ELEMENTS OF THEORETICAL MECHANICS
CONTENTS A BRIEF HISTORY OF CLASSICAL MECHANICS INTRODUCTION GREEK ANTIQUITY MIDDLE AGES AGE OF ENLIGHTENMENT VARIATIONAL METHODS IN PHYSICS THE 20??CENTURY REVOLUTION IN PHYSICS REVIEW OF NEWTONIAN MECHANICS NEWTON’S LAWS OF MOTION INERTIAL FRAMES OF REFERENCE FIRST-ORDER INTEGRALS IN NEWTONIAN MECHANICS LINEAR MOMENTUM ANGULAR MOMENTUM KINETIC ENERGY CONSERVATION LAWS IN CLASSICAL MECHANICS MOTION OF ?NITE-SIZED AND MANY-BODY SYSTEMS CENTER OF MASS OF A MANY-BODY SYSTEM TOTAL LINEAR MOMENTUM OF A MANY-BODY SYSTEM CENTER-OF-MASS DECOMPOSITION EQUATIONS OF MOTION ANGULAR MOMENTUM OF A MANY-BODY SYSTEM 2 10 WORK AND KINETIC ENERGY FOR A MANY-BODY SYSTEM 2 10 1 CENTER-OF-MASS KINETIC ENERGY 2 10 2 CONSERVATIVE FORCES AND POTENTIAL ENERGY 2 10 3 TOTAL MECHANICAL ENERGY 2 10 4 TOTAL MECHANICAL ENERGY FOR CONSERVATIVE SYSTEMS 2 11 VIRIAL THEOREM 2 12 APPLICATIONS OF NEWTON’S EQUATIONS OF MOTION 2 12 1 CONSTANT FORCE PROBLEMS 2 12 2 LINEAR RESTORING FORCE 2 12 3 POSITION-DEPENDENT CONSERVATIVE FORCES 2 12 4 CONSTRAINED MOTION 2 12 5 VELOCITY DEPENDENT FORCES 2 12 6 SYSTEMS WITH VARIABLE MASS 2 12 7 RIGID-BODY ROTATION ABOUT A BODY-?XED ROTATION AXIS 2 12 8 TIME DEPENDENT FORCES 2 13 SOLUTION OF MANY-BODY EQUATIONS OF MOTION 2 13 1 ANALYTIC SOLUTION 2 13 2 SUCCESSIVE APPROXIMATION 2 13 3 PERTURBATION METHOD 2 14 NEWTON’S LAW OF GRAVITATION 2 14 1 GRAVITATIONAL AND INERTIAL MASS 2 14 2 GRAVITATIONAL POTENTIAL ENERGY ? 2 14 3 GRAVITATIONAL POTENTIAL ? 2 14 4 POTENTIAL THEORY 2 14 5 CURL OF THE GRAVITATIONAL ?ELD 2 14 6 GAUSS’S LAW FOR GRAVITATION 2 14 7 CONDENSED FORMS OF NEWTON’S LAW OF GRAVITATION 2 15 SUMMARY LINEAR OSCILLATORS LINEAR RESTORING FORCES LINEARITY AND SUPERPOSITION GEOMETRICAL REPRESENTATIONS OF DYNAMICAL MOTION CON?GURATION SPACE (???????) STATE SPACE, (????????) PHASE SPACE, (???????) PLANE PENDULUM LINEARLY-DAMPED FREE LINEAR OSCILLATOR GENERAL SOLUTION ENERGY DISSIPATION SINUSOIDALLY-DRIVE, LINEARLY-DAMPED, LINEAR OSCILLATOR TRANSIENT RESPONSE OF A DRIVEN OSCILLATOR STEADY STATE RESPONSE OF A DRIVEN OSCILLATOR COMPLETE SOLUTION OF THE DRIVEN OSCILLATOR RESONANCE ENERGY ABSORPTION WAVE EQUATION TRAVELLING AND STANDING WAVE SOLUTIONS OF THE WAVE EQUATION WAVEFORM ANALYSIS HARMONIC DECOMPOSITION THE FREE LINEARLY-DAMPED LINEAR OSCILLATOR DAMPED LINEAR