AN ELEMENTARY TREATISE ON DYNAMICS, CONTAINING APPLICATIONS TO THERMODYNAMICS

By J. NANIL

CONTENTS SKETCH OF LAGRANGIAN FORMALISM NEWTON’S EQUATION GALILEAN TRANSFORMATIONS: PRINCIPLE OF GALILEAN RELATIVITY POINCARÉ AND LORENTZ TRANSFORMATIONS: THE PRINCIPLE OF SPECIAL RELATIVITY PRINCIPLE OF LEAST ACTION VARIATIONAL ANALYSIS GENERALIZED COORDINATES, COORDINATE TRANSFORMATIONS AND SYMMETRIES OF AN ACTION EXAMPLES OF CONTINUOUS (FIELD) SYSTEMS ACTION OF A CONSTRAINED SYSTEM THE RECIPE JUSTI?CATION OF THE RECIPE DESCRIPTION OF CONSTRAINED SYSTEM BY SINGULAR ACTION KINETIC VERSUS POTENTIAL ENERGY: FORCELESS MECHANICS OF HERTZ ELECTROMAGNETIC FIELD IN LAGRANGIAN FORMALISM MAXWELL EQUATIONS NONSINGULAR LAGRANGIAN ACTION OF ELECTRODYNAMICS MANIFESTLY POINCARÉ-INVARIANT FORMULATION IN TERMS OF A SINGULAR LAGRANGIAN ACTION NOTION OF LOCAL (GAUGE) SYMMETRY LORENTZ TRANSFORMATIONS OF THREE-DIMENSIONAL POTENTIAL: ROLE OF GAUGE SYMMETRY RELATIVISTIC PARTICLE IN ELECTROMAGNETIC FIELD SPEED OF LIGHT AND CRITICAL SPEED IN EXTERNAL FIELD POINCARÉ TRANSFORMATIONS OF ELECTRIC AND MAGNETIC FIELDS HAMILTONIAN FORMALISM DERIVATION OF HAMILTONIAN EQUATIONS PRELIMINARIES FROM LAGRANGIAN TO HAMILTONIAN EQUATIONS SHORT PRESCRIPTION FOR HAMILTONIZATION PROCEDURE, PHYSICAL INTERPRETATION OF HAMILTONIAN INVERSE PROBLEM: FROM HAMILTONIAN TO LAGRANGIAN FORMULATION POISSON BRACKET AND SYMPLECTIC MATRIX GENERAL SOLUTION TO HAMILTONIAN EQUATIONS PICTURE OF MOTION IN PHASE SPACE CONSERVED QUANTITIES AND THE POISSON BRACKET PHASE SPACE TRANSFORMATIONS AND HAMILTONIAN EQUATIONS DE?NITION OF CANONICAL TRANSFORMATION GENERALIZED HAMILTONIAN EQUATIONS: EXAMPLE OF NON-CANONICAL POISSON BRACKET HAMILTONIAN ACTION FUNCTIONAL SCHRÖDINGER EQUATION AS THE HAMILTONIAN SYSTEM LAGRANGIAN ACTION ASSOCIATED WITH THE SCHRÖDINGER EQUATION ANALOGIES BETWEEN QUANTUM MECHANICS AND ELECTRODYNAMICS PROBABILITY AS A CONSERVED CHARGE VIA THE NOETHER THEOREM FIRST-ORDER ACTION FUNCTIONAL, ROUTHIAN AND ALL THAT HAMILTONIZATION OF A THEORY WITH HIGHER-ORDER DERIVATIVES FIRST-ORDER TRICK OSTROGRADSKY METHOD CANONICAL TRANSFORMATIONS OF TWO-DIMENSIONAL PHASE SPACE TIME-INDEPENDENT CANONICAL TRANSFORMATIONS TIME-INDEPENDENT CANONICAL TRANSFORMATIONS AND SYMPLECTIC MATRIX GENERATING FUNCTION TIME-DEPENDENT CANONICAL TRANSFORMATIONS CANONICAL TRANSFORMATIONS AND SYMPLECTIC MATRIX PROPERTIES OF CANONICAL TRANSFORMATIONS INVARIANCE OF THE POISSON BRACKET (SYMPLECTIC MATRIX) IN?NITESIMAL CANONICAL TRANSFORMATIONS: HAMILTONIAN AS A GENERATOR OF EVOLUTION GENERATOR OF IN?NITESIMAL CANONICAL TRANSFORMATION GENERATING FUNCTION OF CANONICAL TRANSFORMATION FREE CANONICAL TRANSFORMATION AND ITS FUNCTION F Q0; P0; / GENERATING