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CONTENTS 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 AF?NE 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 ?ELDS 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 ?UIDS 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 ?RST 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 AF?NE 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 AF?NE TENSORS PART 3 ADVANCED TOPICS CHAPTER 15 AF?NE 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 AF?NE SPACE BY TRANSPORT 15 5 CONSTRUCTION OF THE AF?NE TANGENT SPACE 15 6 PARTICLE DERIVATIVE AND AF?NE FUNCTIONS CHAPTER 16 GALILEAN, BARGMANNIAN AND POINCAREAN STRUCTURES ON A MANIFOLD 16 1 TOUPINIAN STRUCTURE 16 2 NORMALIZER OF GALILEO’S GROUP IN THE AF?NE 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 ?ELDS 18 2 LIE GROUP 18 3 FOLIATION 18 4 EXTERIOR ALGEBRA 18 5 CURVATURE TENSOR