Cover -- Title Page -- Copyright -- Contents -- Foreword -- Introduction -- I.1. A geometrical viewpoint -- I.2. Overview -- I.2.1. Part 1: particles and rigid bodies -- I.2.2. Part 2: continuous media -- I.2.3. Part 3: advanced topics -- I.3. Historical background and key concepts -- PART 1: Particles and Rigid Bodies -- Chapter 1: Galileo's Principle of Relativity -- 1.1. Events and space-time -- 1.2. Event coordinates -- 1.2.1. When? -- 1.2.2. Where? -- 1.3. Galilean transformations -- 1.3.1. Uniform straight motion -- 1.3.2. Principle of relativity -- 1.3.3. Space-time structure and velocity addition -- 1.3.4. Organizing the calculus -- 1.3.5. About the units of measurement -- 1.4. Comments for experts -- Chapter 2: Statics -- 2.1. Introduction -- 2.2. Statical torsor -- 2.2.1. Two-dimensional model -- 2.2.2. Three-dimensional model -- 2.2.3. Statical torsor and transport law of the moment -- 2.3. Statics equilibrium -- 2.3.1. Resultant torsor -- 2.3.2. Free body diagram and balance equation -- 2.3.3. External and internal forces -- 2.4. Comments for experts -- Chapter 3: Dynamics of Particles -- 3.1. Dynamical torsor -- 3.1.1. Transformation law and invariants -- 3.1.2. Boost method -- 3.2. Rigid body motions -- 3.2.1. Rotations -- 3.2.2. Rigid motions -- 3.3. Galilean gravitation -- 3.3.1. How to model the gravitational forces? -- 3.3.2. Gravitation -- 3.3.3. Galilean gravitation and equation of motion -- 3.3.4. Transformation laws of the gravitation and acceleration -- 3.4. Newtonian gravitation -- 3.5. Other forces -- 3.5.1. General equation of motion -- 3.5.2. Foucault's pendulum -- 3.5.3. Thrust -- 3.6. Comments for experts -- Chapter 4: Statics of Arches, Cables and Beams -- 4.1. Statics of arches -- 4.1.1. Modeling of slender bodies -- 4.1.2. Local equilibrium equations of arches -- 4.1.3. Corotational equilibrium equations of arches.

4.1.4. Equilibrium equations of arches in Fresnet's moving frame -- 4.2. Statics of cables -- 4.3. Statics of trusses and beams -- 4.3.1. Traction of trusses -- 4.3.2. Bending of beams -- Chapter 5: Dynamics of Rigid Bodies -- 5.1. Kinetic co-torsor -- 5.1.1. Lagrangian coordinates -- 5.1.2. Eulerian coordinates -- 5.1.3. Co-torsor -- 5.2. Dynamical torsor -- 5.2.1. Total mass and mass-center -- 5.2.2. The rigid body as a particle -- 5.2.3. The moment of inertia matrix -- 5.2.4. Kinetic energy of a body -- 5.3. Generalized equations of motion -- 5.3.1. Resultant torsor of the other forces -- 5.3.2. Transformation laws -- 5.3.3. Equations of motion of a rigid body -- 5.4. Motion of a free rigid body around it -- 5.5. Motion of a rigid body with a contact point (Lagrange's top) -- 5.6. Comments for experts -- Chapter 6: Calculus of Variations -- 6.1. Introduction -- 6.2. Particle subjected to the Galilean gravitation -- 6.2.1. Guessing the Lagrangian expression -- 6.2.2. The potentials of the Galilean gravitation -- 6.2.3. Transformation law of the potentials of the gravitation -- 6.2.4. How to manage holonomic constraints? -- Chapter 7: Elementary Mathematical Tools -- 7.1. Maps -- 7.2. Matrix calculus -- 7.2.1. Columns -- 7.2.2. Rows -- 7.2.3. Matrices -- 7.2.4. Block matrix -- 7.3. Vector calculus in R3 -- 7.4. Linear algebra -- 7.4.1. Linear space -- 7.4.2. Linear form -- 7.4.3. Linear map -- 7.5. Affine geometry -- 7.6. Limit and continuity -- 7.7. Derivative -- 7.8. Partial derivative -- 7.9. Vector analysis -- 7.9.1. Gradient -- 7.9.2. Divergence -- 7.9.3. Vector analysis in R3 and curl -- PART 2: Continuous Media -- Chapter 8: Statics of 3D Continua -- 8.1. Stresses -- 8.1.1. Stress tensor -- 8.1.2. Local equilibrium equations -- 8.2. Torsors -- 8.2.1. Continuum torsor -- 8.2.2. Cauchy's continuum -- 8.3. Invariants of the stress tensor.

