Galilean Mechanics and Thermodynamics of Continua

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