FUNCTION S Q; Q0; / EXAMPLES OF CANONICAL TRANSFORMATIONS EVOLUTION AS A CANONICAL TRANSFORMATION: INVARIANCE OF PHASE-SPACE VOLUME CANONICAL TRANSFORMATIONS IN PERTURBATION THEORY COORDINATES ADJUSTED TO A SURFACE TRANSFORMATION PROPERTIES OF THE HAMILTONIAN ACTION SUMMARY: EQUIVALENT DE?NITIONS FOR CANONICAL TRANSFORMATION HAMILTON–JACOBI EQUATION ACTION FUNCTIONAL AS A GENERATING FUNCTION OF EVOLUTION INTEGRAL INVARIANTS POINCARÉ-CARTAN INTEGRAL INVARIANT LINE INTEGRAL OF A VECTOR FIELD, HAMILTONIAN ACTION, POINCARÉ-CARTAN AND POINCARÉ INTEGRAL INVARIANTS INVARIANCE OF THE POINCARɖCARTAN INTEGRAL UNIVERSAL INTEGRAL INVARIANT OF POINCARÉ SOME MECHANICAL PROBLEMS IN A GEOMETRIC SETTING ANALYSIS OF TRAJECTORIES AND THE PRINCIPLE OF MAUPERTUIS TRAJECTORY: SEPARATION OF KINEMATICS FROM DYNAMICS EQUATIONS FOR TRAJECTORY IN THE HAMILTONIAN FORMULATION THE PRINCIPLE OF MAUPERTUIS FOR TRAJECTORIES LAGRANGIAN ACTION FOR TRAJECTORIES DESCRIPTION OF A POTENTIAL MOTION IN TERMS OF A PAIR OF RIEMANN SPACES BASIC NOTIONS OF RIEMANN GEOMETRY RIEMANN SPACE COVARIANT DERIVATIVE AND RIEMANN CONNECTION PARALLEL TRANSPORT: NOTIONS OF COVARIANCE AND COORDINATE INDEPENDENCE DE?NITION OF COVARIANT DERIVATIVE THROUGH PARALLEL TRANSPORT: FORMAL SOLUTION TO THE PARALLEL TRANSPORT EQUATION THE GEODESIC LINE AND ITS REPARAMETRIZATION COVARIANT EQUATION REPARAMETRIZATION COVARIANT EQUATION OF THE GEODESIC LINE EXAMPLE: A SURFACE EMBEDDED IN EUCLIDEAN SPACE SHORTEST LINE AND GEODESIC LINE: ONE MORE EXAMPLE OF A SINGULAR ACTION FORMAL GEOMETRIZATION OF MECHANICS THREE-DIMENSIONAL ACCELERATION AND SPEED OF LIGHT IN GENERAL RELATIVITY TRANSFORMATIONS, SYMMETRIES AND NOETHER THEOREM THE NOTION OF INVARIANT ACTION FUNCTIONAL COORDINATE TRANSFORMATION, INDUCED TRANSFORMATION OF DYNAMICAL VARIABLES AND SYMMETRIES OF AN ACTION EXAMPLES OF INVARIANT ACTIONS, GALILEO GROUP POINCARÉ GROUP, RELATIVISTIC PARTICLE SYMMETRIES OF EQUATIONS OF MOTION NOETHER THEOREM IN?NITESIMAL SYMMETRIES DISCUSSION OF THE NOETHER THEOREM USE OF NOETHER CHARGES FOR REDUCTION OF THE ORDER OF EQUATIONS OF MOTION EXAMPLES REPARAMETRIZATION INVARIANCE AS A LOCAL SYMMETRY SYMMETRIES OF HAMILTONIAN ACTION IN?NITESIMAL SYMMETRIES GIVEN BY CANONICAL TRANSFORMATION AND BY CHARGE STRUCTURE OF IN?NITESIMAL SYMMETRY OF A GENERAL FORM HAMILTONIAN VERSUS LAGRANGIAN GLOBAL SYMMETRY HAMILTONIAN FORMALISM FOR SINGULAR THEORIES HAMILTONIZATION OF A SINGULAR THEORY: THE RECIPE TWO TOY MODELS DIRAC PROCEDURE JUSTI?CATION OF THE HAMILTONIZATION RECIPE CON?GURATION-VELOCITY SPACE HAMILTONIZATION COMPARISON WITH THE DIRAC RECIPE CLASSI?CATION OF CONSTRAINTS CLASSICAL OBSERVABLES AND PHYSICAL CONTENT OF A DEGENERATE THEORY