Introduction to Symmetry Analysis Paperback with CD-ROM

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

ISBN: 9780521777407 | 0521777402
Cover: Multimedia
Copyright: 9/23/2002

Rent

(Recommended)

$92.02

Term

Due

Price

Short Term

Aug 13

$92.02

*This item is part of an exclusive publisher rental program and requires an additional convenience fee. This fee will be reflected in the shopping bag.
Buy New

Special Order: 1-2 Weeks

$108.79
eBook

Available Instantly

Online: 180 Days

Downloadable: 180 Days

*To support the delivery of the digital material to you, a digital delivery fee of $3.99 will be charged on each digital item.

$108.48*

Symmetry analysis based on Lie group theory is the most important method for solving nonlinear problems aside from numerical computation. The method can be used to find the symmetries of almost any system of differential equations and the knowledge of these symmetries can be used to reduce the complexity of physical problems governed by the equations. This is a broad, self-contained, introduction to the basics of symmetry analysis for first and second year graduate students in science, engineering and applied mathematics. Mathematica-based software for finding the Lie point symmetries and Lie-Bäcklund symmetries of differential equations is included on a CD along with more than forty sample notebooks illustrating applications ranging from simple, low order, ordinary differential equations to complex systems of partial differential equations. MathReader 4.0 is included to let the user read the sample notebooks and follow the procedure used to find symmetries.

Author's Preface

xvii

Historical Preface

xxi

Rise of the Academies

xxi

Abel and Galois

xxiv

Lie and Klein

xxvi

1870

xxviii

Lie's Arrest

xxix

Gauss, Riemann, and the New Geometry

xxx

The Erlangen Program

xxxiv

Lie's Career at Leipzig

xxxvi

A Falling Out

xxxvi

Lie's Final Return to Norway

xxxvii

After 1900

xxxviii

The Ariadne Thread

xxxix

Suggested Reading

Introduction to Symmetry

(32)

Symmetry in Nature

(2)

Some Background

(1)

The Discrete Symmetries of Objects

(6)

The Twelvefold Discrete Symmetry Group of a Snowflake

(6)

The Principle of Covariance

(2)

Continuous Symmetries of Functions and Differential Equations

(11)

One-Parameter Lie Groups in the Plane

(1)

Invariance of Functions, ODEs, and PDEs under Lie Groups

(9)

Some Notation Conventions

(2)

Concluding Remarks

(4)

Exercises

(4)

References

(2)

Dimensional Analysis

(22)

Introduction

(1)

The Two-Body Problem in a Gravitational Field

(4)

The Drag on a Sphere

(6)

Some Further Physical Considerations

(1)

The Drag on a Sphere in High-Speed Gas Flow

(3)

Buckingham's Pi Theorem -- The Dimensional-Analysis Algorithm

(3)

Concluding Remarks

(1)

Exercises

(5)

References

(1)

Systems of ODEs and First-Order PDEs; State-Space Analysis

(41)

Autonomous Systems of ODEs in the Plane

(1)

Characteristics

(1)

First-Order Ordinary Differential Equations

(8)

Perfect Differentials

(2)

The Integrating Factor; Pfaff's Theorem

(1)

Nonsolvability of the Integrating Factor

(2)

Examples of Integrating Factors

(3)

Thermodynamics; The Legendre Transformation

(3)

Incompressible Flow in Two Dimensions

(1)

Fluid Flow in Three Dimensions -- The Dual Stream Function

(5)

The Method of Lagrange

(2)

The Integrating Factor in Three and Higher Dimensions

(1)

Incompressible Flow in Three Dimensions

(1)

Nonlinear First-Order PDEs -- The Method of Lagrange and Charpit

(5)

The General and Singular Solutions

(2)

Characteristics in n Dimensions

(4)

Nonlinear First-Order PDEs in n Dimensions

(1)

State-Space Analysis in Two and Three Dimensions

(7)

Critical Points

(1)

Matrix Invariants

(1)

Linear Flows in Two Dimensions

(3)

Linear Flows in Three Dimensions

(2)

Concluding Remarks

(1)

Exercises

(5)

References

(1)

Classical Dynamics

(25)

Introduction

(2)

Hamilton's Principle

(3)

Hamilton's Equations

101

(4)

Poisson Brackets

103

(2)

The Hamilton--Jacobi Equation

105

(3)

Examples

108

(11)

Concluding Remarks

119

(1)

Exercises

119

(2)

References

120

(1)

Introduction to One-Parameter Lie Groups

121

(28)

The Symmetry of Functions

121

(1)

