ISBN-10:
1848216424
ISBN-13:
9781848216426
Pub. Date:
02/08/2016
Publisher:
Wiley
Galilean Mechanics and Thermodynamics of Continua / Edition 1

Galilean Mechanics and Thermodynamics of Continua / Edition 1

by Gery de Saxce, Claude Vallee
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Product Details

ISBN-13: 9781848216426
Publisher: Wiley
Publication date: 02/08/2016
Series: Iste
Pages: 446
Product dimensions: 6.50(w) x 9.50(h) x 1.10(d)

About the Author

Géry de Saxcé is Professor at Lille 1 University - Science and Technology, France.

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Table of Contents

Foreword xiii

Introduction xxi

Part 1. Particles and Rigid Bodies 1

Chapter 1. Galileo’s Principle of Relativity 3

1.1. Events and space–time 3

1.2. Event coordinates 3

1.2.1. When? 3

1.2.2. Where? 4

1.3. Galilean transformations 6

1.3.1. Uniform straight motion 6

1.3.2. Principle of relativity 9

1.3.3. Space–time structure and velocity addition 10

1.3.4. Organizing the calculus 11

1.3.5. About the units of measurement 12

1.4. Comments for experts 14

Chapter 2. Statics 15

2.1. Introduction 15

2.2. Statical torsor 16

2.2.1. Two-dimensional model 16

2.2.2. Three-dimensional model 17

2.2.3. Statical torsor and transport law of the moment 18

2.3. Statics equilibrium 20

2.3.1. Resultant torsor 20

2.3.2. Free body diagram and balance equation 20

2.3.3. External and internal forces 23

2.4. Comments for experts 25

Chapter 3. Dynamics of Particles 27

3.1. Dynamical torsor 27

3.1.1. Transformation law and invariants 27

3.1.2. Boost method 30

3.2. Rigid body motions 32

3.2.1. Rotations 32

3.2.2. Rigid motions 34

3.3. Galilean gravitation 36

3.3.1. How to model the gravitational forces? 36

3.3.2. Gravitation 38

3.3.3. Galilean gravitation and equation of motion 40

3.3.4. Transformation laws of the gravitation and acceleration 42

3.4. Newtonian gravitation 46

3.5. Other forces 51

3.5.1. General equation of motion 51

3.5.2. Foucault’s pendulum 52

3.5.3. Thrust 55

3.6. Comments for experts 56

Chapter 4. Statics of Arches, Cables and Beams 57

4.1. Statics of arches 57

4.1.1. Modeling of slender bodies 57

4.1.2. Local equilibrium equations of arches 59

4.1.3. Corotational equilibrium equations of arches 62

4.1.4. Equilibrium equations of arches in Fresnet’s moving frame 63

4.2. Statics of cables 67

4.3. Statics of trusses and beams 69

4.3.1. Traction of trusses 69

4.3.2. Bending of beams 71

Chapter 5. Dynamics of Rigid Bodies 75

5.1. Kinetic co-torsor 75

5.1.1. Lagrangian coordinates 75

5.1.2. Eulerian coordinates 76

5.1.3. Co-torsor 76

5.2. Dynamical torsor 80

5.2.1. Total mass and mass-center 80

5.2.2. The rigid body as a particle 81

5.2.3. The moment of inertia matrix 84

5.2.4. Kinetic energy of a body 87

5.3. Generalized equations of motion 88

5.3.1. Resultant torsor of the other forces 88

5.3.2. Transformation laws 89

5.3.3. Equations of motion of a rigid body 91

5.4. Motion of a free rigid body around it 93

5.5. Motion of a rigid body with a contact point (Lagrange’s top) 95

5.6. Comments for experts 103

Chapter 6. Calculus of Variations 105

6.1. Introduction 105

6.2. Particle subjected to the Galilean gravitation 109

6.2.1. Guessing the Lagrangian expression 109

6.2.2. The potentials of the Galilean gravitation 110

6.2.3. Transformation law of the potentials of the gravitation 113

6.2.4. How to manage holonomic constraints? 116

Chapter 7. Elementary Mathematical Tools 117

7.1. Maps 117

7.2. Matrix calculus 118

7.2.1. Columns 118

7.2.2. Rows 119

7.2.3. Matrices 120

7.2.4. Block matrix 124

7.3. Vector calculus in R3 125

7.4. Linear algebra 127

7.4.1. Linear space 127

7.4.2. Linear form 129

7.4.3. Linear map 130

7.5. Affine geometry 132

7.6. Limit and continuity 135

7.7. Derivative 136

7.8. Partial derivative 136

7.9. Vector analysis 137

7.9.1. Gradient 137

7.9.2. Divergence 139

7.9.3. Vector analysis in R3 and curl 139

Part 2. Continuous Media 141

Chapter 8. Statics of 3D Continua 143

8.1. Stresses 143

8.1.1. Stress tensor 143

8.1.2. Local equilibrium equations 148

8.2. Torsors 150

8.2.1. Continuum torsor 150

8.2.2. Cauchy’s continuum 153

8.3. Invariants of the stress tensor 155

Chapter 9. Elasticity and Elementary Theory of Beams 157

9.1. Strains 157

9.2. Internal work and power 162

9.3. Linear elasticity 164

9.3.1. Hooke’s law 164

