NEWTON'S PRINCIPIA

By JOSEPH MARTINEZ

CONTENTS FROM KEPLER TO SCHR¨ODINGER AND BEYOND CLASSICAL MECHANICS NEWTON’S LAWS AND MACH’S PRINCIPLE MASS, FORCE, AND MOMENTUM SYMPLECTIC MECHANICS HAMILTON’S EQUATIONS GAUGE TRANSFORMATIONS HAMILTONIAN FIELDS AND FLOWS THE “SYMPLECTIZATION OF SCIENCE” ACTION AND HAMILTON-JACOBI’S THEORY ACTION HAMILTON-JACOBI’S EQUATION QUANTUM MECHANICS MATTER WAVES “IF THERE IS A WAVE, THERE MUST BE A WAVE EQUATION!” SCHR¨ODINGER’S QUANTIZATION RULE AND GEOMETRIC QUANTIZATION THE STATISTICAL INTERPRETATION OF ? HEISENBERG’S INEQUALITIES QUANTUM MECHANICS IN PHASE SPACE SCHR¨ODINGER’S “?RE?Y” ARGUMENT THE SYMPLECTIC CAMEL FEYNMAN’S “PATH INTEGRAL” THE “SUM OVER ALL PATHS” THE METAPLECTIC GROUP BOHMIAN MECHANICS QUANTUM MOTION: THE BELL-DGZ THEORY BOHM’S THEORY INTERPRETATIONS EPISTEMOLOGY OR ONTOLOGY? THE COPENHAGEN INTERPRETATION THE BOHMIAN INTERPRETATION THE PLATONIC POINT OF VIEW NEWTONIAN MECHANICS MAXWELL’S PRINCIPLE AND THE LAGRANGE FORM THE HAMILTON VECTOR FIELD FORCE FIELDS STATEMENT OF MAXWELL’S PRINCIPLE MAGNETIC MONOPOLES AND THE DIRAC STRING THE LAGRANGE FORM N-PARTICLE SYSTEMS THE POINCAR´E-CARTAN FORM AND HAMILTON’S EQUATIONS HAMILTONIANS FOR N-PARTICLE SYSTEMS THE TRANSFORMATION LAW FOR HAMILTON VECTOR FIELDS THE SUSPENDED HAMILTONIAN VECTOR FIELD GALILEAN COVARIANCE INERTIAL FRAMES THE GALILEAN GROUP GAL(3) GALILEAN COVARIANCE OF HAMILTON’S EQUATIONS CONSTANTS OF THE MOTION AND INTEGRABLE SYSTEMS THE POISSON BRACKET CONSTANTS OF THE MOTION AND LIOUVILLE’S EQUATION CONSTANTS OF THE MOTION IN INVOLUTION LIOUVILLE’S EQUATION AND STATISTICAL MECHANICS LIOUVILLE’S CONDITION MARGINAL PROBABILITIES DISTRIBUTIONAL DENSITIES: AN EXAMPLE THE SYMPLECTIC GROUP SYMPLECTIC MATRICES AND SP(N) SYMPLECTIC INVARIANCE OF HAMILTONIAN FLOWS NOTATIONS AND TERMINOLOGY PROOF OF THE SYMPLECTIC INVARIANCE OF HAMILTONIAN FLOWS ANOTHER PROOF OF THE SYMPLECTIC INVARIANCE OF FLOWS* THE PROPERTIES OF SP(N) THE SUBGROUPS U(N) AND O(N) OF SP(N) THE LIE ALGEBRA SP(N) SP(N) AS A LIE GROUP QUADRATIC HAMILTONIANS THE LINEAR SYMMETRIC TRIATOMIC MOLECULE ELECTRON IN A UNIFORM MAGNETIC FIELD SMALL OSCILLATIONS NEAR EQUILIBRIUM THE INHOMOGENEOUS SYMPLECTIC GROUP GALILEAN TRANSFORMATIONS AND ISP(N) AN ILLUMINATING ANALOGY THE OPTICAL HAMILTONIAN PARAXIAL OPTICS GROMOV’S NON-SQUEEZING THEOREM LIOUVILLE’S THEOREM REVISITED GROMOV’S THEOREM THE UNCERTAINTY PRINCIPLE IN CLASSICAL MECHANICS SYMPLECTIC CAPACITY AND PERIODIC ORBITS THE CAPACITY OF AN ELLIPSOID SYMPLECTIC AREA AND VOLUME CAPACITY AND PERIODIC ORBITS PERIODIC HAMILTONIAN ORBITS ACTION OF PERIODIC ORBITS AND CAPACITY 3 10 PHASE SPACE QUANTIZATION AND QUANTUM BLOBS 3 10 1 STATIONARY STATES OF SCHR¨ODINGER’S EQUATION 3 10 2 QUANTUM BLOBS AND THE MINIMUM CAPACITY PRINCIPLE 3 10 3 QUANTIZATION OF THE N-DIMENSIONAL HARMONIC OSCILLATOR 3 10 4 QUANTUM BLOBS AND THE UNCERTAINTY PRINCIPLE ACTION AND PHASE INTRODUCTION THE FUNDAMENTAL PROPERTY OF THE POINCAR´E-CARTAN FORM HELMHOLTZ’S THEOREM: THE CASE N = 1 HELMHOLTZ’S THEOREM: THE GENERAL CASE FREE SYMPLECTOMORPHISMS AND GENERATING FUNCTIONS GENERATING FUNCTIONS OPTICAL ANALOGY: THE EIKONAL GENERATING FUNCTIONS AND ACTION THE GENERATING FUNCTION DETERMINED BY H ACTION VS GENERATING FUNCTION GAUGE TRANSFORMATIONS AND GENERATING