OSCILLATOR SUBJECT TO AN ARBITRARY PERIODIC FORCE 3 10 SIGNAL PROCESSING 3 11 WAVE PROPAGATION 3 11 1 PHASE, GROUP, AND SIGNAL VELOCITIES OF WAVE PACKETS 3 11 2 FOURIER TRANSFORM OF WAVE PACKETS 3 11 3 WAVE-PACKET UNCERTAINTY PRINCIPLE 3 12 SUMMARY NONLINEAR SYSTEMS AND CHAOS WEAK NONLINEARITY BIFURCATION, AND POINT ATTRACTORS LIMIT CYCLES POINCARÉ-BENDIXSON THEOREM VAN DER POL DAMPED HARMONIC OSCILLATOR HARMONICALLY-DRIVEN, LINEARLY-DAMPED, PLANE PENDULUM CLOSE TO LINEARITY ONSET OF COMPLICATION PERIOD DOUBLING AND BIFURCATION ROLLING MOTION ONSET OF CHAOS DI?ERENTIATION BETWEEN ORDERED AND CHAOTIC MOTION LYAPUNOV EXPONENT BIFURCATION DIAGRAM POINCARÉ SECTION WAVE PROPAGATION FOR NON-LINEAR SYSTEMS PHASE, GROUP, AND SIGNAL VELOCITIES SOLITON WAVE PROPAGATION SUMMARY CALCULUS OF VARIATIONS EULER’S DI?ERENTIAL EQUATION APPLICATIONS OF EULER’S EQUATION SELECTION OF THE INDEPENDENT VARIABLE FUNCTIONS WITH SEVERAL INDEPENDENT VARIABLES ??(?) EULER’S INTEGRAL EQUATION CONSTRAINED VARIATIONAL SYSTEMS HOLONOMIC CONSTRAINTS GEOMETRIC (ALGEBRAIC) EQUATIONS OF CONSTRAINT KINEMATIC (DI?ERENTIAL) EQUATIONS OF CONSTRAINT ISOPERIMETRIC (INTEGRAL) EQUATIONS OF CONSTRAINT PROPERTIES OF THE CONSTRAINT EQUATIONS TREATMENT OF CONSTRAINT FORCES IN VARIATIONAL CALCULUS GENERALIZED COORDINATES IN VARIATIONAL CALCULUS LAGRANGE MULTIPLIERS FOR HOLONOMIC CONSTRAINTS ALGEBRAIC EQUATIONS OF CONSTRAINT INTEGRAL EQUATIONS OF CONSTRAINT 5 10 GEODESIC 5 11 VARIATIONAL APPROACH TO CLASSICAL MECHANICS 5 12 SUMMARY LAGRANGIAN DYNAMICS NEWTONIAN PLAUSIBILITY ARGUMENT FOR LAGRANGIAN MECHANICS LAGRANGE EQUATIONS FROM D’ALEMBERT’S PRINCIPLE D’ALEMBERT’S PRINCIPLE OF VIRTUAL WORK TRANSFORMATION TO GENERALIZED COORDINATES LAGRANGIAN LAGRANGE EQUATIONS FROM HAMILTON’S ACTION PRINCIPLE CONSTRAINED SYSTEMS CHOICE OF GENERALIZED COORDINATES MINIMAL SET OF GENERALIZED COORDINATES LAGRANGE MULTIPLIERS APPROACH GENERALIZED FORCES APPROACH APPLYING THE EULER-LAGRANGE EQUATIONS TO CLASSICAL MECHANICS APPLICATIONS TO UNCONSTRAINED SYSTEMS APPLICATIONS TO SYSTEMS INVOLVING HOLONOMIC CONSTRAINTS APPLICATIONS INVOLVING NON-HOLONOMIC CONSTRAINTS 6 10 VELOCITY-DEPENDENT LORENTZ FORCE 6 11 TIME-DEPENDENT FORCES 6 12 IMPULSIVE FORCES 6 13 THE LAGRANGIAN VERSUS THE NEWTONIAN APPROACH TO CLASSICAL MECHANICS 6 14 SUMMARY SYMMETRIES, INVARIANCE AND THE