9.3.2. Isotropic materials 166

9.3.3. Elasticity problems 170

9.4. Elementary theory of elastic trusses and beams 171

9.4.1. Multiscale analysis: from the beam to the elementary volume 171

9.4.2. Transversely rigid body model 176

9.4.3. Calculating the local fields 179

9.4.4. Multiscale analysis: from the elementary volume to the beam 183

Chapter 10. Dynamics of 3D Continua and Elementary Mechanics of Fluids 187

10.1. Deformation and motion 187

10.2. Flash-back: Galilean tensors 192

10.3. Dynamical torsor of a 3D continuum 196

10.4. The stress–mass tensor 198

10.4.1. Transformation law and invariants 198

10.4.2. Boost method 200

10.5. Euler’s equations of motion 202

10.6. Constitutive laws in dynamics 206

10.7. Hyperelastic materials and barotropic fluids 210

Chapter 11. Dynamics of Continua of Arbitrary Dimensions 215

11.1. Modeling the motion of one-dimensional (1D) material bodies 215

11.2. Group of the 1D linear Galilean transformations 217

11.3. Torsor of a continuum of arbitrary dimension 219

11.4. Force–mass tensor of a 1D material body 220

11.5. Full torsor of a 1D material body 222

11.6. Equations of motion of a continuum of arbitrary dimension 224

11.7. Equation of motion of 1D material bodies 225

11.7.1. First group of equations of motion 226

11.7.2. Multiscale analysis 227

11.7.3. Secong group of equations of motion 231

Chapter 12. More About Calculus of Variations 235

12.1. Calculus of variation and tensors 235

12.2. Action principle for the dynamics of continua 237

12.3. Explicit form of the variational equations 240

12.4. Balance equations of the continuum 244

12.5. Comments for experts . 245

Chapter 13. Thermodynamics of Continua 247

13.1. Introduction 247

13.2. An extra dimension 248

13.3. Temperature vector and friction tensor 251

13.4. Momentum tensors and first principle 253

13.5. Reversible processes and thermodynamical potentials 258

13.6. Dissipative continuum and heat transfer equation 263

13.7. Constitutive laws in thermodynamics 268

13.8. Thermodynamics and Galilean gravitation 272

13.9. Comments for experts 279

Chapter 14. Mathematical Tools 281

14.1. Group 281

14.2. Tensor algebra 282

14.2.1. Linear tensors 282

14.2.2. Affine tensors 288

14.2.3. G-tensors and Euclidean tensors 292

14.3. Vector analysis 295

14.3.1. Divergence 295

14.3.2. Laplacian 296

14.3.3. Vector analysis in R3 and curl 296

14.4. Derivative with respect to a matrix 297

14.5. Tensor analysis 297

14.5.1. Differential manifold 297

14.5.2. Covariant differential of linear tensors 300

14.5.3. Covariant differential of affine tensors 303

Part 3. Advanced Topics 307

Chapter 15. Affine Structure on a Manifold 309

15.1. Introduction 309

15.2. Endowing the structure of linear space by transport 310

15.3. Construction of the linear tangent space 311

15.4. Endowing the structure of affine space by transport 313

15.5. Construction of the affine tangent space 316

15.6. Particle derivative and affine functions 319

Chapter 16. Galilean, Bargmannian and Poincarean Structures on a Manifold 321

16.1. Toupinian structure 321

16.2. Normalizer of Galileo’s group in the affine group 323

16.3. Momentum tensors 325

16.4. Galilean momentum tensors 328

16.4.1. Coadjoint representation of Galileo’s group 328

16.4.2. Galilean momentum transformation law 329

16.4.3. Structure of the orbit of a Galilean momentum torsor 335

16.5. Galilean coordinate systems 338

16.5.1. G-structures 338

16.5.2. Galilean coordinate systems 338

16.6. Galilean curvature 341

16.7. Bargmannian coordinates 346

16.8. Bargmannian torsors 349

16.9. Bargmannian momenta 352

16.10. Poincarean structures 357

16.11. Lie group statistical mechanics 362

Chapter 17. Symplectic Structure on a Manifold 367

17.1. Symplectic form 367

17.2. Symplectic group 370

17.3. Momentum map 371

17.4. Symplectic cohomology 373

17.5. Central extension of a group 375

17.6. Construction of a central extension from the symplectic cocycle 377

17.7. Coadjoint orbit method 383

17.8. Connections 385

17.9. Factorized symplectic form 387

17.10. Application to classical mechanics 393

17.11. Application to relativity 396

Chapter 18. Advanced Mathematical Tools 399

18.1. Vector fields 399

18.2. Lie group 400

18.3. Foliation 402

18.4. Exterior algebra 402

18.5. Curvature tensor 405

Bibliography 407

Index 411

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