E <V0) . . . . . . . . . . . . . . . . . . . . . . . . 407
14.4 BarrierPotential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
14.4.1 BarrierPotential(E >V0) . . . . . . . . . . . . . . . . . . . . . . . 409
14.4.2 BarrierPotential(E <V0) . . . . . . . . . . . . . . . . . . . . . . . 410
14.5 SummaryofStationaryStates . . . . . . . . . . . . . . . . . . . . . . . . . 411
14.6 WavePacket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
14.6.1 ReflectionofWavePacket . . . . . . . . . . . . . . . . . . . . . . . 417
15 SchrödingerEquationinOneDimension(II):BoundStates 425
15.1 DiscreteSpectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
15.2 InfinitePotentialWell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
15.3 FinitePotentialWell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
15.4 SeriesSolutionandHermiteEquation . . . . . . . . . . . . . . . . . . . . . 436
15.5 LinearHarmonicOscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
15.6 HomogeneousField . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
15.7 MorsePotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
15.8 BoundNonstationaryStates . . . . . . . . . . . . . . . . . . . . . . . . . . 449
15.9 Two-stateSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
16 SchrödingerEquationinThreeDimensions 465
16.1 Central-forceProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
16.2 SphericalHarmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
16.3 AngularMomentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
16.4 CoulombPotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
16.5 HydrogenAtom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
16.5.1 ElectricPotentialDuetotheElectron . . . . . . . . . . . . . . . . . 492
16.5.2 HybridBondOrbitals. . . . . . . . . . . . . . . . . . . . . . . . . . 494
16.6 InfiniteSphericalWell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
17 QuantumStatistics 509
17.1 StatisticalDistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
17.2 Maxwell–BoltzmannStatistics . . . . . . . . . . . . . . . . . . . . . . . . . 512
Contents IX
17.3 IdealBoseGas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
17.3.1 LowDensityandVirialExpansion . . . . . . . . . . . . . . . . . . . 518
17.3.2 Bose–EinsteinCondensationatLowTemperature . . . . . . . . . . . 522
17.4 IdealFermiGas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
17.4.1 LowDensityandVirialExpansion . . . . . . . . . . . . . . . . . . . 530
17.4.2 SpecificHeatofaMetalatLowTemperature . . . . . . . . . . . . . 531
17.5 RelativisticGases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
18 GeneralRelativity 545
18.1 BasicFormulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
18.2 NewtonianLimit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
18.2.1 GravitationalRedshift . . . . . . . . . . . . . . . . . . . . . . . . . 551
18.3 SchwarzschildSolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
18.4 Robertson–WalkerMetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
18.4.1 EvolutionoftheUniverse . . . . . . . . . . . . . . . . . . . . . . . 565
Appendix
A PhysicalandAstrophysicalConstants 577
B MathematicalNotes 579
B.1 LegendreEquationandSeriesSolutions . . . . . . . . . . . . . . . . . . . . 579
B.2 WhittakerFunctionandHypergeometricSeries . . . . . . . . . . . . . . . . 583
B.2.1 HarmonicOscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 584
B.2.2 MorsePotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
B.2.3 CoulombPotential . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
B.3 Clausius–MossottiEquation . . . . . . . . . . . . . . . . . . . . . . . . . . 589
B.4 Bose–EinsteinIntegralFunction . . . . . . . . . . . . . . . . . . . . . . . . 592
B.5 EmbeddingFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
Index 599
Preface
Physicsisguidedbysimpleprinciples,butformanytopicsthephysicstendstobeobscuredin
theprofusionofmathematics.Asinteractivesoftwareforcomputeralgebra,Maple†canassist
educatorsand students to overcomethe obstacle of mathematicaldifficulties. The objective
ofthisbookistointroduceMaplebothforteachingandlearningphysicsbytakingadvantage
ofthemathematicalpowerofsymboliccomputation,sothatonecanconcentrateonapplying
theprinciplesofsettingequations,insteadoftechnicaldetailsofsolvingequations.