An Example and a Counterexample

122

(1)

Translation along Horizontal Lines

122

(1)

A Reflection and a Translation

123

(1)

One-Parameter Lie Groups

123

(2)

Invariant Functions

125

(2)

Infinitesimal Form of a Lie Group

127

(1)

Lie Series, the Group Operator, and the Infinitesimal Invariance Condition for Functions

128

(2)

Group Operators and Vector Fields

130

(1)

Solving the Characteristic Equation Xφ[x] = 0

130

(2)

Invariant Points

132

(1)

Reconstruction of a Group from Its Infinitesimals

132

(2)

Multiparameter Groups

134

(2)

The Commutator

135

(1)

Lie Algebras

136

(3)

The Commutator Table

137

(1)

Lie Subalgebras

138

(1)

Abelian Lie Algebras

138

(1)

Ideal Lie Subalgebras

139

(1)

Solvable Lie Algebras

139

(3)

Some Remarks on Lie Algebras and Vector Spaces

142

(3)

Concluding Remarks

145

(1)

Exercises

145

(4)

References

148

(1)

First-Order Ordinary Differential Equations

149

(29)

Invariant Families

149

(3)

Invariance Condition for a Family

152

(3)

First-Order ODEs; The Integrating Factor

155

(2)

Using Groups to Integrate First-Order ODEs

157

(5)

Canonical Coordinates

162

(4)

Invariant Solutions

166

(6)

Elliptic Curves

172

(2)

Criterion for a First-Order ODE to Admit a Given Group

174

(2)

Concluding Remarks

176

(1)

Exercises

176

(2)

References

177

(1)

Differential Functions and Notation

178

(13)

Introduction

179

(8)

Superscript Notation for Dependent and Independent Variables

181

(1)

Subscript Notation for Derivatives

181

(2)

Curly-Brace Subscript Notation for Functions That Transform Derivatives

183

(1)

The Total Differentiation Operator

184

(1)

Definition of a Differential Function

185

(1)

Total Differentiation of Differential Functions

186

(1)

Contact Conditions

187

(2)

One Dependent and One Independent Variable

187

(1)

Several Dependent and Independent Variables

188

(1)

Concluding Remarks

189

(1)

Exercises

190

(1)

References

190

(1)

Ordinary Differential Equations

191

(46)

Extension of Lie Groups in the Plane

191

(10)

Finite Transformation of First Derivatives

191

(2)

The Extended Transformation Is a Group

193

(2)

Finite Transformation of the Second Derivative

195

(1)

Finite Transformation of Higher Derivatives

195

(2)

Infinitesimal Transformation of the First Derivative

197

(1)

Infinitesimal Transformation of the Second Derivative

198

(1)

Infinitesimal Transformation of Higher-Order Derivatives

199

(1)

Invariance of the Contact Conditions

200

(1)

Expansion of an ODE in a Lie Series; The Invariance Condition for ODEs

201

(2)

What Does It Take to Transform a Derivative?

202

(1)

Group Analysis of Ordinary Differential Equations

203

(1)

Failure to Solve for the Infinitesimals That Leave a First-Order ODE Invariant

203

(1)

Construction of the General First-Order ODE That Admits a Given Group -- The Ricatti Equation

204

(3)

Second-Order ODEs and the Determining Equations of the Group

207

(6)

Projective Group of the Simplest Second-Order ODE

208

(5)

Construction of the General Second-Order ODE that Admits a Given Group

213

(1)

Higher-Order ODEs

213

(2)

Construction of the General pth-Order ODE That Admits a Given Group

214

(1)

Reduction of Order by the Method of Canonical Coordinates

215

(1)

Reduction of Order by the Method of Differential Invariants

216

(1)

Succesive Reduction of Order; Invariance under a Multiparameter Group with a Solvable Lie Algebra

217

(11)

Two-Parameter Group of the Blasius Equation

219

(9)

Group Interpretation of the Method of Variation of Parameters

228

(5)

Reduction to Quadrature

230

(1)

Solution of the Homogeneous Problem

231

(2)

Concluding Remarks

233

(1)

Exercises

233

(4)

References

236

(1)

Partial Differential Equations

237

(40)

Finite Transformation of Partial Derivatives

238

(9)

Finite Transformation of the First Partial Derivative

238

(1)

Finite Transformation of Second and Higher Partial Derivatives

239

(2)

Variable Count

241

(1)

Infinitesimal Transformation of First Partial Derivatives

242

(1)

Infinitesimal Transformation of Second and Higher Partial Derivatives

243

(2)

Invariance of the Contact Conditions

245

(2)

