Galilean Mechanics and Thermodynamics of Continua

Géry de Saxcé is Professor at Lille 1 University – Science and Technology, France.Claude Vallée was Emeritus Professor at the University of Poitiers, France.

• Foreword  xiiiIntroduction  xxiPart 1. Particles and Rigid Bodies 1Chapter 1. Galileo’s Principle of Relativity  31.1. Events and space–time  31.2. Event coordinates 31.2.1. When? 31.2.2. Where? 41.3. Galilean transformations 61.3.1. Uniform straight motion 61.3.2. Principle of relativity 91.3.3. Space–time structure and velocity addition 101.3.4. Organizing the calculus  111.3.5. About the units of measurement 121.4. Comments for experts 14Chapter 2. Statics  152.1. Introduction  152.2. Statical torsor 162.2.1. Two-dimensional model 162.2.2. Three-dimensional model 172.2.3. Statical torsor and transport law of the moment  182.3. Statics equilibrium 202.3.1. Resultant torsor  202.3.2. Free body diagram and balance equation 202.3.3. External and internal forces 232.4. Comments for experts 25Chapter 3. Dynamics of Particles 273.1. Dynamical torsor 273.1.1. Transformation law and invariants  273.1.2. Boost method 303.2. Rigid body motions  323.2.1. Rotations  323.2.2. Rigid motions 343.3. Galilean gravitation 363.3.1. How to model the gravitational forces? 363.3.2. Gravitation 383.3.3. Galilean gravitation and equation of motion 403.3.4. Transformation laws of the gravitation and acceleration 423.4. Newtonian gravitation 463.5. Other forces 513.5.1. General equation of motion  513.5.2. Foucault’s pendulum 523.5.3. Thrust  553.6. Comments for experts 56Chapter 4. Statics of Arches, Cables and Beams 574.1. Statics of arches  574.1.1. Modeling of slender bodies 574.1.2. Local equilibrium equations of arches 594.1.3. Corotational equilibrium equations of arches 624.1.4. Equilibrium equations of arches in Fresnet’s moving frame 634.2. Statics of cables  674.3. Statics of trusses and beams 694.3.1. Traction of trusses 694.3.2. Bending of beams 71Chapter 5. Dynamics of Rigid Bodies 755.1. Kinetic co-torsor 755.1.1. Lagrangian coordinates 755.1.2. Eulerian coordinates 765.1.3. Co-torsor 765.2. Dynamical torsor 805.2.1. Total mass and mass-center 805.2.2. The rigid body as a particle 815.2.3. The moment of inertia matrix 845.2.4. Kinetic energy of a body 875.3. Generalized equations of motion 885.3.1. Resultant torsor of the other forces 885.3.2. Transformation laws 895.3.3. Equations of motion of a rigid body 915.4. Motion of a free rigid body around it 935.5. Motion of a rigid body with a contact point (Lagrange’s top) 955.6. Comments for experts 103Chapter 6. Calculus of Variations 1056.1. Introduction  1056.2. Particle subjected to the Galilean gravitation 1096.2.1. Guessing the Lagrangian expression 1096.2.2. The potentials of the Galilean gravitation 1106.2.3. Transformation law of the potentials of the gravitation  1136.2.4. How to manage holonomic constraints? 116Chapter 7. Elementary Mathematical Tools 1177.1. Maps  1177.2. Matrix calculus 1187.2.1. Columns 1187.2.2. Rows 1197.2.3. Matrices 1207.2.4. Block matrix 1247.3. Vector calculus in R3 1257.4. Linear algebra 1277.4.1. Linear space  1277.4.2. Linear form 1297.4.3. Linear map 1307.5. Affine geometry  1327.6. Limit and continuity  1357.7. Derivative  1367.8. Partial derivative  1367.9. Vector analysis 1377.9.1. Gradient  1377.9.2. Divergence 1397.9.3. Vector analysis in R3 and curl 139Part 2. Continuous Media 141Chapter 8. Statics of 3D Continua 1438.1. Stresses 1438.1.1. Stress tensor  1438.1.2. Local equilibrium equations 1488.2. Torsors 1508.2.1. Continuum torsor 1508.2.2. Cauchy’s continuum  1538.3. Invariants of the stress tensor 155Chapter 9. Elasticity and Elementary Theory of Beams 1579.1. Strains  1579.2. Internal work and power 1629.3. Linear elasticity  1649.3.1. Hooke’s law 1649.3.2. Isotropic materials 1669.3.3. Elasticity problems  1709.4. Elementary theory of elastic trusses and beams 1719.4.1. Multiscale analysis: from the beam to the elementary volume 1719.4.2. Transversely