THEORETICAL MECHANICS FOR SIXTH FORMS
CONTENTS CONCEPTUAL BASIS OF CLASSICAL MECHANICS NEWTON’S THREE LAWS HOMOGENEITY AND ISOTROPY SOLUTION PROCESS CONSERVATIVE FORCES SIMPLE HARMONIC MOTION DAMPED AND FORCED OSCILLATOR DAMPED OSCILLATOR FORCED OSCILLATOR THE SIMPLE PENDULUM PROBLEM CONSERVATION PRINCIPLES CONSERVATION OF LINEAR MOMENTUM CONSERVATION OF ANGULAR MOMENTUM CONSERVATION OF ENERGY PERTURBATIVE ANALYSIS AND THE QUARTIC OSCILLATOR 1 10 REWRITING NEWTON’S SECOND LAW IN TERMS OF KINETIC AND POTENTIAL ENERGY IN A CONSERVATIVE SYSTEM 1 11 SUMMARY CENTRAL FORCE PROBLEMS INERTIAL AND GRAVITATIONAL MASS: PRINCIPLE OF EQUIVALENCE DERIVATION OF KEPLER’S THREE LAWS PROPERTIES AND EQUATIONS OF ORBITS INTEGRAL REPRESENTATIONS A GENERAL CLASS OF POWER LAW POTENTIALS MAPPING THE GENERAL CLASS OF POTENTIALS: ORBIT EQUATION FOR THE INVERSE SQUARE LAW PROBLEM COULOMB AND ISOTROPIC OSCILLATOR POTENTIALS LAPLACE–RUNGE–LENZ VECTOR SUMMARY LAGRANGIAN FORMULATION IN MECHANICS CONSTRAINTS AND GENERALIZED COORDINATES FORMULATION OF D’ALEMBERT’S PRINCIPLE KINETIC ENERGY OF A HOLONOMIC SYSTEM LAGRANGE’S EQUATIONS OF MOTION LAGRANGE’S EQUATIONS FOR SOME SIMPLE SYSTEMS PLANE PENDULUM SPHERICAL PENDULUM BINARY STAR SYSTEM A SYSTEM WITH FOUR DEGREES OF FREEDOM THE PROBLEM OF A DAMPED OSCILLATOR A CONSERVATIVE SCLERONOMIC SYSTEM IGNORABLE COORDINATES: ROUTH’S PROCEDURE OF SOLUTION LIOUVILLE’S CLASS OF LAGRANGIANS SMALL OSCILLATIONS HAMILTONIAN AND POISSON BRACKET THE HAMILTONIAN HAMILTONIAN CANONICAL EQUATIONS OF MOTION POISSON BRACKET PROPERTIES OF POISSON BRACKET POISSON THEOREM ANGULAR MOMENTUM LIOUVILLE’S THEOREM THE CASE OF SINGULAR LAGRANGIANS HIGHER DERIVATIVE CLASSICAL SYSTEMS 4 10 THE PAIS–UHLENBECK OSCILLATOR 4 11 SUMMARY DYNAMICAL SYSTEMS: AN OVERVIEW BASIC NOTIONS AND PRELIMINARIES SIMPLE EXAMPLES FROM CLASSICAL MECHANICS ANALYSIS OF A LINEAR SYSTEM NONLINEAR SYSTEMS: PROCESS OF LINEARIZATION LOTKA–VOLTERRA MODEL STABILITY OF SOLUTIONS: LYAPUNOV FUNCTION VAN DER POL OSCILLATOR AND LIMIT CYCLES BIFURCATIONS ACTION PRINCIPLES THE PRINCIPLE OF STATIONARY ACTION COROLLARIES CONTINUOUS SYSTEMS: UNIFORM STRING PROBLEM NORMAL MODES OF OSCILLATION EXTENDED POINT TRANSFORMATION AND ?VARIATION ?AND ? VARIATIONS BRACHISTOCHRONE PROBLEM MOTION IN NONINERTIAL COORDINATE SYSTEMS ROTATING FRAMES BASIC EQUATIONS SOME REMARKS ON THE CORIOLIS FORCE E?ECTIVE GRAVITATIONAL CONSTANT FOUCAULT’S PENDULUM NONPOTENTIAL FORCE EXAMPLES SYMMETRIES AND CONSERVED QUANTITIES