Expansion of a PDE in a Lie Series -- Invariance Condition for PDEs

247

(8)

Isolating the Determining Equations of the Group -- The Lie Algorithm

247

(1)

The Classical Point Group of the Heat Equation

248

(7)

Invariant Solutions and the Characteristic Function

255

(2)

Impulsive Source Solutions of the Heat Equation

257

(6)

A Modified Problem of an Instantaneous Heat Source

263

(6)

Nonclassical Symmetries

269

(3)

A Non-classical Point Group of the Heat Equation

270

(2)

Concluding Remarks

272

(1)

Exercises

273

(4)

References

275

(2)

Laminar Boundary Layers

277

(41)

Background

277

(2)

The Boundary-Layer Formulation

279

(2)

The Blasius Boundary Layer

281

(12)

Similarity Variables

282

(2)

Reduction of Order; The Phase Plane

284

(7)

Numerical Solution of the Blasius Equation as a Cauchy Initial-Value Problem

291

(2)

Temperature Gradient Shocks in Nonlinear Diffusion

293

(8)

First Try: Solution of a Cauchy Initial-Value Problem -- Uniqueness

294

(1)

Second Try: Solution Using Group Theory

295

(3)

The Solution

298

(1)

Exact Thermal Analogy of the Blasius Boundary Layer

299

(2)

Boundary Layers with Pressure Gradient

301

(4)

The Falkner--Skan Boundary Layers

305

(8)

Falkner--Skan Sink Flow

310

(3)

Concluding Remarks

313

(1)

Exercises

313

(5)

References

317

(1)

Incompressible Flow

318

(46)

Invariance Group of the Navier--Stokes Equations

318

(3)

Frames of Reference

321

(2)

Two-Dimensional Viscous Flow

323

(2)

Viscous Flow in a Diverging Channel

325

(4)

Transition in Unsteady Jets

329

(24)

The Impulse Integral

320

(5)

Starting-Vortex Formation in an Impulsively Started Jet

325

(28)

Elliptic Curves and Three-Dimensional Flow Patterns

353

(4)

Acceleration Field in the Round Jet

353

(4)

Classification of Falkner--Skan Boundary Layers

357

(3)

Concluding Remarks

360

(1)

Exercises

361

(3)

References

362

(2)

Compressible Flow

364

(31)

Invariance Group of the Compressible Euler Equations

365

(4)

Isentropic Flow

369

(1)

Sudden Expansion of a Gas Cloud into a Vacuum

370

(5)

The Gasdynamic--Shallow-Water Analogy

371

(1)

Solutions

372

(3)

Propagation of a Strong Spherical Blast Wave

375

(9)

Effect of the Ratio of Specific Heats

382

(2)

Compressible Flow Past a Thin Airfoil

384

(7)

Subsonic Flow, M∞ < 1

386

(3)

Supersonic Similarity, M∞ > 1

389

(1)

Transonic Similarity, M∞ ~ 1

390

(1)

Concluding Remarks

391

(1)

Exercises

392

(3)

References

393

(2)

Similarity Rules for Turbulent Shear Flows

395

(51)

Introduction

395

(2)

Reynolds-Number Invariance

397

(4)

Group Interpretation of Reynolds-Number Invariance

401

(9)

One-Parameter Flows

401

(2)

Temporal Similarity Rules

403

(1)

Frames of Reference

404

(1)

Spatial Similarity Rules

405

(2)

Reynolds Number

407

(1)

Fine-Scale Motions

408

(2)

The Inertial Subrange

410

(14)

Application: Experiment to Measure Small Scales in a Turbulent Vortex Ring

413

(2)

Similarity Rules for the Turbulent Vortex Ring

415

(2)

Particle Paths in the Turbulent Vortex Ring

417

(2)

Estimates of Microscales

419

(1)

Vortex-Ring Formation

420

(2)

Apparatus Design

422

(2)

The Geometry of Dissipating Fine-Scale Motion

424

(13)

Transport Equation for the Velocity Gradient Tensor

425

(12)

Concluding Remarks

437

(1)

Exercises

438

(8)

References

444

(2)

Lie--Backlund Transformations

446

(52)

Lie--Backlund Transformations -- Infinite Order Structure

447

(6)

Infinitesimal Lie--Backlund Transformation

449

(3)

Reconstruction of the Finite Lie--Backlund Transformation

452

(1)

Lie Contact Transformations

453

(4)

Contact Transformations and the Hamilton--Jacobi Equation

457

(1)

Equivalence Classes of Transformations

457

(7)