Mostphysicstextbookswerewrittenbeforeadvancedcomputersoftwarebecameconveniently
available.Theconventionalapproachtoatopicplacesemphasisontheoryandformalism,de-
voting manyparagraphsto performingalgebraicoperationsin derivingequationsmanually;
otherthansomewellknownexamples,mostapplicationsoftheoryareomitted. Onereason
that those examplesare well knownis that they admit analytic solution: they typically rep-
resent simplified situations that generally fail to fully reflect the reality. In most situations,
analytic solutions simply do not exist, and one cannot proceed without the assistance of a
computer. Althoughsomebookshavesectionsdiscussingnumericalmethods,manyofthem
contain just the theory of numerical methods, and one is required to possess programming
skillforpractice;thispartishencegenerallyneglected.Essentiallyallexperimentsinphysics
measurenumbers,soanyformulationmusteventuallybereducibletonumbers.Underacon-
ventional curriculum, a student’s ability to calculate and to extract numerical results from
the formalism is somehow inadequate. The result is not surprising: a studentmay be weak
in those areas, and he or she thus achievesonly partialcomprehensionbecause of technical
difficulties.
Maple can remedy some deficiency or weakness in traditional training. Using Maple, one
canmanipulateequationsanddiminishtediouspaperworkthatdistractsfromthemainfocus
of learningphysics. It is particularlyuseful in problemsthat require extensive calculations,
such as problems in calculus involving the chain rule, change of variables, and integration
by parts. Maple is such a powerfulsoftware that an educator can introducemore advanced
topicswithoutbeingrestricted to thepresumedmathematicalbackgroundof students, anda
studentcanexploremoreadvancedapplicationswithoutfearofmathematicaldifficulty.From
an analytic solution one can obtain numericalresults by substituting numerical values. For
equations that admit no analytic solution, one can, in practice, solve them numerically if
proper initial conditionsare supplied. An importantfeature of Maple is that it can produce
†MapleisaregisteredtrademarkofMaplesoft,adivisionofWaterlooMapleInc.;
see .
(cid:1)(cid:2)(cid:2)(cid:3)(cid:4)(cid:5)(cid:5)(cid:6)(cid:6)(cid:6)(cid:7)(cid:8)(cid:9)(cid:3)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:2)(cid:7)(cid:15)(cid:13)(cid:8)
PhysicswithMaple™:TheComputerAlgebraResourceforMathematicalMethodsinPhysics.FrankY.Wang
Copyright©2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:3-527-40640-9
XII Preface
instantgraphicstoserveasavisualguide,whichisgenerallythebestwaytounderstandthe
underlyingphysics.
TaketheBessel functionsasanexample: manystudentsandevenresearchersfind theseso-
calledspecialfunctionsalien,despitetheirfrequentrecurrenceindiversebranchesofphysics.
OnereasonisthatthetreatmentofBesselfunctionsrequiresnumericalmethods,withwhich
noteverybodyis adequatelyfamiliar. Many physicists have never produceda numberfrom
suchfunctions,andaworkedexampleofexpansioninBesselfunctionsislackingfrommost
commonlyused textbooks. Can we imagine a studentlearningthe relationsforthe trigono-
metricfunctionsandFourierserieswithouteverproducinganumber?Maplereducesroutines
oftheBesselfunctionstoasimplecommand,whichsparesonefromatediousandprotracted
processofprogramminganddebugging,andcanproduceplotsinteractively. Themosteffec-
tivewaytolearnsuchfunctionsistopractisethem,bycalculatingtheexpansioncoefficients,
similartothoseintheFourierseries,andobservingthegraphicaloutput.Maplecanservenot
onlyforpedagogicalpurposes:inpracticeitismuchmoreconvenienttoevaluatecoefficients
forexpansionsinorthogonalfunctionswithacomputerthanitisbylongmanualcalculations.
TherealreadyexistmanybooksonMaple,whichindicatesthatMapleisacommonbutintrin-
sicallycomplicatedsoftware. Maplecontainsliterallythousandsofcommandsandoperators,
fromthemostelementarytothequitecomplicated. Fewpeopleareproficientineveryaspect
ofthissoftware,noteventheauthor! Thepurposeofthisbookistousebasiccommandsin
Maple,so thatoneisnotdauntedbythesoftwareitself, todemonstratewhatcanbeaccom-
plished. From worked examples, a reader can develop a sense of knowing which problems
are amenableto the assistance of Maple. Thisbookis notintendedfor someonewho seeks
to explore diverse Maple commands; on the contrary, we generally limit ourselves to basic
ones. AlthoughMapleisapowerfulsoftware,itisnottheonlytoolnorisittheperfecttool
inmathematicalphysics. ForsomeproblemsMaplecanbeoftremendoushelp,whereasfor
othersan alternativeapproachmightbe more appropriate. Identifyingthe typesof problem
thatarewellsuitedtothecapabilityofMapleisanimportantskill,anditisthemainpurpose
ofthisvolume.
