A TREATISE ON STATICS: FUNDAMENTAL PRINCIPLES OF ELECTROSTATICS AND ELASTICITY

By C. CAZAU

CONTENTS PART I INTRODUCTION TO LINEAR ALGEBRA AND DIFFERENTIAL GEOMETRY VECTOR SPACE AND LINEAR MAPS DE?NITION OF VECTOR SPACE DIMENSION OF A VECTOR SPACE BASIS CHANGES VECTOR SUBSPACES ALGEBRAS EXAMPLES LINEAR MAPS EXERCISES TENSOR ALGEBRA LINEAR FORMS AND DUAL VECTOR SPACE BIDUALITY COVARIANT 2-TENSORS R; S/-TENSORS CONTRACTION AND CONTRACTED MULTIPLICATION SKEW-SYMMETRIC TENSORS AND EXTERIOR ALGEBRA SKEW-SYMMETRIC (0, 2)-TENSORS SKEW-SYMMETRIC 0; R/-TENSORS EXTERIOR ALGEBRA ORIENTED VECTOR SPACES EUCLIDEAN AND SYMPLECTIC VECTOR SPACES REPRESENTATION THEOREMS FOR SYMMETRIC AND SKEW-SYMMETRIC 0; 2/-TENSORS DEGENERATE AND NONDEGENERATE 0; 2/-TENSORS PSEUDO-EUCLIDEAN VECTOR SPACES EUCLIDEAN VECTOR SPACES EIGENVECTORS OF EUCLIDEAN 2-TENSORS ORTHOGONAL TRANSFORMATIONS SYMPLECTIC VECTOR SPACES DUALITY AND EUCLIDEAN TENSORS DUALITY EUCLIDEAN TENSORS THE LEVI–CIVITA TENSOR DIFFERENTIABLE MANIFOLDS DIFFERENTIABLE FUNCTIONS AND CURVES ON MANIFOLDS TANGENT VECTOR SPACE COTANGENT VECTOR SPACE DIFFERENTIAL AND CODIFFERENTIAL OF A MAP TANGENT AND COTANGENT FIBER BUNDLES RIEMANNIAN MANIFOLDS GEODESICS OF A RIEMANNIAN MANIFOLD ONE-PARAMETER GROUPS OF DIFFEOMORPHISMS GLOBAL AND LOCAL ONE-PARAMETER GROUPS LIE’S DERIVATIVE EXTERIOR DERIVATIVE AND INTEGRATION EXTERIOR DERIVATIVE CLOSED AND EXACT DIFFERENTIAL FORMS PROPERTIES OF EXTERIOR DERIVATIVE AN INTRODUCTION TO THE INTEGRATION OF R-FORMS ABSOLUTE DIFFERENTIAL CALCULUS PRELIMINARY CONSIDERATIONS AF?NE CONNECTION ON MANIFOLDS PARALLEL TRANSPORT AND AUTOPARALLEL CURVES COVARIANT DIFFERENTIAL OF TENSOR FIELDS TORSION AND CURVATURE TENSORS RIEMANNIAN CONNECTION DIFFERENTIAL OPERATORS ON A RIEMANNIAN MANIFOLD CONTENTS AN OVERVIEW OF DYNAMICAL SYSTEMS MODELING AND DYNAMICAL SYSTEMS GENERAL DE?NITIONS AND CAUCHY’S PROBLEM PRELIMINARY CONSIDERATIONS ABOUT STABILITY DE?NITIONS OF STABILITY ANALYSIS OF STABILITY: THE DIRECT METHOD POINCARE’S PERTURBATION METHOD INTRODUCING THE SMALL PARAMETER WEIERSTRASS’ QUALITATIVE ANALYSIS PART II MECHANICS KINEMATICS OF A POINT PARTICLE SPACE-TIME FRAMES OF REFERENCE OF CLASSICAL KINEMATICS TRAJECTORY OF A POINT PARTICLE VELOCITY AND ACCELERATION VELOCITY AND ACCELERATION IN PLANE MOTIONS CIRCULAR MOTION AND HARMONIC MOTION COMPOUND MOTIONS EXERCISE KINEMATICS OF RIGID BODIES CHANGE OF THE FRAME OF REFERENCE VELOCITY FIELD OF A RIGID MOTION TRANSLATIONAL AND ROTATIONAL MOTIONS SPHERICAL RIGID MOTIONS RIGID PLANAR MOTIONS TWO EXAMPLES RELATIVE MOTIONS PRINCIPLES OF DYNAMICS INTRODUCTION PRINCIPLE OF INERTIA AND INERTIAL FRAMES DYNAMICAL INTERACTIONS AND FORCE LAWS NEWTON’S SECOND LAW MECHANICAL DETERMINISM AND GALILEAN RELATIVITY PRINCIPLE BALANCE EQUATIONS KINETIC ENERGY AND K¨ONIG’S THEOREM KINETIC ENERGY THEOREM DYNAMICS IN NONINERTIAL FRAMES CONSTRAINED DYNAMICS DYNAMICS OF A MATERIAL POINT CENTRAL FORCES NEWTONIAN FORCES TWO-BODY PROBLEM KEPLER’S LAWS SCATTERING BY NEWTONIAN FORCES VERTICAL MOTION OF A HEAVY PARTICLE IN AIR CURVILINEAR MOTION OF A HEAVY PARTICLE IN AIR TERRESTRIAL DYNAMICS SIMPLE PENDULUM ROTATING