HAMILTONIAN GENERALIZED MOMENTUM INVARIANT TRANSFORMATIONS AND NOETHER’S THEOREM ROTATIONAL INVARIANCE AND CONSERVATION OF ANGULAR MOMENTUM CYCLIC COORDINATES KINETIC ENERGY IN GENERALIZED COORDINATES GENERALIZED ENERGY AND THE HAMILTONIAN FUNCTION GENERALIZED ENERGY THEOREM GENERALIZED ENERGY AND TOTAL ENERGY 7 10 HAMILTONIAN INVARIANCE 7 11 HAMILTONIAN FOR CYCLIC COORDINATES 7 12 SYMMETRIES AND INVARIANCE 7 13 HAMILTONIAN IN CLASSICAL MECHANICS 7 14 SUMMARY HAMILTONIAN MECHANICS LEGENDRE TRANSFORMATION BETWEEN LAGRANGIAN AND HAMILTONIAN MECHANICS HAMILTON’S EQUATIONS OF MOTION CANONICAL EQUATIONS OF MOTION HAMILTONIAN IN DI?ERENT COORDINATE SYSTEMS CYLINDRICAL COORDINATES ????? SPHERICAL COORDINATES, ????? APPLICATIONS OF HAMILTONIAN DYNAMICS ROUTHIAN REDUCTION R??????- ROUTHIAN IS A HAMILTONIAN FOR THE CYCLIC VARIABLES R?????????- ROUTHIAN IS A HAMILTONIAN FOR THE NON-CYCLIC VARIABLES VARIABLE-MASS SYSTEMS ROCKET PROPULSION MOVING CHAINS HAMILTON’S ACTION PRINCIPLE STATIONARY-ACTION PRINCIPLE IN LAGRANGIAN MECHANICS STATIONARY-ACTION PRINCIPLE IN HAMILTONIAN MECHANICS ABBREVIATED ACTION HAMILTON’S PRINCIPLE APPLIED USING INITIAL BOUNDARY CONDITIONS STANDARD LAGRANGIAN GAUGE INVARIANCE OF THE STANDARD LAGRANGIAN NON-STANDARD LAGRANGIANS INVERSE VARIATIONAL CALCULUS APPLICATION OF HAMILTON’S ACTION PRINCIPLE TO MECHANICS 10 NONCONSERVATIVE SYSTEMS 10 1 INTRODUCTION 10 2 ORIGINS OF NONCONSERVATIVE MOTION 10 3 ALGEBRAIC MECHANICS FOR NONCONSERVATIVE SYSTEMS 10 4 RAYLEIGH’S DISSIPATION FUNCTION 10 4 1 GENERALIZED DISSIPATIVE FORCES FOR LINEAR VELOCITY DEPENDENCE 10 4 2 GENERALIZED DISSIPATIVE FORCES FOR NONLINEAR VELOCITY DEPENDENCE 10 4 3 LAGRANGE EQUATIONS OF MOTION 10 4 4 HAMILTONIAN MECHANICS 10 5 DISSIPATIVE LAGRANGIANS 10 6 SUMMARY 11 CONSERVATIVE TWO-BODY CENTRAL FORCES 11 1 INTRODUCTION 11 2 EQUIVALENT ONE-BODY REPRESENTATION FOR TWO-BODY MOTION 11 3 ANGULAR MOMENTUM L 11 4 EQUATIONS OF MOTION 11 5 DI?ERENTIAL ORBIT EQUATION 11 6 HAMILTONIAN 11 7 GENERAL FEATURES OF THE ORBIT SOLUTIONS 11 8 INVERSE-SQUARE, TWO-BODY, CENTRAL FORCE 11 8 1 BOUND ORBITS 11 8 2 KEPLER’S LAWS FOR BOUND PLANETARY MOTION 11 8 3 UNBOUND ORBITS 11 8 4 ECCENTRICITY VECTOR 11 9 ISOTROPIC, LINEAR, TWO-BODY, CENTRAL FORCE 11 9 1 POLAR COORDINATES 11 9 2 CARTESIAN COORDINATES 11 9 3 SYMMETRY TENSOR A0 11 10CLOSED-ORBIT