Chapter 9: Elasticity and Elementary Theory of Beams -- 9.1. Strains -- 9.2. Internal work and power -- 9.3. Linear elasticity -- 9.3.1. Hooke's law -- 9.3.2. Isotropic materials -- 9.3.3. Elasticity problems -- 9.4. Elementary theory of elastic trusses and beams -- 9.4.1. Multiscale analysis: from the beam to the elementary volume -- 9.4.2. Transversely rigid body model -- 9.4.3. Calculating the local fields -- 9.4.4. Multiscale analysis: from the elementary volume to the beam -- Chapter 10: Dynamics of 3D Continua and Elementary Mechanics of Fluids -- 10.1. Deformation and motion -- 10.2. Flash-back: Galilean tensors -- 10.3. Dynamical torsor of a 3D continuum -- 10.4. The stress-mass tensor -- 10.4.1. Transformation law and invariants -- 10.4.2. Boost method -- 10.5. Euler's equations of motion -- 10.6. Constitutive laws in dynamics -- 10.7. Hyperelastic materials and barotropic fluids -- Chapter 11: Dynamics of Continua of Arbitrary Dimensions -- 11.1. Modeling the motion of one-dimensional (1D) material bodies -- 11.2. Group of the 1D linear Galilean transformations -- 11.3. Torsor of a continuum of arbitrary dimension -- 11.4. Force-mass tensor of a 1D material body -- 11.5. Full torsor of a 1D material body -- 11.6. Equations of motion of a continuum of arbitrary dimension -- 11.7. Equation of motion of 1D material bodies -- 11.7.1. First group of equations of motion -- 11.7.2. Multiscale analysis -- 11.7.3. Secong group of equations of motion -- Chapter 12: More About Calculus of Variations -- 12.1. Calculus of variation and tensors -- 12.2. Action principle for the dynamics of continua -- 12.3. Explicit form of the variational equations -- 12.4. Balance equations of the continuum -- 12.5. Comments for experts -- Chapter 13: Thermodynamics of Continua -- 13.1. Introduction -- 13.2. An extra dimension.

13.3. Temperature vector and friction tensor -- 13.4. Momentum tensors and first principle -- 13.5. Reversible processes and thermodynamical potentials -- 13.6. Dissipative continuum and heat transfer equation -- 13.7. Constitutive laws in thermodynamics -- 13.8. Thermodynamics and Galilean gravitation -- 13.9. Comments for experts -- Chapter 14: Mathematical Tools -- 14.1. Group -- 14.2. Tensor algebra -- 14.2.1. Linear tensors -- 14.2.2. Affine tensors -- 14.2.3. G-tensors and Euclidean tensors -- 14.3. Vector analysis -- 14.3.1. Divergence -- 14.3.2. Laplacian -- 14.3.3. Vector analysis in R3 and curl -- 14.4. Derivative with respect to a matrix -- 14.5. Tensor analysis -- 14.5.1. Differential manifold -- 14.5.2. Covariant differential of linear tensors -- 14.5.3. Covariant differential of affine tensors -- PART 3: Advanced Topics -- Chapter 15: Affine Structure on a Manifold -- 15.1. Introduction -- 15.2. Endowing the structure of linear space by transport -- 15.3. Construction of the linear tangent space -- 15.4. Endowing the structure of affine space by transport -- 15.5. Construction of the affine tangent space -- 15.6. Particle derivative and affine functions -- Chapter 16: Galilean, Bargmannian and Poincarean Structures on a Manifold -- 16.1. Toupinian structure -- 16.2. Normalizer of Galileo's group in the affine group -- 16.3. Momentum tensors -- 16.4. Galilean momentum tensors -- 16.4.1. Coadjoint representation of Galileo's group -- 16.4.2. Galilean momentum transformation law -- 16.4.3. Structure of the orbit of a Galilean momentum torsor -- 16.5. Galilean coordinate systems -- 16.5.1. G-structures -- 16.5.2. Galilean coordinate systems -- 16.6. Galilean curvature -- 16.7. Bargmannian coordinates -- 16.8. Bargmannian torsors -- 16.9. Bargmannian momenta -- 16.10. Poincarean structures -- 16.11. Lie group statistical mechanics.

Chapter 17: Symplectic Structure on a Manifold -- 17.1. Symplectic form -- 17.2. Symplectic group -- 17.3. Momentum map -- 17.4. Symplectic cohomology -- 17.5. Central extension of a group -- 17.6. Construction of a central extension from the symplectic cocycle -- 17.7. Coadjoint orbit method -- 17.8. Connections -- 17.9. Factorized symplectic form -- 17.10. Application to classical mechanics -- 17.11. Application to relativity -- Chapter 18: Advanced Mathematical Tools -- 18.1. Vector fields -- 18.2. Lie group -- 18.3. Foliation -- 18.4. Exterior algebra -- 18.5. Curvature tensor -- Bibliography -- Index.