SIMPLE PENDULUM FOUCAULT’S PENDULUM GENERAL PRINCIPLES OF RIGID BODY DYNAMICS MASS DENSITY AND CENTER OF MASS LINEAR MOMENTUM, ANGULAR MOMENTUM, AND KINETIC ENERGY OF A RIGID BODY TENSOR OF INERTIA SOME PROPERTIES OF THE TENSOR OF INERTIA ELLIPSOID OF INERTIA ACTIVE AND REACTIVE FORCES BALANCE EQUATIONS FOR A RIGID BODY: THEOREM OF KINETIC ENERGY DYNAMICS OF A RIGID BODY RIGID BODY WITH A SMOOTH FIXED AXIS COMPOUND PENDULUM SOLID WITH A FIXED POINT: EULER’S EQUATIONS POINSOT’S MOTIONS UNIFORM ROTATIONS POINSOT’S MOTIONS IN A GYROSCOPE HEAVY GYROSCOPE SOME REMARKS ON THE HEAVY GYROSCOPE TORQUE OF TERRESTRIAL FICTITIOUS FORCES FOUCAULT’S GYROSCOPE AND GYROSCOPE COMPASS FREE HEAVY SOLID LAGRANGIAN DYNAMICS DYNAMICS OF CONSTRAINED SYSTEMS CON?GURATION SPACE VIRTUAL VELOCITIES AND NONHOLONOMIC CONSTRAINTS CON?GURATION SPACE-TIME SIMPLE EXAMPLE PRINCIPLE OF VIRTUAL POWER LAGRANGE’S EQUATIONS CONSERVATIVE FORCES FIRST INTEGRALS SYMMETRIES AND FIRST INTEGRALS LAGRANGE’S EQUATIONS FOR NONHOLONOMIC CONSTRAINTS SMALL OSCILLATIONS HAMILTON’S PRINCIPLE GEOMETRIC FORMULATIONS OF LAGRANGIAN DYNAMICS LEGENDRE’S TRANSFORMATION HAMILTONIAN DYNAMICS SYMPLECTIC STRUCTURE OF THE PHASE SPACE CANONICAL COORDINATES GENERATING FUNCTIONS OF CANONICAL TRANSFORMATIONS ISOMORPHISM BETWEEN VECTOR FIELDS AND DIFFERENTIAL FORMS GLOBAL AND LOCAL HAMILTONIAN VECTOR FIELDS POISSON BRACKETS FIRST INTEGRALS AND SYMMETRIES POINCAR´E’S ABSOLUTE AND RELATIVE INTEGRAL INVARIANTS TWO FUNDAMENTAL THEOREMS VOLUME FORM ON INVARIANT SUBMANIFOLDS THE HAMILTON-JACOBI THEORY THE HAMILTON–JACOBI EQUATION SEPARATION METHOD EXAMPLES OF SEPARABLE SYSTEMS HAMILTON’S PRINCIPAL FUNCTION COMPLETELY INTEGRABLE SYSTEMS ARNOLD-LIOUVILLE’S THEOREM ANGLE-ACTION VARIABLES FUNDAMENTAL FREQUENCIES AND ORBITS ANGLE-ACTION VARIABLES OF THE HARMONIC OSCILLATOR ANGLE-ACTION VARIABLES FOR A MATERIAL POINT IN NEWTONIAN FIELDS BOHR–SOMMERFELD QUANTIZATION RULES A SKETCH OF THE HAMILTONIAN PERTURBATION THEORY OVERVIEW OF KAM THEOREM ELEMENTS OF STATISTICAL MECHANICS OF EQUILIBRIUM KINETIC THEORY OF GASES BOLTZMANN–GIBBS DISTRIBUTION EQUIPARTITION OF ENERGY IMPULSIVE DYNAMICS BALANCE EQUATIONS OF IMPULSIVE DYNAMICS FUNDAMENTAL PROBLEM OF IMPULSIVE DYNAMICS LAGRANGIAN FORMULATION OF IMPULSIVE DYNAMICS ANALYSIS OF THE LAGRANGE EQUATIONS INTRODUCTION TO FLUID MECHANICS KINEMATICS OF A CONTINUOUS SYSTEM VELOCITY GRADIENT RIGID, IRROTATIONAL, AND ISOCHORIC MOTIONS MASS CONSERVATION EQUATIONS OF BALANCE OF A PERFECT FLUID STATICS OF FLUIDS FUNDAMENTAL THEOREMS OF FLUID DYNAMICS BOUNDARY VALUE PROBLEMS FOR A PERFECT FLUID TWO-DIMENSIONAL STEADY FLOW OF A PERFECT FLUID D’ALEMBERT’S PARADOX AND THE KUTTA–JOUKOWKY THEOREM WAVES IN PERFECT GASES AN INTRODUCTION TO SPECIAL RELATIVITY GALILEAN RELATIVITY OPTICAL ISOTROPY PRINCIPLE SIMPLE CONSEQUENCES OF LORENTZ’S TRANSFORMATIONS RELATIVISTIC COMPOSITION OF VELOCITIES AND ACCELERATIONS PRINCIPLE OF RELATIVITY COVARIANCE OF MAXWELL’S EQUATIONS RELATIVISTIC DYNAMICS MINKOWSKI’S SPACE-TIME PHYSICAL MEANING OF MINKOWSKI’S SPACE-TIME FOUR-DIMENSIONAL EQUATION OF MOTION TENSOR FORMULATION OF ELECTROMAGNETISM IN VACUUM