STABILITY 11 11THE THREE-BODY PROBLEM 11 12TWO-BODY SCATTERING 11 12 1TOTAL TWO-BODY SCATTERING CROSS SECTION 11 12 2DI?ERENTIAL TWO-BODY SCATTERING CROSS SECTION 11 12 3IMPACT PARAMETER DEPENDENCE ON SCATTERING ANGLE 11 12 4RUTHERFORD SCATTERING 11 13TWO-BODY KINEMATICS 11 14SUMMARY 12 NON-INERTIAL REFERENCE FRAMES 12 1 INTRODUCTION 12 2 TRANSLATIONAL ACCELERATION OF A REFERENCE FRAME 12 3 ROTATING REFERENCE FRAME 12 3 1 SPATIAL TIME DERIVATIVES IN A ROTATING, NON-TRANSLATING, REFERENCE FRAME 12 3 2 GENERAL VECTOR IN A ROTATING, NON-TRANSLATING, REFERENCE FRAME 12 4 REFERENCE FRAME UNDERGOING ROTATION PLUS TRANSLATION 12 5 NEWTON’S LAW OF MOTION IN A NON-INERTIAL FRAME 12 6 LAGRANGIAN MECHANICS IN A NON-INERTIAL FRAME 12 7 CENTRIFUGAL FORCE 12 8 CORIOLIS FORCE 12 9 ROUTHIAN REDUCTION FOR ROTATING SYSTEMS 12 10E?ECTIVE GRAVITATIONAL FORCE NEAR THE SURFACE OF THE EARTH 12 11FREE MOTION ON THE EARTH 12 12WEATHER SYSTEMS 12 12 1LOW-PRESSURE SYSTEMS 12 12 2HIGH-PRESSURE SYSTEMS 12 13FOUCAULT PENDULUM 12 14SUMMARY 13 RIGID-BODY ROTATION 13 1 INTRODUCTION 13 2 RIGID-BODY COORDINATES 13 3 RIGID-BODY ROTATION ABOUT A BODY-?XED POINT 13 4 INERTIA TENSOR 13 5 MATRIX AND TENSOR FORMULATIONS OF RIGID-BODY ROTATION 13 6 PRINCIPAL AXIS SYSTEM 13 7 DIAGONALIZE THE INERTIA TENSOR 13 8 PARALLEL-AXIS THEOREM 13 9 PERPENDICULAR-AXIS THEOREM FOR PLANE LAMINAE 13 10GENERAL PROPERTIES OF THE INERTIA TENSOR 13 10 1INERTIAL EQUIVALENCE 13 10 2ORTHOGONALITY OF PRINCIPAL AXES 13 11ANGULAR MOMENTUM L AND ANGULAR VELOCITY ? VECTORS 13 12KINETIC ENERGY OF ROTATING RIGID BODY 13 13EULER ANGLES 13 14ANGULAR VELOCITY ? 13 15KINETIC ENERGY IN TERMS OF EULER ANGULAR VELOCITIES 13 16ROTATIONAL INVARIANTS 13 17EULER’S EQUATIONS OF MOTION FOR RIGID-BODY ROTATION 13 18LAGRANGE EQUATIONS OF MOTION FOR RIGID-BODY ROTATION 13 19HAMILTONIAN EQUATIONS OF MOTION FOR RIGID-BODY ROTATION 13 20TORQUE-FREE ROTATION OF AN INERTIALLY-SYMMETRIC RIGID ROTOR 13 20 1EULER’S EQUATIONS OF MOTION 13 20 2LAGRANGE EQUATIONS OF MOTION 13 21TORQUE-FREE ROTATION OF AN ASYMMETRIC RIGID ROTOR 13 22STABILITY OF TORQUE-FREE ROTATION OF AN ASYMMETRIC BODY 13 23SYMMETRIC RIGID ROTOR SUBJECT TO TORQUE ABOUT A ?XED POINT 13 24THE ROLLING WHEEL 13 25DYNAMIC BALANCING OF WHEELS 13 26ROTATION OF DEFORMABLE BODIES 13 27SUMMARY 14 COUPLED LINEAR OSCILLATORS 14 1 INTRODUCTION 