CONDITION OF INVARIANCE AND NOETHER’S THEOREM OPERATOR APPROACH SYMMETRY OPERATOR PARITY TRANSFORMATION TIME-REVERSAL SYMMETRY VIRIAL THEOREM HAMILTON–JACOBI EQUATION AND ACTION-ANGLE VARIABLES CANONICAL TRANSFORMATION SYMPLECTIC PROPERTY IDEA OF A GENERATING FUNCTION TYPES OF TIME-DEPENDENT CANONICAL TRANSFORMATIONS TYPE I CANONICAL TRANSFORMATION TYPE II CANONICAL TRANSFORMATION TYPE III CANONICAL TRANSFORMATION TYPE IV CANONICAL TRANSFORMATION IN?NITESIMAL CANONICAL TRANSFORMATIONS HAMILTON–JACOBI EQUATION TIME INDEPENDENT HAMILTON–JACOBI EQUATION: HAMILTON’S CHARACTERISTIC FUNCTION OTHER VARIANTS OF HAMILTON–JACOBI EQUATION ACTION-ANGLE VARIABLES MOTION OF A PARTICLE IN A 2-DIMENSIONAL RECTANGULAR WELL POSSIBLE TRAJECTORIES PERIODIC TRAJECTORIES SOME EXPLICIT EXAMPLES FOR PERIODIC TRAJECTORIES OPEN TRAJECTORIES SPECIAL TRAJECTORIES WHEN THE BILLIARD BALL HITS A CORNER REFERENCES CHAPTER REWRITING NEWTON’S SECOND LAW IN TERMS OF KINETIC AND POTENTIAL ENERGY IN A CONSERVATIVE SYSTEM CLASSICAL MECHANICS IS THE STUDY OF PHYSICAL LAWS THAT CONTROL THE MOTION OF MATERIAL OBJECTS WHICH ARE UNDER THE ACTION OF A FORCE OR SYSTEM OF FORCES IT PROVIDES THE BASIS FOR THE GROWTH OF MODERN SCIENCE IT HAS APPLICATIONS THAT COVER AREAS SUCH AS PHYSICS, CHEMISTRY, APPLIED MATHEMATICS, BIOLOGY AND EN- GINEERING SCIENCES IN PARTICULAR, IT SEEKS TO ADDRESS AND EXPLAIN THE DYNAMICS OF PARTICLES AND RIGID BODIES, GENERAL CLASSES OF INTERACTIVE SYSTEMS, ROTATING EARTH PROBLEMS, MOTION OF CHARGED OBJECTS, PLANETARY MOTIONS AROUND THE SUN AND MODELING OF BIOLOGICAL SYSTEMS CLASSICAL MECHANICS HAS AN EXTRAORDINARILY RICH HISTORY THAT BEGAN ABOUT THE TIME OF GALILEO (1564–1642) ALTHOUGH ITS BA- SIC FOUNDATIONS WERE LAID LATER BY NEWTON (1642–1727) IN HIS FAMOUS TREATISE, THE PRINCIPIA HE ENUMERATED A SET OF THREE AXIOMS WHICH BECAME THE CORNER- STONE IN EXPLAINING MOST OF THE QUALITATIVE FEATURES OF CLASSICAL MECHANICS IN THIS CHAPTER WE DISCUSS THESE LAWS AND PROVIDE A MULTI-LAYERED PERSPECTIVE ON THEM NEWTON’S THREE LAWS NEWTON’S THREE LAWS MAY BE STATED AS FOLLOWS: 1 EVERY BODY CONTINUES IN ITS STATE OF REST OR OF UNIFORM MOTION IN A STRAIGHT LINE UNLESS IT IS COMPELLED BY EXTERNAL FORCES TO CHANGE THE STATE 2 THE RATE OF CHANGE OF MOMENTUM IS PROPORTIONAL TO THE IMPRESSED FORCE AND TAKES PLACE ALONG THE DIRECTION IN WHICH THE FORCE ACTS 3 TO EVERY ACTION THERE IS AN EQUAL AND OPPOSITE REACTION THE ?RST LAW PROVIDES THE CONCEPT OF INERTIA IT IS ALSO REFERRED