ANALYTICAL MECHANICS: COMPRISING THE KINETICS AND STATICS OF SOLIDS AND FLUIDS

By I. BEORR

CONTENTS PHYSICAL QUANTITIES AND UNITS OF MEASUREMENT PHYSICS AND THE SCIENTI?C METHOD MEASURABLE QUANTITIES AND THE INTERNATIONAL SYSTEM DISTANCES TIMES MASSES MEASURING PHYSICAL QUANTITIES DIRECT AND INDIRECT MEASUREMENTS THE DISTANCE LADDER MASS MEASUREMENTS OUT OF RANGE FOR SCALES FROM PICO TO TERA THE CONCEPT OF SPACE AND TIME THE EUCLIDEAN SPACE NEWTONIAN SPACE AND TIME FOUR-DIMENSIONAL SPACETIME (*) COSMOLOGICAL TIME TIME SYNCHRONIZATION AND DISSEMINATION TIME MEASUREMENTS WITHOUT PERIODIC PHENOMENA (*) THE LAW OF RADIOACTIVE DECAY RADIOCARBON DATING DIMENSIONAL ANALYSIS QUESTIONS AND EXERCISES VECTORS AND OPERATIONS WITH VECTORS INTRODUCTION: SCALARS, VECTORS AND TENSORS CARTESIAN COORDINATE SYSTEMS RIGHT-HANDED COORDINATE SYSTEM AND RIGHT-HAND RULE REPRESENTATION OF VECTORS VECTORS IN INTRINSIC REPRESENTATION VECTORS IN CARTESIAN REPRESENTATION UNIT VECTORS IN CARTESIAN REPRESENTATION PRODUCT OF A SCALAR AND A VECTOR SUM AND DIFFERENCE OF VECTORS SCALAR PRODUCT SCALAR PRODUCT IN INTRINSIC REPRESENTATION SCALAR PRODUCT IN CARTESIAN REPRESENTATION VECTOR PRODUCT BETWEEN VECTORS VECTOR PRODUCT IN INTRINSIC REPRESENTATION VECTOR PRODUCT IN CARTESIAN REPRESENTATION AREA AND VOLUME IN VECTOR SPACES (*) THE EQUALS ARE NOT ALL EQUAL LAWS AND PRINCIPLES, PHYSICS AND MATHEMATICS QUESTIONS KINEMATICS OF THE PARTICLE INTRODUCTION AND DE?NITIONS UNIFORM AND UNIFORMLY ACCELERATED MOTION EQUATION OF MOTION, VELOCITY AND ACCELERATION EQUATION OF MOTION VELOCITY, DE?NITION ACCELERATION, DE?NITION THE DIRECT KINEMATICS PROBLEM IN?NITESIMAL DISPLACEMENT AND IN?NITESIMAL PATH COMPOSITION OF MOTIONS AND TRAJECTORY THE INVERSE KINEMATICS PROBLEM UNIFORM AND NON-UNIFORM CIRCULAR MOTION UNIFORM CIRCULAR MOTION THE CARTESIAN REPRESENTATION NON-UNIFORM CIRCULAR MOTION POLAR PLANE COORDINATES CIRCULAR MOTION WITH CO-MOVING UNIT VECTORS DE?NITION OF POLAR AND CYLINDRICAL COORDINATES THE INTRINSIC COORDINATES POISSON’S RULES FOR MOVING UNIT VECTORS MOTION ON ANY TRAJECTORY (*) FORCES AND THE DYNAMICS OF THE PARTICLE INTRODUCTION MEASURING FORCES WEIGHT AND ANTHROPOMORPHIC “EFFORT” THE DYNAMOMETER THE WEIGHT IS A FORCE THE VECTOR NATURE OF FORCES CONSTRAINT FORCES CONTACT FRICTION BETWEEN SOLIDS NEWTON’S FIRST LAW OF MOTION NEWTON’S SECOND LAW OF MOTION KINEMATIC EFFECTS OF SOME FORCES WEIGHT FALL WITH THE PRESENCE OF VISCOUS FRICTION HARMONIC OSCILLATORS THE SIMPLE PENDULUM THE ELASTIC FORCE AND THE HARMONIC OSCILLATOR WHAT DO WE KNOW TODAY ABOUT FORCES FUNDAMENTAL INTERACTIONS FORCE FIELDS THE ARROW OF TIME FRAMES OF REFERENCE IN RELATIVE MOTION GALILEAN