THE PROJECT PHYSICS COURSE: CONCEPTS OF MOTION

By SARAH MOORE

CONTENTS ELEMENTARY NEWTONIAN MECHANICS NEWTON’S LAWS OF MOTION SINGLE-PARTICLE MECHANICS WORK WORK-ENERGY THEOREM CONSERVATIVE FORCES ANGULAR MOMENTUM SYSTEMS OF PARTICLES CENTER OF MASS CONSERVATION LAWS NON-INERTIAL REFERENCE FRAMES TRANSLATIONAL MOTION ROTATIONAL MOTION COMBINED TRANSLATIONAL AND ROTATIONAL MOTION FOUCAULT’S PENDULUM CONSTRAINED SYSTEMS SUGGESTED REFERENCES ADDITIONAL PROBLEMS PRINCIPLE OF VIRTUAL WORK AND LAGRANGE’S EQUATIONS NEWTONIAN APPROACH TO CONSTRAINED SYSTEMS TYPES OF CONSTRAINTS HOLONOMIC CONSTRAINTS NON-HOLONOMIC CONSTRAINTS TESTING WHETHER THE CONSTRAINTS ARE HOLONOMIC PRINCIPLE OF VIRTUAL WORK A NEW PRINCIPLE OF MECHANICS METHOD OF LAGRANGE MULTIPLIERS D’ALEMBERT’S PRINCIPLE LAGRANGE’S EQUATIONS OF THE 1ST KIND SOLVING LAGRANGE’S EQUATIONS OF THE 1ST KIND LAGRANGE’S EQUATIONS OF THE 2ND KIND GENERALIZED POTENTIALS HAMILTON’S PRINCIPLE AND ACTION INTEGRALS HAMILTON’S PRINCIPLE PROOF OF HAMILON’S PRINCIPLE CONSTRAINED VARIATIONS CONSERVATION LAWS REVISITED HAMILTON’S EQUATIONS LEGENDRE TRANSFORM HAMILTONIAN 1ST-ORDER ACTION FOR HAMILTON’S EQUATIONS POISSON BRACKETS AND CANONICAL TRANSFORMATIONS POISSON BRACKETS CANONICAL TRANSFORMATIONS APPLICATIONS OF CANONICAL TRANSFORMATIONS IN?NITESIMAL CANONICAL TRANSFORMATIONS SYMMETRIES AND CONSERVED QUANTITIES TRANSITION TO QUANTUM MECHANICS CENTRAL FORCE PROBLEMS GENERAL FORMALISM ORBIT EQUATION INTEGRABLE SOLUTIONS OF THE ORBIT EQUATION KEPLER’S LAWS GRAVITATION EFFECTIVE ONE-BODY PROBLEM CLASSI?CATION OF ORBITS EQUATIONS OF MOTION AND THEIR SOLUTIONS KEPLER’S EQUATION FOURIER SERIES SOLUTION TO KEPLER’S EQUATION VIRIAL THEOREM EQUATIONS OF STATE CLOSED ORBITS DRIVEN HARMONIC OSCILLATOR EQUATION FOR U 1=R NEARLY CIRCULAR ORBITS (1ST-ORDER PERTURBATIONS) HIGHER-ORDER PERTURBATIONS ANOTHER CONSERVED QUANTITY FOR INVERSE-SQUARE-LAW FORCES MOTION OF THE MOMEMTUM VECTOR IN MOMENTUM SPACE LAPLACE-RUNGE-LENZ VECTOR THREE-BODY PROBLEM SCATTERING REVIEW OF COLLISIONS IN?NITELY-MASSIVE SECOND OBJECT FINITE-MASS SECOND OBJECT BARYCENTER (CENTER-OF-MASS) FRAME THE HARD SPHERE CENTRAL POTENTIAL SCATTERING DIFFERENTIAL CROSS SECTION SCATTERING OF A BEAM OF INCIDENT PARTICLES GRAVITATIONAL SCATTERING RUTHERFORD SCATTERING EXAMPLE: GRAVITATIONAL SLINGSHOT TRANSFORMATION TO THE LAB FRAME ELASTIC SCATTERING IN THE LAB FRAME INELASTIC SCATTERING IN THE LAB FRAME PHENOMENOLOGICAL TREATMENT OF INELASTIC SCATTERING RIGID BODY KINEMATICS GENERALIZED COORDINATES FOR A RIGID BODY ORTHOGONAL TRANSFORMATIONS, ROTATION MATRICES AND EULER ANGLES PASSIVE VERSUS ACTIVE TRANSFORMATIONS ORTHOGONAL GROUP AND SPECIAL ORTHOGONAL