TO AS THE LAW OF INERTIA THE IDEA OF ABSOLUTENESS OF SPACE (EUCLIDEAN) AND TIME IS INHER- ENT IN THE NEWTONIAN SCHEME OF THINGS BOTH ARE LOOKED UPON AS INDEPENDENT ENTITIES AS NEWTON NOTED IN HIS PRINCIPIA: “ABSOLUTE SPACE, IN ITS OWN NATURE, WITHOUT REGARD TO ANYTHING EXTERNAL, REMAINS ALWAYS SIMILAR AND IMMOVABLE ” ABOUT ABSOLUTENESS OF TIME, WHICH IS MEASURED VIS-A-VIS THE MOTION OF DI?ER- ENT OBJECTS AND WHICH EVOLVES CONTINUOUSLY, NEWTON OPINED THAT “ABSOLUTE, TRUE, AND MATHEMATICAL TIME, OF ITSELF, AND FROM ITS OWN NATURE, ?OWS EQUABLY WITHOUT REGARD TO ANYTHING EXTERNAL, AND BY ANOTHER NAME IS CALLED DURATION ” DESPITE THE INDEPENDENCE OF SPACE AND TIME, THEIR CORRELATION CANNOT BE DE- NIED IN THAT FOR A MATERIAL POINT CHARACTERIZED ONLY BY ITS LOCATION WITHOUT ANY REFERENCE TO TIME IS NOT OF MUCH SIGNI?CANCE CONVERSELY, THE HAPPENING OF AN EVENT AT A PARTICULAR POINT OF TIME OR FOR A CERTAIN DURATION WITH A PRIORI NO KNOWLEDGE OF ITS LOCATION LOOKS CONCEPTUALLY ABSURD THERE ARE ACTUALLY TWO ASPECTS TO THE ?RST LAW THAT MAY BE NECESSARY TO BE KEPT IN MIND IT AT ONCE SEEKS TO ADDRESS BOTH A STATIONARY OBJECT AND A MOVING ONE FOR THE STATIONARY OBJECT IT SAYS THAT AN OBJECT AT REST CONTINUES TO DO SO UNLESS ACTED UPON BY AN EXTERNAL FORCE FOR A MOVING OBJECT THE LAW HOLDS THAT A PARTICLE IN MOTION REMAINS IN MOTION (IN A STRAIGHT LINE) WITH THE SAME SPEED AND DIRECTION UNLESS OF COURSE IT IS INTERACTED UPON BY AN EXTERNAL FORCE NEWTON’S ?RST LAW ALSO DE?NES A CLASS OF REFERENCE FRAMES WHICH ARE CALLED INERTIAL FRAMES THESE REFERENCE FRAMES ARE NOT SUPPOSED TO UNDERGO ANY TYPE OF ACCELERATION IN OTHER WORDS, AN INERTIAL FRAME MOVES WITH A CONSTANT VE- LOCITY OR IS AT REST WITH RESPECT TO ANY OTHER INERTIAL FRAME IN 1886 LUDWIG LANGE, A GERMAN PHYSICIST, DE?NED AN INERTIAL FRAME TO BE: “A REFERENCE FRAME IN WHICH A MASS POINT THROWN FROM THE SAME POINT IN THREE DI?ERENT NON- COPLANAR DIRECTIONS FOLLOWS RECTILINEAR PATHS EACH TIME IT IS THROWN, IS CALLED AN INERTIAL FRAME ” AN INERTIAL FRAME MAY BE LOOKED UPON AS THE ONE THAT IS ?XED RELATIVE TO THE AVERAGE POSITION OF A ?XED STAR OR THAT IS MOVING WITH A CONSTANT VELOCITY AND WITHOUT ANY ROTATION RELATIVE TO IT ACCORDING TO NEW- TON, ABSOLUTE SPACE AND TIME FORM A CONVENIENT BACKGROUND AGAINST WHICH WE VISUALIZE OCCURRENCE OF PHYSICAL PHENOMENA BUT THE BACKGROUND ITSELF CANNOT BE IN?UENCED BY PHYSICAL EVENTS THEMSELVES TO DESCRIBE A PHYSICAL EVENT, THEORETICAL MECHANICS FOR SIXTH FORMS