Every Lie Point Operator Has an Equivalent Lie--Backlund Operator

459

(1)

Equivalence of Lie--Backlund Transformations

459

(2)

Equivalence of Lie--Backlund and Lie Contact Operators

461

(1)

The Extended Infinitesimal Lie--Backlund Group

461

(1)

Proper Lie--Backlund Transformations

462

(1)

Lie Series Expansion of Differential Functions and the Invariance Condition

462

(2)

Applications of Lie--Backlund Transformations

464

(14)

Third-Order ODE Governing a Family of Parabolas

466

(5)

The Blasius Equation yxxx + yyxx = 0

471

(3)

A Particle Moving Under the Influence of a Spherically Symmetric Inverse-Square Body Force

474

(4)

Recursion Operators

478

(18)

Linear Equations

479

(2)

Nonlinear Equations

481

(15)

Concluding Remarks

496

(1)

Exercises

496

(2)

References

497

(1)

Variational Symmetries and Conservation Laws

498

(17)

Introduction

498

(6)

Transformation of Integrals by Lie--Backlund Groups

499

(1)

Transformation of the Differential Volume

499

(1)

Invariance Condition for Integrals

500

(4)

Examples

504

(7)

Concluding Remarks

511

(1)

Exercises

512

(3)

References

513

(2)

Backlund Transformations and Nonlocal Groups

515

(37)

Two Classical Examples

517

(4)

The Liouville Equation

517

(2)

The Sine--Gordon Equation

519

(2)

Symmetries Derived from a Potential Equation; Nonlocal Symmetries

521

(26)

The General Solution of the Burgers Equation

522

(11)

Solitary-Wave Solutions of the Korteweg--de Vries Equation

533

(14)

Concluding Remarks

547

(1)

Exercises

548

(4)

References

550

(2)

Appendix 1 Review of Calculus and the Theory of Contact

552

(6)

A1.1 Differentials and the Chain Rule

552

(3)

A1.1.1 A Problem with Notation

553

(1)

A1.1.2 The Total Differentiation Operator

554

(1)

A1.1.3 The Inverse Total Differentiation Operator

555

(1)

A1.2 The Theory of Contact

555

(3)

A1.2.1 Finite-Order Contact between a Curve and a Surface

555

(3)

Appendix 2 Invariance of the Contact Conditions under Lie Point Transformation Groups

558

(11)

A2.1 Preservation of Contact Conditions -- One Dependent and One Independent Variable

558

(4)

A2.1.1 Invariance of the First-Order Contact Condition

558

(1)

A2.1.2 Invariance of the Second-Order Contact Condition

559

(1)

A2.1.3 Invariance of Higher-Order Contact Conditions

560

(2)

A2.2 Preservation of the Contact Conditions -- Several Dependent and Independent Variables

562

(7)

A2.2.1 Invariance of the First-Order Contact Condition

562

(2)

A2.2.2 Invariance of the Second-Order Contact Conditions

564

(2)

A2.2.3 Invariance of Higher-Order Contact Conditions

566

(3)

Appendix 3 Infinite-Order Structure of Lie--Backlund Transformations

569

(15)

A3.1 Lie Point Groups

569

(1)

A3.2 Lie--Backlund Groups

570

(1)

A3.3 Lie Contact Transformations

570

(4)

A3.3.1 The Case m > 1

573

(1)

A3.3.2 The Case m = 1

573

(1)

A3.4 Higher-Order Tangent Transformation Groups

574

(6)

A3.5 One Dependent Variable and One Independent Variable

580

(1)

A3.6 Infinite-Order Structure

581

(3)

A3.6.1 Infinitesimal Transformation

582

(1)

Reference

583

(1)

Appendix 4 Symmetry Analysis Software

584

(17)

A4.1 Summary of the Theory

586

(2)

A4.2 The Program

588

(6)

A4.2.1 Getting Started

589

(2)

A4.2.2 Using the Program

591

(2)

A4.2.3 Solving the Determining Equations and Viewing the Results

593

(1)

A4.3 Timing, Memory and Saving Intermediate Data

594

(7)

A4.3.1 Why Give the Output in the Form of Strings?

597

(1)

A4.3.2 Summary of Program Functions

597

(3)

References

600

(1)

Author Index

601

(3)

Subject Index

604

Please wait while the item is added to your bag...

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Introduction to Symmetry Analysis Paperback with CD-ROM

Summary

Table of Contents

Supplemental Materials

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Introduction to Symmetry Analysis Paperback with CD-ROM

Summary

Table of Contents

Supplemental Materials

Digital License