Galilean Mechanics and Thermodynamics of Continua
1. Auflage Januar 2016
448 Seiten, Hardcover
Wiley & Sons Ltd
ISBN:
978-1-84821-642-6
John Wiley & Sons
Kurzbeschreibung
This title proposes a unified approach to continuum mechanics which is consistent with Galilean relativity. Based on the notion of affine tensors, a simple generalization of the classical tensors, this approach allows gathering the usual mechanical entities . mass, energy, force, moment, stresses, linear and angular momentum . in a single tensor.
This title proposes a unified approach to continuum mechanics which is consistent with Galilean relativity. Based on the notion of affine tensors, a simple generalization of the classical tensors, this approach allows gathering the usual mechanical entities . mass, energy, force, moment, stresses, linear and angular momentum . in a single tensor.
Starting with the basic subjects, and continuing through to the most advanced topics, the authors' presentation is progressive, inductive and bottom-up. They begin with the concept of an affine tensor, a natural extension of the classical tensors. The simplest types of affine tensors are the points of an affine space and the affine functions on this space, but there are more complex ones which are relevant for mechanics . torsors and momenta. The essential point is to derive the balance equations of a continuum from a unique principle which claims that these tensors are affine-divergence free.
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
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