Thisbookisorganizedaccordingtothefieldsofphysics,coveringclassicalmechanics,elec-
tromagnetism, relativity, quantummechanicsand statistical mechanics. We select problems
that we consider suitable for Maple, and each is representative of its kind so that one can
modifyandadaptaworksheetforasimilarproblem. Ourphilosophyofsolvingproblemsis
toapplyMaple’scapabilitytoattackthemathematicsinadirectfashion,sothatweavoiddi-
gressionintointricatemathematicalmanipulation. Becausemostproblemsadmitnoanalytic
solution,weparticularlyemphasizeformingplotsbasedonnumericalsolution.Agraphicpre-
sentationofasolutionprovidesthemostenduringimpression,andbyexperimentingthrough
varying values of parameters and observing the graphic output, one can develop a sense of
intuitionandorderofmagnitude.Astrongphysicalintuitiontowardaproblemisarguablythe
mostimportantassetofanyphysicistorengineer.
Inourpresentation,therelevantformulationprecedeseachproblem;wedevoteparticularat-
tentiontosubjectsthatarelesscommonlypresentedinconventionaltextbooksbutarecrucial
for computation. Mathematicalformulasof the problemand results of calculationsare out-
Preface XIII
lined, omitting the details of intermediate steps. Our intentionis to guide the reader with a
clearmathematicalobjectivethroughconventionalequationsandtheirsymbols. Theomitted
portionofcalculationsislistedintheattachedMapleworksheet,withashortexplanationifit
isnotself-explanatory. Inaworksheet,weattempttoperformmostcalculationsusingbasic
Maplecommands.Otherthanusinga“FOR”loopinsomeexamples,wedonotexplicitlyuti-
lizetheprogrammingabilityofMaple. Insomesituations,rearrangementandsimplification
of an expression are done manually; we avoid unnatural steps in Maple that might confuse
readers. For most physical problems an alternative solution is practicable: we emphasize
directnessandconsistency,notelegance.
Mapleisinteractivesoftware, thuspresentingworksheetsina static formconstitutesa great
challenge. Because the feature of this book is to use simple commands to solve physical
problems,mostMapleplotsaregeneratedinthedefaultmode.Withoutoptionalcommandsto
refinetheplots,someofthemmightappearlesssatisfactory.Ourcompromiseisbasedonour
contentionthatcommandspurelyforgraphornamentationarelessimportantandpotentially
distracting for the purpose of this book. When one tries the worksheets on a computer, it
is easy to discern the plots. In the same spirit, we believe that no matter how detailed the
worksheetmay be, the best way to learn Maple is to experimentwith examples, and in this
process one naturally learns the commands which are new or unfamiliar to one when first
encounteredinprintedpages. Additionally,oneshouldtakeadvantageofthecomprehensive
indexofMaplecommandsusedinthisbook,whichgreatlyfacilitateslearningbyexamples.
The first chapter is an illustration of basic algebraic operations with Maple, through their
applicationtophysics. BecausemostMaplecommandsareeasytounderstand,wehopethat,
even if one is unfamiliar with this software, one can follow those examples and develop a
senseofthepotentialofMaple.
We begin our treatment of classical mechanics with oscillatory motion. Problems such as
solving a system of equations and solving differential equations with constant coefficients,
canbereadilyaccomplishedwithMaple.WethenintroduceLagrangianmechanics:thistopic
providesaperfectexampleforwhichMaplecanbeofgreatassistance. Therequiredmathe-
maticsinvolvesfindinga functionthatextremizesanintegral: thistypeofproblemis called
the calculus of variations, and calculations are typically extensive even for simple systems.
WedevelopinMapleamethodtoderivetheequationofmotionwithoutinvokinganexternal
library. We furtheruse Maple’s capabilityof solving differentialequations, symbolicallyor
numerically,tofindtheactualmotionofaparticle.Withthismethodwecanpracticallysolve
anyprobleminclassicalmechanicsforwhichtheLagrangianfunctionisknown.
Achapteronexpansioninorthogonalfunctionsservesasapreparationforsubsequentchap-
ters. We start with the Fourierseries; a task suchas calculatingthe Fouriercoefficientsis a
particularlyvaluableapplicationofMaple. ThereisanevengreaterbenefitinusingMaplefor
expansionsinvolvingotherorthogonalfunctions,mostnotablytheBesselfunctions:thelatter
topicisacommonweaknessamongstudents. Wepresentindetailmanyworkedexamplesto
demonstrateMaple’sgreatutilityforthispurpose.