RELATIVITY PRINCIPLE INERTIAL FRAMES OF REFERENCE NON-INERTIAL FRAMES OF REFERENCE PSEUDO-FORCES IN NON-INERTIAL FRAMES DYNAMICS IN AN ACCELERATED VEHICLE DYNAMICS IN A ROTATING FRAME EARTH’S DRAG ACCELERATION CORIOLIS FORCE IN GUGLIELMINI’S EXPERIMENT (*) THE CORIOLIS FORCE ON THE MOTION OF TERRESTRIAL FLUIDS GENERALIZED PRINCIPLES OF RELATIVITY WORK AND ENERGY DE?NITION OF WORK WORK IN VECTORS’ INTRINSIC REPRESENTATION WORK IN CARTESIAN COORDINATES MEASUREMENT UNITS OF WORK THE WORK-ENERGY THEOREM EXAMPLES OF CALCULATING THE WORK OF A FORCE WORK OF THE WEIGHT FORCE WORK OF THE ELASTIC FORCE WORK OF THE DYNAMIC FRICTION ANTHROPOMORPHIC WORK (“WORK” IN COMMON PARLANCE) CONSERVATIVE AND NON-CONSERVATIVE FORCES POTENTIAL ENERGY MECHANICAL ENERGY AND ITS CONSERVATION DE?NITION OF MECHANICAL ENERGY MECHANICAL ENERGY WITH NON-CONSERVATIVE FORCES EXACT DIFFERENTIALS AND POTENTIAL ENERGY FORCE IS THE NEGATIVE GRADIENT OF POTENTIAL ENERGY DIFFERENTIAL OPERATORS: DIVERGENCE, CURL AND LAPLACIAN (*) DIVERGENCE CURL LAPLACIAN SCHWARZ’S THEOREM CONSERVATIVE FORCES: NULL CURL CENTRAL FORCES ARE CONSERVATIVE POWER TOWARDS THE PRINCIPLE OF ENERGY CONSERVATION DYNAMICS OF MECHANICAL SYSTEMS MASS DISTRIBUTION: DISCRETE AND CONTINUOUS SYSTEMS SYSTEMS OF MATERIAL POINTS CONTINUOUS SYSTEMS DEGREES OF FREEDOM CENTER OF MASS THE CENTER OF MASS OF A SYSTEM OF POINTS THE CENTER OF MASS OF A CONTINUOUS BODY TORQUE LINEAR MOMENTUM AND ANGULAR MOMENTUM LINEAR MOMENTUM OF A SYSTEM ANGULAR MOMENTUM OF A SYSTEM CONSERVATION OF LINEAR MOMENTUM EULER’S FIRST LAW CONSERVATION OF ANGULAR MOMENTUM EULER’S SECOND LAW NEWTON’S THIRD LAW OF MOTION COMMENT ON NEWTON’S LAWS PROPERTIES OF THE CENTER OF MASS FRAME LINEAR MOMENTUM P’ IN THE C M FRAME INTRINSIC ANGULAR MOMENTUM (SPIN) L’ IN THE C M FRAME EULER’S SECOND LAW IN THE C M FRAME KINETIC ENERGY T’ IN THE C M FRAME EQUILIBRIUM OF A RIGID BODY COLLISIONS AND DECAYS IMPULSIVE FORCES ELASTIC COLLISION BETWEEN TWO BODIES THE ONE-DIMENSIONAL CASE TWO-DIMENSIONAL CASE AND CONSTRAINING FORCES LET US RE?ECT ON THE NATURE OF THE CONSTRAINING FORCES COLLISIONS IN THE CENTER-OF-MASS FRAME ONE-DIMENSIONAL CASE IN THE CENTER-OF-MASS FRAME MOTION OF A ROCKET ENERGY-MASS CONSERVATION IN COLLISIONS AND DECAYS CONSERVATION LAWS IN TWO OR THREE BODY DECAYS ALPHA DECAY (TWO-BODY PROCESS) (*) BETA DECAY (THREE-BODY PROCESS) (*) PARTIALLY ELASTIC COLLISIONS CONSIDERATIONS ON VECTORS ISOTROPY AND HOMOGENEITY OF THE UNIVERSE TRANSLATION OF REFERENCE FRAMES ROTATION OF REFERENCE FRAMES RE?EXION OF REFERENCE FRAMES NOT ALL TRIPLETS ARE VECTORS POLAR AND AXIAL VECTORS HOW CAN AXIAL VECTORS BE DISTINGUISHED SCALAR AND PSEUDOSCALAR