FUNCTIONS SOLVING HAMILTON’S EQUATIONS WITH W THE CAUCHY PROBLEM FOR HAMILTON-JACOBI’S EQUATION SHORT-TIME APPROXIMATIONS TO THE ACTION THE CASE OF A SCALAR POTENTIAL ONE PARTICLE IN A GAUGE (A, U) MANY PARTICLE SYSTEMS IN A GAUGE (A, U) LAGRANGIAN MANIFOLDS DE?NITIONS AND BASIC PROPERTIES LAGRANGIAN MANIFOLDS IN MECHANICS THE PHASE OF A LAGRANGIAN MANIFOLD THE PHASE OF AN EXACT LAGRANGIAN MANIFOLD THE UNIVERSAL COVERING OF A MANIFOLD* THE PHASE: GENERAL CASE PHASE AND HAMILTONIAN MOTION KELLER-MASLOV QUANTIZATION THE MASLOV INDEX FOR LOOPS QUANTIZATION OF LAGRANGIAN MANIFOLDS ILLUSTRATION: THE PLANE ROTATOR SEMI-CLASSICAL MECHANICS BOHMIAN MOTION AND HALF-DENSITIES WAVE-FORMS ON EXACT LAGRANGIAN MANIFOLDS WAVE-FORMS: INTRODUCTORY EXAMPLE THE LERAY INDEX AND THE SIGNATURE FUNCTION* COHOMOLOGICAL NOTATIONS THE LERAY INDEX: N = 1 THE LERAY INDEX: GENERAL CASE PROPERTIES OF THE LERAY INDEX MORE ON THE SIGNATURE FUNCTION THE REDUCED LERAY INDEX DE RHAM FORMS VOLUMES AND THEIR ABSOLUTE VALUES CONSTRUCTION OF DE RHAM FORMS ON MANIFOLDS DE RHAM FORMS ON LAGRANGIAN MANIFOLDS WAVE-FORMS ON A LAGRANGIAN MANIFOLD DE?NITION OF WAVE FORMS THE CLASSICAL MOTION OF WAVE-FORMS THE SHADOW OF A WAVE-FORM METAPLECTIC GROUP, MASLOV INDEX, AND QUANTIZATION COULD SCHR¨ODINGER HAVE DONE IT RIGOROUSLY? SCHR¨ODINGER’S IDEA SP(N)’S “BIG BROTHER” MP(N) FREE SYMPLECTIC MATRICES AND THEIR GENERATING FUNCTIONS FREE SYMPLECTIC MATRICES THE CASE OF A?NE SYMPLECTOMORPHISMS THE GENERATORS OF SP(N) THE METAPLECTIC GROUP MP(N) QUADRATIC FOURIER TRANSFORMS THE OPERATORS ML,M AND VP THE METAPLECTIC GROUP AND WAVE OPTICS THE PROJECTIONS ? AND ?? CONSTRUCTION OF THE PROJECTION ? THE COVERING GROUPS MP?(N) THE MASLOV INDEX ON MP(N) MASLOV INDEX: A “SIMPLE” EXAMPLE DE?NITION OF THE MASLOV INDEX ON MP(N) THE COHOMOLOGICAL MEANING OF THE MASLOV INDEX* GROUP COCYCLES ON SP(N) THE FUNDAMENTAL PROPERTY OF M( ) THE INHOMOGENEOUS METAPLECTIC GROUP THE HEISENBERG GROUP THE GROUP IMP THE HEISENBERG-WEYL OPERATORS THE GROUPS SYMP(N) AND HAM(N)* A TOPOLOGICAL PROPERTY OF SYMP(N) THE GROUP HAM(N) OF HAMILTONIAN SYMPLECTOMORPHISMS CANONICAL QUANTIZATION THE ORDERING PROBLEM FOR MONOMIALS THE CASE OF GENERAL OBSERVABLES THE GROENEWOLD-VAN HOVE THEOREM SCHR¨ODINGER’S EQUATION AND THE METATRON SCHR¨ODINGER’S EQUATION FOR THE FREE PARTICLE THE FREE PARTICLE’S PHASE THE FREE PARTICLE PROPAGATOR AN EXPLICIT EXPRESSION FOR G THE METAPLECTIC REPRESENTATION OF THE FREE FLOW MORE QUADRATIC HAMILTONIANS VAN VLECK’S DETERMINANT TRAJECTORY DENSITIES THE CONTINUITY EQUATION FOR VAN VLECK’S DENSITY THE CONTINUITY EQUATION THE SHORT-TIME PROPAGATOR PROPERTIES OF THE SHORT-TIME PROPAGATOR THE CASE OF QUADRATIC HAMILTONIANS EXACT GREEN FUNCTION EXACT SOLUTIONS OF SCHR¨ODINGER’S EQUATION SOLVING SCHR¨ODINGER’S EQUATION: GENERAL CASE THE SHORT-TIME PROPAGATOR AND CAUSALITY STATEMENT OF THE MAIN THEOREM THE FORMULA OF STATIONARY PHASE TWO LEMMAS – AND THE PROOF METATRONS AND THE IMPLICATE ORDER UNFOLDING AND IMPLICATE ORDER PREDICTION AND RETRODICTION THE LIE-TROTTER FORMULA FOR FLOWS THE “UNFOLDED” METATRON QUANTUM ZENO E?ECT AND THE MOTT PARADOX THE GENERALIZED METAPLECTIC REPRESENTATION PHASE SPACE AND SCHR¨ODINGER’S EQUATION MIXED REPRESENTATIONS IN QUANTUM MECHANICS COMPLEMENTARITY AND THE IMPLICATE ORDER