rigid body model  1769.4.3. Calculating the local fields  1799.4.4. Multiscale analysis: from the elementary volume to the beam 183Chapter 10. Dynamics of 3D Continua and Elementary Mechanics of Fluids 18710.1. Deformation and motion 18710.2. Flash-back: Galilean tensors 19210.3. Dynamical torsor of a 3D continuum 19610.4. The stress–mass tensor 19810.4.1. Transformation law and invariants 19810.4.2. Boost method 20010.5. Euler’s equations of motion 20210.6. Constitutive laws in dynamics 20610.7. Hyperelastic materials and barotropic fluids  210Chapter 11. Dynamics of Continua of Arbitrary Dimensions 21511.1. Modeling the motion of one-dimensional (1D) material bodies 21511.2. Group of the 1D linear Galilean transformations 21711.3. Torsor of a continuum of arbitrary dimension 21911.4. Force–mass tensor of a 1D material body 22011.5. Full torsor of a 1D material body  22211.6. Equations of motion of a continuum of arbitrary dimension  22411.7. Equation of motion of 1D material bodies 22511.7.1. First group of equations of motion 22611.7.2. Multiscale analysis  22711.7.3. Secong group of equations of motion 231Chapter 12. More About Calculus of Variations  23512.1. Calculus of variation and tensors  23512.2. Action principle for the dynamics of continua  23712.3. Explicit form of the variational equations 24012.4. Balance equations of the continuum  24412.5. Comments for experts . 245Chapter 13. Thermodynamics of Continua  24713.1. Introduction  24713.2. An extra dimension 24813.3. Temperature vector and friction tensor 25113.4. Momentum tensors and first principle 25313.5. Reversible processes and thermodynamical potentials  25813.6. Dissipative continuum and heat transfer equation 26313.7. Constitutive laws in thermodynamics 26813.8. Thermodynamics and Galilean gravitation  27213.9. Comments for experts  279Chapter 14. Mathematical Tools 28114.1. Group 28114.2. Tensor algebra  28214.2.1. Linear tensors 28214.2.2. Affine tensors 28814.2.3. G-tensors and Euclidean tensors  29214.3. Vector analysis  29514.3.1. Divergence  29514.3.2. Laplacian 29614.3.3. Vector analysis in R3 and curl 29614.4. Derivative with respect to a matrix 29714.5. Tensor analysis  29714.5.1. Differential manifold  29714.5.2. Covariant differential of linear tensors 30014.5.3. Covariant differential of affine tensors 303Part 3. Advanced Topics  307Chapter 15. Affine Structure on a Manifold 30915.1. Introduction  30915.2. Endowing the structure of linear space by transport 31015.3. Construction of the linear tangent space  31115.4. Endowing the structure of affine space by transport 31315.5. Construction of the affine tangent space  31615.6. Particle derivative and affine functions 319Chapter 16. Galilean, Bargmannian and Poincarean Structures on a Manifold 32116.1. Toupinian structure  32116.2. Normalizer of Galileo’s group in the affine group  32316.3. Momentum tensors  32516.4. Galilean momentum tensors 32816.4.1. Coadjoint representation of Galileo’s group  32816.4.2. Galilean momentum transformation law  32916.4.3. Structure of the orbit of a Galilean momentum torsor  33516.5. Galilean coordinate systems 33816.5.1. G-structures 33816.5.2. Galilean coordinate systems  33816.6. Galilean curvature  34116.7. Bargmannian coordinates  34616.8. Bargmannian torsors 34916.9. Bargmannian momenta 35216.10. Poincarean structures  35716.11. Lie group statistical mechanics  362Chapter 17. Symplectic Structure on a Manifold  36717.1. Symplectic form 36717.2. Symplectic group 37017.3. Momentum map 37117.4. Symplectic cohomology 37317.5. Central extension of a group  37517.6. Construction of a central extension from the symplectic cocycle 37717.7. Coadjoint orbit method 38317.8. Connections  38517.9. Factorized symplectic form 38717.10. Application to classical mechanics  39317.11. Application to relativity  396Chapter 18. Advanced Mathematical Tools 39918.1. Vector fields  39918.2. Lie group 40018.3. Foliation  40218.4. Exterior algebra 40218.5. Curvature tensor 405Bibliography 407Index 411