Mechanics and Dynamical Systems With

This book addresses modeling, qualitative analysis, and simulation of real physical systems using ordinary differential equations. The scientific computation aspects are presented using the software program Mathematica for exercises and examples. The book provides a unified treatment of the subject; modeling; analytic and computational methods that are linked and organized in a synergistic fashion.

Preface

ix

I Mathematical Methods for Differential Equations

Models and Differential Equations

1

(18)

Introduction

1

(1)

Mathematical Models and Computation

2

(6)

Examples of Mathematical Models

8

(8)

Validation, Determinism, and Stochasticity

16

(3)

Models and Mathematical Problems

19

(38)

Introduction

19

(1)

Classification of Models

20

(6)

Statement of Problems

26

(3)

Solution of Initial-Value Problems

29

(11)

Representation of the Dynamic Response

40

(6)

On the Solution of Boundary-Value Problems

46

(4)

Problems

50

(7)

Stability and Perturbation Methods

57

(36)

Introduction

57

(6)

Stability Definitions

63

(2)

Linear Stability Methods

65

(6)

Nonlinear Stability

71

(5)

Regular Perturbation Methods

76

(11)

Problems

87

(6)

II Mathematical Methods of Classical Mechanics

Newtonian Dynamics

93

(46)

Introduction

93

(2)

Principles of Newtonian Mechanics

95

(7)

Balance Laws for Systems of Point Masses

102

(4)

Active and Reactive Forces

106

(16)

Constraints and reactive forces

107

(8)

Active forces and force fields

115

(7)

Applications

122

(14)

Dynamics of simple pendulum

123

(2)

Particle subject to a central force

125

(4)

Heavy particle falling in air

129

(4)

Three-point masses subject to elastic forces

133

(3)

Problems

136

(3)

Rigid Body Dynamics

139

(44)

Introduction

139

(1)

Rigid Body Models

140

(8)

Active and Reactive Forces in Rigid Body Dynamics

148

(4)

Constrained Rigid Body Models

152

(10)

Articulated Systems

162

(3)

Applications

165

(13)

Rigid body model of a vehicle and plane dynamics

166

(1)

Compound pendulum

167

(3)

Uniform rotations

170

(1)

Free rotations of a gyroscope

171

(3)

Ball on an inclined plane

174

(4)

Problems

178

(5)

Energy Methods and Lagrangian Mechanics

183

(50)

Introduction

183

(2)

Elementary and Virtual Work

185

(4)

Energy Theorems

189

(5)

The Method of Lagrange Equations

194

(6)

Potential and First Integrals

200

(7)

Energy Methods and Stability

207

(4)

Applications

211

(16)

Three body articulated system

212

(2)

Stability of Duffing's model

214

(3)

Free rotations or Poinsot's motion

217

(3)

Heavy gyroscope

220

(4)

The rolling coin

224

(3)

Problems

227

(6)

III Bifurcations, Chaotic Dynamics, Stochastic Models, and Discretization of Continuous Models

Deterministic and Stochastic Models in Applied Sciences

233

(34)

Introduction

233

(1)

Mathematical Modeling in Applied Sciences

234

(5)

Examples of Mathematical Models

239

(5)

Further Remarks on Modeling

244

(2)

Mathematical Modeling and Stochasticity

246

(17)

Random variables and stochastic calculus

251

(8)

Moment representation of the dynamic response

259

(2)

Statistical representation of large systems

261

(2)

Problems

263

(4)

Chaotic Dynamics, Stability, and Bifurcations

267

(46)

Introduction

267

(1)

Stability Diagrams

268

(7)

Stability Diagrams and Potential Energy

275

(7)

Limit Cycles

282

(5)

Hopf Bifurcation

287

(5)

Chaotic Motions

292

(2)

Applications

294

(15)

Ring on a rotating wire

294

(4)

Metallic meter

298

(3)

Line galloping model

301

(3)

Flutter instability model

304

(2)

Models presenting transition to chaos

306

(3)

Problems

309

(4)

Discrete Models of Continuous Systems

313

(28)

Introduction

313

(4)

Diffusion Models

317

(4)

Mathematical Models of Traffic Flow

321

(5)

Mathematical Statement of Problems

326

(3)

Discretization of Continuous Models

329

(7)

Problems

336

(5)

Appendix I. Numerical Methods for Ordinary Differential Equations

341

(16)

1 Introduction

341

(1)

2 Numerical Methods for Initial-Value Problems

342

(7)

3 Numerical Methods for Boundary-Value Problems

349

(8)

Appendix II. Kinematics, Applied Forces, Momentum and Mechanical Energy

357

(20)

1 Introduction

357

(1)

2 Systems of Applied Forces

358

(3)

3 Fundamental of Kinematics

361

(2)

4 Center of Mass

363

(2)

5 Tensor of Inertia

365

(4)

6 Linear Momentum

369

(1)

7 Angular Momentum

370

(2)

8 Kinetic Energy

372

(5)

Appendix III. Scientific Programs

377

(32)

1 Introduction to Programming

377

(5)

2 Scientific Programs

382

(27)

References

409

(2)

Subject Index

411

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Mechanics and Dynamical Systems With Mathematica

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