INDICE: 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
- ISBN: 978-1-84821-642-6
- Editorial: ISTE Ltd.
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- Fecha Publicación: 08/01/2016
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