CLASSICAL MECHANICS

By H. LISIR

CONTENTS 1 INTRODUCTION SPACE AND TIME NEWTON’S LAWS THE CONCEPTS OF MASS AND FORCE EXTERNAL FORCES SUMMARY 2 LINEAR MOTION CONSERVATIVE FORCES; CONSERVATION OF ENERGY MOTION NEAR EQUILIBRIUM; THE HARMONIC OSCILLATOR COMPLEX REPRESENTATION THE LAW OF CONSERVATION OF ENERGY THE DAMPED OSCILLATOR OSCILLATOR UNDER SIMPLE PERIODIC FORCE GENERAL PERIODIC FORCE IMPULSIVE FORCES; THE GREEN’S FUNCTION METHOD COLLISION PROBLEMS 3 ENERGY AND ANGULAR MOMENTUM ENERGY; CONSERVATIVE FORCES PROJECTILES MOMENTS; ANGULAR MOMENTUM CENTRAL FORCES; CONSERVATION OF ANGULAR MOMENTUM POLAR CO-ORDINATES THE CALCULUS OF VARIATIONS HAMILTON’S PRINCIPLE; LAGRANGE’S EQUATIONS 4 CENTRAL CONSERVATIVE FORCES THE ISOTROPIC HARMONIC OSCILLATOR THE CONSERVATION LAWS THE INVERSE SQUARE LAW ORBITS SCATTERING CROSS-SECTIONS MEAN FREE PATH RUTHERFORD SCATTERING 5 ROTATING FRAMES ANGULAR VELOCITY; RATE OF CHANGE OF A VECTOR PARTICLE IN A UNIFORM MAGNETIC FIELD ACCELERATION; APPARENT GRAVITY CORIOLIS FORCE LARMOR E?ECT ANGULAR MOMENTUM AND THE LARMOR E?ECT 6 POTENTIAL THEORY GRAVITATIONAL AND ELECTROSTATIC POTENTIALS THE DIPOLE AND QUADRUPOLE SPHERICAL CHARGE DISTRIBUTIONS EXPANSION OF POTENTIAL AT LARGE DISTANCES THE SHAPE OF THE EARTH THE TIDES THE FIELD EQUATIONS CONTENTS 7 THE TWO-BODY PROBLEM CENTRE-OF-MASS AND RELATIVE CO-ORDINATES THE CENTRE-OF-MASS FRAME ELASTIC COLLISIONS CM AND LAB CROSS-SECTIONS 8 MANY-BODY SYSTEMS MOMENTUM; CENTRE-OF-MASS MOTION ANGULAR MOMENTUM; CENTRAL INTERNAL FORCES THE EARTH–MOON SYSTEM LAGRANGE’S EQUATIONS 9 RIGID BODIES BASIC PRINCIPLES ROTATION ABOUT AN AXIS PERPENDICULAR COMPONENTS OF ANGULAR MOMENTUM PRINCIPAL AXES OF INERTIA CALCULATION OF MOMENTS OF INERTIA E?ECT OF A SMALL FORCE ON THE AXIS INSTANTANEOUS ANGULAR VELOCITY ROTATION ABOUT A PRINCIPAL AXIS EULER’S ANGLES 10 LAGRANGIAN MECHANICS GENERALIZED CO-ORDINATES; HOLONOMIC SYSTEMS PRECESSION OF A SYMMETRIC TOP PENDULUM CONSTRAINED TO ROTATE ABOUT AN AXIS CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD THE STRETCHED STRING 11 SMALL OSCILLATIONS AND NORMAL MODES ORTHOGONAL CO-ORDINATES EQUATIONS OF MOTION FOR SMALL OSCILLATIONS NORMAL MODES COUPLED OSCILLATORS OSCILLATIONS OF PARTICLES ON A STRING NORMAL MODES OF A STRETCHED STRING 12 HAMILTONIAN MECHANICS HAMILTON’S EQUATIONS CONSERVATION OF ENERGY IGNORABLE CO-ORDINATES GENERAL MOTION OF THE SYMMETRIC TOP LIOUVILLE’S THEOREM SYMMETRIES AND CONSERVATION LAWS GALILEAN TRANSFORMATIONS 13 DYNAMICAL SYSTEMS AND THEIR GEOMETRY PHASE SPACE AND PHASE PORTRAITS FIRST-ORDER SYSTEMS — THE PHASE LINE (N = 1) SECOND-ORDER SYSTEMS — THE PHASE PLANE (N = 2) PREY–PREDATOR, COMPETING-SPECIES SYSTEMS AND WAR LIMIT CYCLES SYSTEMS OF THIRD (AND HIGHER) ORDER SENSITIVITY TO INITIAL CONDITIONS AND PREDICTABILITY 14 ORDER AND CHAOS IN HAMILTONIAN SYSTEMS INTEGRABILITY SURFACES OF SECTION ACTION/ANGLE VARIABLES SOME HAMILTONIAN SYSTEMS WHICH EXHIBIT CHAOS SLOW CHANGE OF PARAMETERS — ADIABATIC INVARIANCE NEAR-INTEGRABLE SYSTEMS