Theoretical Mechanics of Particles and Continua

, by Fetter, Alexander L.; Walecka, John Dirk

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ISBN: 9780486432618 | 0486432610
Cover: Paperback
Copyright: 12/16/2003

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This two-part text supplies a lucid, self-contained account of classical mechanics and provides a natural framework for introducing advanced mathematical concepts in physics. Topics include Lagrangian dynamics, Hamiltonian dynamics, fluids and sound and surface waves, more. 165 figures. 2 tables. 1980 edition.

Preface

Significant Names in Mechanics and Mathematical Physics

xvii

Basic Principles

(30)

Newton's Laws

(4)

Statement of Newton's Laws

(2)

Conservation Laws

(2)

Systems of Particles

(5)

Center-of-Mass Motion

(1)

Angular Momentum

(2)

Energy

(2)

Central Forces

(6)

Conservation Laws

(1)

Effective Potential

(2)

Inverse-Square Force: Kepler's Laws

(3)

Two-Body Motion with a Central Potential

(2)

Scattering

(13)

Hyperbolic Orbits in Gravitational Potential

(3)

General Scattering Orbits

(1)

Cross Section

(2)

Rutherford Scattering

(2)

Scattering by a Hard Sphere

(4)

Accelerated Coordinate Systems

(18)

Rotating Coordinate Systems

(2)

Infinitesimal Rotations

(3)

Accelerations

(1)

Translations and Rotations

(1)

Newton's Laws in Accelerated Coordinate Systems

(1)

Motion on the Surface of the Earth

(5)

Particle on a Scale

(1)

Falling Particle

(2)

Horizontal Motion

(1)

Foucault Pendulum

(5)

Lagrangian Dynamics

(37)

Constrained Motion and Generalized Coordinates

(3)

Constraints

(1)

Generalized Coordinates

(1)

Virtual Displacements

(1)

D'Alembert's Principle

(1)

Lagrange's Equations

(5)

Examples

(2)

Pendulum

(1)

Bead on a Rotating Wire Hoop

(1)

Calculus of Variations

(6)

Hamilton's Principle

(2)

Forces of Constraint

(10)

Pendulum

(2)

Atwood's Machine

(1)

One Cylinder Rolling on Another

(4)

Generalized Momenta and the Hamiltonian

(8)

Symmetry Principles and Conserved Quantities

(1)

The Hamiltonian

(7)

Small Oscillations

(48)

Formulation

(3)

Normal Modes

(12)

Simplest Case

(2)

Coupled Problem: Formulation

(1)

Linear Equations: A Review

(1)

Coupled Problem: Eigenvectors and Eigenvalues

(2)

Coupled Problem: General Solution

(1)

Matrix Notation

(2)

Modal Matrix

(1)

Normal Coordinates

(2)

Example: Coupled Pendulums

101

(7)

Example: Many Degrees of Freedom

108

(11)

Two N-Body Problems

108

(2)

Normal Modes

110

(9)

Transition from Discrete to Continuous Systems

119

(15)

Passage to the Continuum Limit

120

(1)

Direct Treatment of a Continuous String

120

(2)

General Solution to the Wave Equation with Specified Initial Conditions

122

(3)

Lagrangian for a Continuous String

125

(1)

Normal Coordinates

126

(2)

Hamilton's Principle for Continuous Systems

128

(6)

Rigid Bodies

134

(39)

General Theory

134

(9)

Motion with One Arbitrary Fixed Point

134

(3)

General Motion with No Fixed Point

137

(2)

Inertia Tensor

139

(1)

Principal Axes

140

(3)

Euler's Equations

143

(1)

Applications

144

(10)

Compound Pendulum: Kater's Pendulum and the Center of Percussion

144

(5)

Rolling and Sliding Billiard Ball

149

(2)

Torque-free Motion: Symmetric Top

151

(2)

Torque-free Motion: Asymmetric Top

153

(1)

Euler Angles

154

(2)

Symmetric Top: Torque-free Motion

156

(5)

Equations of Motion and First Integrals

157

(1)

Description of Motion in Inertial Frame

158

(3)