QUANTITIES PSEUDOSCALAR QUANTITIES IN NUCLEAR PHENOMENA NEWTON’S LAW OF GRAVITATION PRE-GALILEAN ASTRONOMICAL MEASUREMENTS EARTH’S RADIUS EARTH-MOON DISTANCE EARTH-SUN DISTANCE THE APPLE AND THE MOON LAW OF UNIVERSAL GRAVITATION THE INVERSE OF THE SQUARE OF THE DISTANCE INERTIAL MASS AND GRAVITATIONAL MASS GRAVITATIONAL POTENTIAL ENERGY POTENTIAL ENERGY OF THE WEIGHT FORCE CLASSICAL LIMITS OF RELATIVITY AND QUANTUM MECHANICS ESCAPE SPEED FROM A CELESTIAL BODY OF MASS M EVENT HORIZON MEASUREMENT OF G: THE TORSION BALANCE SPHERICAL COORDINATES (*) LINE, SURFACE AND VOLUME ELEMENTS GRAVITATIONAL POTENTIAL ENERGY: SECOND METHOD MASS IN THE CENTER OF THE SPHERE (*) MOTIONS IN GRAVITATIONAL FIELDS HISTORICAL INTRODUCTION KEPLER’S EMPIRICAL LAWS THE TWO-BODY SYSTEM ANGULAR MOMENTUM, KEPLER’S 1ST AND 2ND LAWS KEPLER’S FIRST LAW, PLANE ORBITS KEPLER’S SECOND LAW KEPLER’S THIRD LAW MECHANICAL ENERGY OF THE TWO-BODY SYSTEM QUALITATIVE SOLUTIONS FOR THE EFFECTIVE POTENTIAL KEPLER’S FIRST LAW, ELLIPTIC ORBITS (*) CONIC FUNCTIONS FIRST INTEGRAL ECCENTRICITY VERSUS ENERGY AND ANGULAR MOMENTUM ELLIPTIC SOLUTIONS: SEMI-AXES AS A FUNCTION OF E, L DEGENERACY IN CLASSICAL AND QUANTUM PHYSICS KEPLER’S THIRD LAW, REVISED THE BLACK HOLE IN THE CENTER OF THE GALAXY TWO NEUTRON STARS TRIUMPHS AND FALLS OF NEWTONIAN THEORY GRAVITATIONAL INDICATIONS FOR DARK MATTER DYNAMICS OF RIGID BODIES ANGULAR MOMENTUM AND ANGULAR VELOCITY MOMENT OF INERTIA OF RIGID BODIES MOMENT OF INERTIA OF A CYLINDER AND HOLLOW CYLINDER MOMENT OF INERTIA OF A ROD MOMENT OF INERTIA OF A ROD ON A DISK CONSERVATION OF ANGULAR MOMENTUM AND ANGULAR VELOCITY AN APPLICATION OF THE II EULER’S LAW HUYGENS-STEINER THEOREM FOR MOMENTS OF INERTIA CENTER OF GRAVITY PHYSICAL PENDULUM TORSION PENDULUM MOMENT OF INERTIA TENSOR (*) ROTATIONAL KINETIC ENERGY BODY ROLLING WITHOUT SLIDING THE MOTION OF THE WHEEL THE MOTION OF THE SPINNING TOP (*) QUESTIONS AND SUMMARY EXERCISE CONSIDERATIONS ON ENERGY WORK NEEDED TO BUILD A SYSTEM POTENTIAL ENERGY OF A BOUND SPHERICAL SYSTEM AGE OF THE SUN ENERGY CONSERVATION IN STELLAR GRAVITATIONAL COLLAPSE (*) GRAVITATIONAL FIELD AND POTENTIAL FIRST INTEGRAL FROM ENERGY CONSERVATION APPLICATION TO A FALLING PARTICLE MOTION IN A POTENTIAL ENERGY FIELD STABLE AND UNSTABLE EQUILIBRIUM PERMITTED AND FORBIDDEN REGIONS MORE ON HARMONIC MOTION (*) COMPLEX NUMBERS HARMONIC OSCILLATOR WITH COMPLEX NUMBERS MECHANICAL ENERGY OF THE HARMONIC OSCILLATOR DAMPED HARMONIC OSCILLATOR (*) OVERDAMPED MOTION DISCUSSION ON MECHANICAL ENERGY AND DEVELOPMENTS DEVELOPMENTS AND PROBLEMS OF CLASSICAL MECHANICS LAGRANGIAN AND HAMILTONIAN FORMALISMS DETERMINISM IN NEWTONIAN MECHANICS EPILOGUE