THEORY WITH SECOND-CLASS CONSTRAINTS: DIRAC BRACKET EXAMPLES OF THEORIES WITH SECOND-CLASS CONSTRAINTS MECHANICS WITH KINEMATIC CONSTRAINTS SINGULAR LAGRANGIAN ACTION UNDERLYING THE SCHRÖDINGER EQUATION EXAMPLES OF THEORIES WITH FIRST-CLASS CONSTRAINTS CLASSICAL MECHANICS IN REPARAMETRIZATION- INVARIANT FORM AND THE SCHRÖDINGER EQUATION RELATIVISTIC PARTICLE: THREE BASIC FORMULATIONS ELECTRODYNAMICS LOCAL SYMMETRIES AND CONSTRAINTS SYMMETRIES OF LAGRANGIAN AND HAMILTONIAN FORMULATIONS LOCAL SYMMETRY DOES NOT IMPLY A CONSERVED CHARGE FORMALISM OF EXTENDED LAGRANGIAN LOCAL SYMMETRIES OF THE EXTENDED LAGRANGIAN: DIRAC CONJECTURE LOCAL SYMMETRIES OF THE INITIAL LAGRANGIAN CONVERSION OF SECOND-CLASS CONSTRAINTS BY DEFORMATION OF LAGRANGIAN LOCAL SYMMETRIES CONVERSION IN MAXWELL–PROCA LAGRANGIAN FOR MASSIVE VECTOR FIELD CLASSICAL AND QUANTUM RELATIVISTIC MECHANICS OF A SPINNING PARTICLE VECTOR MODEL OF SPINNING PARTICLE: NON RELATIVISTIC SPIN DESCRIPTION OF NON RELATIVISTIC SPIN ON THE BASE OF TWO CONSTRAINTS AN EXAMPLE OF CLASSICAL MECHANICS WITHOUT OBSERVABLE TRAJECTORIES DESCRIPTION OF NON RELATIVISTIC SPIN ON THE BASE OF THREE CONSTRAINTS CANONICAL QUANTIZATION AND PAULI EQUATION THE STRANGE QUANTUM MECHANICS OF DIRAC EQUATION, OR WHY WE NEED A SEMICLASSICAL MODEL OF RELATIVISTIC SPIN? SPIN-TENSOR OF FRENKEL AND LORENTZ COVARIANT FORM OF SPIN FIBER BUNDLE T4 FOUR-DIMENSIONAL SPIN-VECTOR, PAULI-LUBANSKI VECTOR AND BARGMANN-MICHEL-TELEGDI VECTOR SEARCH FOR THE LAGRANGIAN OF RELATIVISTIC SPINNING PARTICLE VARIATIONAL PROBLEM FOR PRESCRIBED DIRAC’S CONSTRAINTS INTERACTION AND THE PROBLEM OF COVARIANT FORMALISM PARTICLE WITH THE FUNDAMENTAL LENGTH SCALE CLASSI?CATION OF VECTOR MODELS INTERACTION WITH ELECTROMAGNETIC FIELD MANIFESTLY COVARIANT HAMILTONIAN FORMULATION LAGRANGIAN EQUATIONS OF MOTION AND COMPARISON WITH APPROXIMATE EQUATIONS OF FRENKEL AND BARGMANN-MICHEL-TELEGDI PARAMETRIZATION OF PHYSICAL TIME AND PHYSICAL HAMILTONIAN FIRST RELATIVISTIC CORRECTIONS AND FINE STRUCTURE OF HYDROGEN SPECTRUM FAST SPINNING PARTICLE AND RAINBOW GEOMETRY INTERACTION WITH GRAVITATIONAL FIELD LAGRANGIAN OF SPINNING PARTICLE WITH GRAVIMAGNETIC MOMENT MATHISSON-PAPAPETROU-TULCZYJEW-DIXON (MPTD) EQUATIONS OF A ROTATING BODY AND SPINNING PARTICLE WITHOUT GRAVIMAGNETIC MOMENT ULTRA-RELATIVISTIC LIMIT: THE PROBLEMS WITH MPTD-EQUATIONS ROTATING BODY WITH GRAVIMAGNETIC MOMENT D1 ONE-PARTICLE QUANTUM MECHANICS OF A SPINNING PARTICLE, CANONICAL FORMALISM RELATIVISTIC COVARIANCE OF CANONICAL FORMALISM RELATIVISTIC QUANTUM MECHANICS OF TWO-COMPONENT KLEIN-GORDON EQUATION COVARIANT OPERATORS OF VECTOR MODEL PROOF OF RELATIVISTIC COVARIANCE RELATION WITH DIRAC EQUATION BIBLIOGRAPHY