Symmetric Top: One Fixed Point in a Gravitational Field

161

(12)

Dynamical Equations

161

(2)

Effective Potential

163

(2)

Small Oscillations about Steady Motion

165

(8)

Hamiltonian Dynamics

173

(34)

Hamilton's Equations

173

(6)

Review of Lagrangian Dynamics

173

(2)

Hamiltonian Dynamics

175

(2)

Derivation of Hamilton's Equations from a Modified Hamilton's Principle

177

(2)

Example: Charged Particle in an Electromagnetic Field

179

(2)

Canonical Transformations

181

(3)

Hamilton-Jacobi Theory

184

(7)

Action-Angle Variables

191

(6)

Poisson Brackets

197

(10)

Basic Formulation

197

(1)

Transition to Quantum Mechanics

198

(9)

Strings

207

(64)

Review of Field Theory

207

(4)

D'Alembert's Solution to the Wave Equation

211

(8)

Solution for an Infinite String

211

(3)

Solution for a Finite String

214

(1)

Equivalence of d'Alembert's and Bernoulli's Solution

215

(4)

Eigenfunction Expansions

219

(7)

Variational Principle

226

(10)

Basic Formulation

227

(2)

Minimum Character of the Functional

229

(3)

Completeness of Eigenfunctions

232

(4)

Estimates of Lowest Eigenvalues; The Rayleigh-Ritz Approximation Method

236

(9)

General Theory

237

(2)

Example: Mass Point on a String

239

(6)

Green's Function in One Dimension

245

(6)

Eigenfunction Expansion

245

(2)

Construction from Solutions to Homogeneous Equations

247

(2)

Example: Uniform String with Fixed Endpoints

249

(2)

Perturbation Theory

251

(7)

General Theory

252

(2)

Expansion for Small Coupling Strength

254

(1)

Example: Mass Point on a String Revisited

255

(3)

Energy Flux

258

(13)

Continuity Equation for the Hamiltonian Density

258

(4)

Example: One-dimensional String

262

(1)

Transmission and Reflection at a Discontinuity in Density

263

(8)

Membranes

271

(19)

General Formulation

271

(3)

Specific Geometries

274

(16)

Rectangular Membrane

274

(5)

Circular Membrane

279

(4)

Variational Estimate of Lowest Drumhead Mode

283

(1)

Perturbation Theory for Nearly Circular Boundary

284

(6)

Sound Waves in Fluids

290

(67)

General Equations of Hydrodynamics

290

(15)

Formulation of Newton's Second Law

291

(3)

Conservation of Matter: The Continuity Equation

294

(2)

Conservation of Momentum: Stress Tensor and Euler's Equation

296

(2)

Conservation of Energy

298

(2)

Bernoulli's Theorem

300

(2)

Thomson's (Lord Kelvin's) Theorem on Circulation

302

(1)

Lagrangian for Isentropic Irrotational Flow

303

(2)

Sound Waves

305

(6)

Fundamental Equations

305

(3)

Standing Waves in Cavities

308

(3)

Fourier Transforms and Green's Functions in Three Dimensions

311

(9)

Screened Poisson Equation

312

(2)

Helmholtz Equation: Causality and Analyticity

314

(3)

Boundaries and the Method of Images

317

(3)

Radiation, Diffraction, and Scattering

320

(19)

Radiation from a Piston in a Wall

320

(5)

Diffraction in Kirchhoff's Approximation

325

(7)

Radiation from an Oscillating Sphere

332

(4)

Scattering by a Rigid Cylinder

336

(3)

Nonlinear Phenomena and Shock Waves

339

(18)

Traveling Waves

340

(3)

Example: Ideal Gas

343

(4)

Shock Waves

347

(10)

Surface Waves on Fluids

357

(49)

Tidal Waves

357

(9)

Equations of Motion

357

(2)

One-dimensional Waves

359

(4)

Two-dimensional Waves

363

(3)

Surface Waves

366

(17)

Formulation for Arbitrary Depths

367

(3)

Dispersion Relation

370

(4)

Energy

374

(2)

Group Velocity

376

(3)