GROUP ROTATION MATRICES EULER’S THEOREM FOR RIGID BODY MOTION FINITE ROTATION OF A VECTOR IN?NITESIMAL ORTHOGONAL TRANSFORMATIONS INSTANTANEOUS ANGULAR VELOCITY VECTOR VELOCITY AND ACCELERATION IN THE INERTIAL AND BODY FRAMES QUATERNION REPRESENTATION OF ROTATIONS QUATERNIONS GIMBAL LOCK AND PARAMETRIZATIONS OF SOÐ3Þ RIGID BODY DYNAMICS ANGULAR MOMENTUM AND KINETIC ENERGY OF A RIGID BODY ROTATIONAL INERTIA TENSOR, PRINCIPAL AXES PARALLEL-AXIS THEOREM PRINCIPAL AXES EULER’S EQUATIONS FOR RIGID BODY MOTION SOLVING EULER’S EQUATIONS FOR SEVERAL EXAMPLES TORQUE-FREE MOTION WITH X ¼ CONST TORQUE-FREE MOTION OF A SYMMETRIC TOP SYMMETRIC TOP WITH ONE POINT FIXED PRECESSION OF THE EQUINOXES SMALL OSCILLATIONS ONE-DIMENSIONAL OSCILLATOR FREE OSCILLATIONS DAMPED OSCILLATIONS DAMPED AND DRIVEN OSCILLATIONS RESONANCE GENERAL FORMALISM—COUPLED OSCILLATIONS SOLVING THE EIGENVALUE/EIGENVECTOR EQUATION NORMAL MODES, NORMAL COORDINATES, AND GENERAL SOLUTION EXAMPLES DOUBLE PENDULUM LINEAR TRIATOMIC MOLECULE LOADED STRING DAMPED AND DRIVEN COUPLED OSCILLATIONS DAMPED SYSTEMS DAMPED AND DRIVEN SYSTEMS WAVE EQUATION TRANSITION FROM DISCRETE TO CONTINUOUS SYSTEMS VIBRATING STRING ONE-DIMENSIONAL WAVE EQUATION EIGENFUNCTION SOLUTION (SEPARATION OF VARIABLES) NORMAL FORM SOLUTION (CHARACTERISTIC COORDINATES) STRING WITH FIXED ENDPOINTS PERIODIC BOUNDARY CONDITIONS EQUIVALENCE OF EIGENFUNCTION AND NORMAL FORM SOLUTIONS IN?NITE BOUNDARY CONDITIONS EQUIVALENCE OF FOURIER TRANSFORM AND NORMAL FORM SOLUTIONS THREE-DIMENSIONAL WAVE EQUATION LAGRANGIAN AND HAMILTONIAN FORMULATIONS FOR CONTINUOUS SYSTEMS AND FIELDS LAGRANGIAN FORMULATION FOR A CONTINUOUS SYSTEM LAGRANGIAN DENSITY EQUATIONS OF MOTION FUNCTIONAL DERIVATIVE NOTATION GENERALIZATION TO MULTIPLE FIELDS AND DIMENSIONS VARIATIONAL DERIVATIVE OF THE LAGRANGIAN DENSITY HAMILTONIAN FORMULATION FOR A CONTINUOUS SYSTEM HAMILTONIAN DENSITY CONSERVED QUANTITIES STRESS-ENERGY TENSOR NOETHER’S THEOREM PROOF OF NOETHER’S THEOREM SOME SIMPLE EXAMPLES SPECIAL RELATIVITY WHY DO WE NEED SPECIAL RELATIVITY? CON?ICT BETWEEN NEWTONIAN MECHANICS AND ELECTRODYNAMICS K CALCULUS THE K FACTOR SOME CONSEQUENCES OF SPECIAL RELATIVITY LACK OF ABSOLUTE SIMULTANEITY TIME DILATION LENGTH CONTRACTION RELATIVISTIC DOPPLER EFFECT LORENTZ TRANSFORMATIONS AND THE POINCARÉ GROUP BOOSTS IN AN ARBITRARY DIRECTION TRANSFORMATION OF THE VELOCITY VECTOR SPACETIME LINE ELEMENT AND 4-VECTORS CAUSAL STRUCTURE 4-VECTORS AND INNER PRODUCT PROPER TIME RELATIVISTIC KINEMATICS 4-VELOCITY 4-ACCELERATION 4-MOMENTUM RELATIVISTIC DYNAMICS RELATIVISTIC LAGRANGIAN FORMALISM FREE PARTICLE FORCES DERIVABLE FROM A POTENTIAL RELATIVISTIC FIELD THEORY