14 2 TWO COUPLED LINEAR OSCILLATORS 14 3 NORMAL MODES 14 4 CENTER OF MASS OSCILLATIONS 14 5 WEAK COUPLING 14 6 GENERAL ANALYTIC THEORY FOR COUPLED LINEAR OSCILLATORS 14 6 1 KINETIC ENERGY TENSOR T 14 6 2 POTENTIAL ENERGY TENSOR V 14 6 3 EQUATIONS OF MOTION 14 6 4 SUPERPOSITION 14 6 5 EIGENFUNCTION ORTHONORMALITY 14 6 6 NORMAL COORDINATES 14 7 TWO-BODY COUPLED OSCILLATOR SYSTEMS 14 8 THREE-BODY COUPLED LINEAR OSCILLATOR SYSTEMS 14 9 MOLECULAR COUPLED OSCILLATOR SYSTEMS 14 10DISCRETE LATTICE CHAIN 14 10 1LONGITUDINAL MOTION 14 10 2TRANSVERSE MOTION 14 10 3 NORMAL MODES 14 10 4TRAVELLING WAVES 14 10 5DISPERSION 14 10 6COMPLEX WAVENUMBER 14 11DAMPED COUPLED LINEAR OSCILLATORS 14 12COLLECTIVE SYNCHRONIZATION OF COUPLED OSCILLATORS 14 13SUMMARY 15 ADVANCED HAMILTONIAN MECHANICS 15 1 INTRODUCTION 15 2 POISSON BRACKET REPRESENTATION OF HAMILTONIAN MECHANICS 15 2 1 POISSON BRACKETS 15 2 2 FUNDAMENTAL POISSON BRACKETS 15 2 3 POISSON BRACKET INVARIANCE TO CANONICAL TRANSFORMATIONS 15 2 4 CORRESPONDENCE OF THE COMMUTATOR AND THE POISSON BRACKET 15 2 5 OBSERVABLES IN HAMILTONIAN MECHANICS 15 2 6 HAMILTON’S EQUATIONS OF MOTION 15 2 7 LIOUVILLE’S THEOREM 15 3 CANONICAL TRANSFORMATIONS IN HAMILTONIAN MECHANICS 15 3 1 GENERATING FUNCTIONS 15 3 2 APPLICATIONS OF CANONICAL TRANSFORMATIONS 15 4 HAMILTON-JACOBI THEORY 15 4 1 TIME-DEPENDENT HAMILTONIAN 15 4 2 TIME-INDEPENDENT HAMILTONIAN 15 4 3 SEPARATION OF VARIABLES 15 4 4 VISUAL REPRESENTATION OF THE ACTION FUNCTION ? 15 4 5 ADVANTAGES OF HAMILTON-JACOBI THEORY 15 5 ACTION-ANGLE VARIABLES 15 5 1 CANONICAL TRANSFORMATION 15 5 2 ADIABATIC INVARIANCE OF THE ACTION VARIABLES 15 6 CANONICAL PERTURBATION THEORY 15 7 SYMPLECTIC REPRESENTATION 15 8 COMPARISON OF THE LAGRANGIAN AND HAMILTONIAN FORMULATIONS 15 9 SUMMARY 16 ANALYTICAL FORMULATIONS FOR CONTINUOUS SYSTEMS 16 1 INTRODUCTION 16 2 THE CONTINUOUS UNIFORM LINEAR CHAIN 16 3 THE LAGRANGIAN DENSITY FORMULATION FOR CONTINUOUS SYSTEMS 16 3 1 ONE SPATIAL DIMENSION 16 3 2 THREE SPATIAL DIMENSIONS 16 4 THE HAMILTONIAN DENSITY FORMULATION FOR CONTINUOUS SYSTEMS 16 5 LINEAR ELASTIC SOLIDS 16 5 1 STRESS TENSOR 16 5 2 STRAIN TENSOR 16 5 3 MODULI OF ELASTICITY 16 5 4 EQUATIONS OF MOTION IN A UNIFORM ELASTIC MEDIA 16 6 ELECTROMAGNETIC ?ELD THEORY 16 6 1 MAXWELL STRESS TENSOR 16 6 2 MOMENTUM IN THE ELECTROMAGNETIC ?ELD 16 7 IDEAL ?UID DYNAMICS 16 7 1 CONTINUITY EQUATION 16 7 2 EULER’S HYDRODYNAMIC EQUATION 16 7 3 IRROTATIONAL ?OW AND BERNOULLI’S EQUATION 16 7 4 GAS ?OW 16 8 VISCOUS ?UID DYNAMICS 16 8 1 NAVIER-STOKES EQUATION 16 8 2 REYNOLDS NUMBER 16 8 3 LAMINAR AND TURBULENT ?UID ?OW 16 9 SUMMARY AND IMPLICATIONS 17 RELATIVISTIC MECHANICS 17 1 INTRODUCTION 17 2 GALILEAN INVARIANCE 17 3 SPECIAL THEORY OF RELATIVITY 17 3 1 EINSTEIN POSTULATES 17 3 2 LORENTZ TRANSFORMATION 17 3 3 TIME DILATION 17 3 4 LENGTH CONTRACTION 17 3 5 SIMULTANEITY 17 4 RELATIVISTIC KINEMATICS 17 4 1 VELOCITY TRANSFORMATIONS 17 4 2 MOMENTUM 17 4 3 CENTER OF MOMENTUM COORDINATE SYSTEM 17 4 4 FORCE 17 4 5 ENERGY 17 5 GEOMETRY OF SPACE-TIME 17 5 1 FOUR-DIMENSIONAL SPACE-TIME 17 5 2 FOUR-VECTOR SCALAR PRODUCTS 17 5 3 MINKOWSKI SPACE-TIME 17 5 4 MOMENTUM-ENERGY FOUR VECTOR 17 6 LORENTZ-INVARIANT FORMULATION OF LAGRANGIAN MECHANICS 17 6 1 PARAMETRIC FORMULATION 17 6 2 EXTENDED LAGRANGIAN 17 6 3 EXTENDED GENERALIZED MOMENTA 17 6 4 EXTENDED LAGRANGE EQUATIONS OF MOTION 17 7 LORENTZ-INVARIANT FORMULATIONS OF HAMILTONIAN MECHANICS 17 7 1 EXTENDED CANONICAL FORMALISM 17 7 2 EXTENDED POISSON BRACKET REPRESENTATION 17 7 3 EXTENDED CANONICAL TRANSFORMATION AND HAMILTON-JACOBI THEORY 17 7 4 VALIDITY OF THE EXTENDED HAMILTON-LAGRANGE FORMALISM 17 8 THE GENERAL THEORY OF RELATIVITY 17 8 1 FUNDAMENTAL CONCEPTS 17 8 2 EINSTEIN’S POSTULATES OF THE GENERAL THEORY OF RELATIVITY 17 8 3 EXPERIMENTAL EVIDENCE 17 9 IMPLICATIONS OF RELATIVISTIC THEORY TO CLASSICAL MECHANICS 17 10SUMMARY 18 THE TRANSITION TO QUANTUM PHYSICS 18 1 INTRODUCTION 18 2 BRIEF SUMMARY OF THE ORIGINS OF QUANTUM THEORY 18 2 1 BOHR MODEL OF THE ATOM 18 2 2 QUANTIZATION 18 2 3 WAVE-PARTICLE DUALITY 18 3 HAMILTONIAN IN QUANTUM THEORY 18 3 1 HEISENBERG’S MATRIX-MECHANICS REPRESENTATION 18 3 2 SCHRÖDINGER’S WAVE-MECHANICS REPRESENTATION 18 4 LAGRANGIAN REPRESENTATION IN QUANTUM THEORY 18 5 CORRESPONDENCE PRINCIPLE 18 6 SUMMARY 19 EPILOGUE CHAPTER GREEK ANTIQUITY THE GREAT PHILOSOPHERS IN ANCIENT GREECE PLAYED A KEY ROLE BY USING THE ASTRONOMICAL WORK OF THE BABYLONIANS TO DEVELOP SCIENTI?C THEORIES OF MECHANICS THALES OF MILETUS (624 - 547BC), THE ?RST OF THE SEVEN GREAT GREEK PHILOSOPHERS, DEVELOPED GEOMETRY, AND IS HAILED AS THE ?RST TRUE MATHEMATICIAN PYTHAGORUS (570 - 495BC) DEVELOPED MATHEMATICS, AND