Inclusion of Surface Tension

379

(4)

Initial-Value Problem

383

(10)

Surface Waves on Deep Water

384

(3)

Method of Stationary Phase

387

(2)

Application to Surface Waves

389

(4)

Solitary Waves

393

(13)

Extended Equation for Tidal Waves

395

(3)

Effective Nonlinear Wave Equation

398

(1)

Solitary Waves

399

(7)

Heat Conduction

406

(28)

Basic Equations

406

(4)

Examples

410

(7)

Separation of Variables

411

(2)

Thermal Waves in a Half Space

413

(2)

Infinite Domain: Fourier Transform

415

(2)

Laplace Transform

417

(17)

Inversion Theorem

417

(2)

Example: Half Space at Fixed Surface Temperature

419

(5)

Example: Sphere Heated Internally

424

(3)

Approximation Methods for Long and Short Times

427

(7)

Viscous Fluids

434

(25)

Viscous Stress Tensor

434

(11)

Basic Formulation

435

(3)

Navier-Stokes Equation

438

(3)

Energy Balance

441

(4)

Examples of Incompressible Flow

445

(6)

Steady Flow in a Channel or Pipe

445

(3)

Tangential Flow in a Half Space

448

(3)

Sound Waves in Viscous Fluids

451

(8)

Elastic Continua

459

(22)

Basic Formulation

459

(11)

Small Deformations

460

(4)

Stress Tensor

464

(4)

Elastic Energy

468

(2)

Dynamical Behavior

470

(11)

Equation of Motion

471

(4)

Elastic Waves in an Unbounded Medium

475

(6)

Appendix A Theory of Functions

481

(31)

A1 Complex Variables

481

(1)

A2 Functions of a Complex Variable

482

(4)

A3 Complex Integration

486

(2)

A4 Cauchy's Theorem

488

(5)

A5 Cauchy's Integral

493

(2)

Morera's Theorem

494

(1)

A6 Uniformly Convergent Series

495

(2)

Power Series

496

(1)

A7 Taylor's Theorem

497

(1)

A8 Laurent Series

498

(2)

A9 Theory of Residues

500

(4)

A10 Zeros of an Analytic Function

504

(2)

A11 Analytic Continuation

506

(6)

Appendix B Curvilinear Orthogonal Coordinates

512

(9)

Gradient

514

(1)

Divergence

514

(2)

Laplacian

516

(1)

Spherical Coordinates

516

(1)

Cylindrical Coordinates

516

(1)

Polar Coordinates in Two Dimensions

517

(4)

Appendix C Separation of Variables

521

(5)

Normal Modes in Polar Coordinates (Two Dimensions)

521

(1)

Normal Modes in Spherical Coordinates (Three Dimensions)

522

(2)

Normal Modes in Cylindrical Coordinates (Three Dimensions)

524

(2)

Appendix D Integral Representations and Special Functions

526

(22)

D1 The Γ Function

526

(5)

D2 Legendre Functions

531

(10)

α = l (an integer)

532

(3)

Arbitrary α

535

(1)

Legendre Functions of the Second Kind

535

(3)

Arbitrary α

538

(3)

D3 Bassel Functions

541

(7)

Appendix E Selected Mathematical Formulas

548

(6)

E1 The Γ Function

548

(1)

E2 Error Function

549

(1)

E3 Legendre Functions

549

(1)

Recursion and General Relations [also for Qα(z)]

549

(1)

Addition Formulas

549

(1)

Explicit Forms for l = 0, 1, 2, 3, . . . , ∞ and integral m

550

(1)

E4 Cylindrical Bessel Functions

550

(2)

Recursion Relations [also for Nv(z)]

550

(1)

Series and Approximate Forms (m is a non-negative integer)

551

(1)

E5 Spherical Bessel Functions

552

(2)

Specific Forms

552

(1)

Recursion Relations [also for nl(z)]

553

(1)

Appendix F Physical Constants

554

(2)

Appendix G Basic Texts and Monographs

556

(1)

Index

557

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Theoretical Mechanics of Particles and Continua

Summary

Table of Contents

Supplemental Materials