POSTULATED THAT THE EARTH IS SPHERICAL DEMOCRITUS (460 - 370BC) HAS BEEN CALLED THE FATHER OF MODERN SCIENCE, WHILE SOCRATES (469 - 399BC) IS RENOWNED FOR HIS CONTRIBUTIONS TO ETHICS PLATO (427-347 B C ) WHO WAS A MATHEMATICIAN AND STUDENT OF SOCRATES, WROTE IMPORTANT PHILOSOPHICAL DIALOGUES HE FOUNDED THE ACADEMY IN ATHENS WHICH WAS THE ?RST INSTITUTION OF HIGHER LEARNING IN THE WESTERN WORLD THAT HELPED LAY THE FOUNDATIONS OF WESTERN PHILOSOPHY AND SCIENCE ARISTOTLE (384-322 B C ) IS AN IMPORTANT FOUNDER OF WESTERN PHILOSOPHY ENCOMPASSING ETHICS, LOGIC, SCIENCE, AND POLITICS HIS VIEWS ON THE PHYSICAL SCIENCES PROFOUNDLY IN?UENCED MEDIEVAL SCHOLARSHIP THAT EXTENDED WELL INTO THE RENAISSANCE HE PRESENTED THE ?RST IMPLIED FORMULATION OF THE PRINCIPLE OF VIRTUAL WORK IN STATICS, AND HIS STATEMENT THAT “WHAT IS LOST IN VELOCITY IS GAINED IN FORCE” IS A VEILED REFERENCE TO KINETIC AND POTENTIAL ENERGY HE ADOPTED AN EARTH CENTERED MODEL OF THE UNIVERSE ARISTARCHUS (310 B C ) ARGUED THAT THE EARTH ORBITED THE SUN AND USED MEASUREMENTS TO IMPLY THE RELATIVE DISTANCES OF THE MOON AND THE SUN THE GREEK PHILOSOPHERS WERE RELATIVELY ADVANCED IN LOGIC AND MATHEMATICS AND DEVELOPED CONCEPTS THAT ENABLED THEM TO CALCULATE AREAS AND PERIMETERS UNFORTUNATELY THEIR PHILOSOPHICAL APPROACH NEGLECTED COLLECTING QUANTITATIVE AND SYSTEMATIC DATA THAT IS AN ESSENTIAL INGREDIENT TO THE ADVANCEMENT OF SCIENCE ARCHIMEDES (287-212 B C ) REPRESENTED THE CULMINATION OF SCIENCE IN ANCIENT GREECE AS AN ENGINEER HE DESIGNED MACHINES OF WAR, WHILE AS A SCIENTIST HE MADE SIGNI?CANT CONTRIBUTIONS TO HYDROSTATICS AND THE PRINCIPLE OF THE LEVER AS A MATHEMATICIAN, HE APPLIED IN?NITESSIMALS IN A WAY THAT IS REMINISCENT OF MODERN INTEGRAL CALCULUS, WHICH HE USED TO DERIVE A VALUE FOR ??UNFORTUNATELY MUCH OF THE WORK OF THE BRILLIANT ARCHIMEDES SUBSEQUENTLY FELL INTO OBLIVION HERO OF ALEXANDRIA (10 - 70 A D ) DESCRIBED THE PRINCIPLE OF RE?ECTION THAT LIGHT TAKES THE SHORTEST PATH THIS IS AN EARLY ILLUSTRATION OF VARIATIONAL PRINCIPLE MIDDLE AGES THE DECLINE AND FALL OF THE ROMAN EMPIRE IN ?410 A D MARKS THE END OF CLASSICAL ANTIQUITY, AND THE BEGINNING OF THE DARK AGES IN WESTERN EUROPE (CHRISTENDOM), WHILE THE MUSLIM SCHOLARS IN EASTERN EUROPE CONTINUED TO MAKE PROGRESS IN ASTRONOMY AND MATHEMATICS FOR EXAMPLE, IN EGYPT, ALHAZEN (965