Lecture Coverage (PHY303, Fall '01)

1.  Aug 27:  CHPT. 0 : course objective: systematic approach to non-relativistic mechanics;  CHPT. 1: defintion of vector and basic operations, scalar product and projection interpretation, vector product and geometric interpretation, determinant, triple products;
2.  Aug 29: CHPT. 1: coordinate system transformations (example: rotation), derivatives of cartesian vectors and scalar/vector products, velocity and acceleration in cartesian and plane polar coordinates;
3.  Aug 31: CHPT. 1: angular coordinate systems in 3-D: position and velocity in cylindrical coordinates, spherical coordinates: position vector and derivation of velocity via an intermediate coordinate transformation to cartesian basis vectors; CHPT. 2: remarks on Newtonian vs. Lagrangian Mechanics, Newton's first law of motion;
4.  Sep 05: CHPT. 2: Newton's 2.law of motion, mass as measure of inertia, linear momentum, Newton's 3.law, action and re-action principle, momentum conservation, general objective of 1-body Newtonian dynamics, 2.order differential equation+ 2 boundary conditions,solution of the equation of motion for a constant force;
5.  Sep 07: CHPT. 2: example: inclined plane with constant kinetic friction force, angle of kinetic friction, position-dependent forces, equation of motion, kinetic and potential energy,  work function, conservation of total (mechanical) energy, solution of the equation of motion for velocity and position, turning points and forbidden regions of the motion in x;
6.  Sep 10: CHPT. 2: example: gravitational force without/with height dependence, velocity dependent forces, linear and quadratic drag constants, approximate values in air, linear drag without external force: equation of motion, velocity v(t) and position x(t), maximal range;
7.  Sep 12: CHPT. 2:equation of motion and its solution for quadratic drag force without external force, linear drag force with constant gravitational force, concept of scaled variables, terminal velocity and characteristic time, time-behavior of initial and terminal velocity components, quadratic drag + constant force: solution of the equation of motion for velocity, comparison to linear drag case;
8.  Sep 14: CHPT. 3: approximation of typical potentials inducing oscillations with quadratic term, simple harmonic motion (SHM): equation of motion and its solution via sine-ansatz, amplitude and phase, generality of solution, velocity;
9.  Sep 17: effect of constant external force on SHM, example probelm (vertical spring), simple pendulum, energy equation for SHM: potential and total energy, velocity as function of displacement, explicit integration of the velocity equation to regenerate the general solution for displacement, turning points;
10.  Sep 21: Damped harmonic motion (DHM) with linear drag force, equation of motion, decomposition via differential operators, general solution, structure of the solutions, overdamping, critical damping (extra solution), underdamping: complex displacement;
11.  Sep 24: requirement of reality for physical displacement, underdamped solution, energy equation, energy loss rate, definition and derivation of quality factor, example;
12.  Sep 26:Forced Harmonic Motion: equation of motion with oscillating external driving force, ansatz for steady-state solution, derivation of frequency-dependence for phase shift and amplitude, resonance phenomenon;
13.  Sep 28: Discussion of frequency-dependence of FHM: maximal amplitude, resonance frequency, loss of resonance behavior, resonance width and quality factor in the weak-damping limit, zero- and high-frequency limits of the solution and physical interpretation, LCR circuit as electrical-mechanical analog;
14.  Oct 01:  MIDTERM I
15.  Oct 03: CHPT. 4: general equation of motion in 3-D (Newton's 2.law), superposition principle, work function as line-integral in 3-D, example, closed loop integral for work function, condition on the force field for its vanishing (path-independence of work function), Stokes' Theorem in 3-D;
16.  Oct 05: surface integral, definition of 3-D differential ('Del') operator and curl of a vector field, definition of conservative force, existence of potential function for conservative force, gradient of potential, path-independence and total differential of work function, total mechanical energy conservation in 3-D, example of Morse-potential in Cartesian and Spherical coordinates, separable forces and ensuing equation of motion for projectile motion, horizontal range;
17.  Oct 08: Harmonic oscillator in 2- and 3-D, vectorial form of Hooke's law, example: 3 orthogonal springs, general solution for the 2-D isotropic SHM, quadratic equation for trajectory, discriminant, ellipse solution, inclination angle, equivalence of 3-D isotropic solution, nonisotropic oscillator, energy equation;
18.  Oct 10: Charged particle in electromagnetic field, Lorentz force, constrained motion of a particle, force of constraint (e.g. from curved surface), example: particle rolling down a sphere; CHPT. 5:  Newton's law in accelerated coordinate systems, example: linear acceleration, angular accelerating (rotating) systems, velocity transformation;
19.  Oct 12: transformation of acceleration from inertial to rotating system, discussion of centripetal, Coriolis and transverse acceleration from the inertial observer's point of view, example: rolling wheel;
20.  Oct 15: example of uniform rotation above z-axis, dynamics in a rotating coordinate system, fictitious forces (Coriolis, transverse and centrifugal), example: bug crawling radially out on rotating disc;
21.  Oct 17: Effects of Earth rotation: static case, deformation of Earth, plumb line, geocentric latitude, dynamic case: projectile motion on the Earth surface, Coriolis force to lowest order in omega, equations of motion to that order , their decoupling and general solution;
22.  Oct 19: Focault pendulum, components of tension force, small angle approximation, equations of motion in x'- and y'-components, ansatz for time-dependent coordinate rotation, decoupling of the differential equations, elliptic motion with superimposed rotation, precession period;
23.  Oct 22:  CHPT. 6: Newton's law of universal gravitation, interchangeablility of source and test mass, central force, spherical mass distribution: integration of total force on test mass outside the sphere, general independence of spherical mass distributions on their radius: total mass as concentrated in the center as a specific feature of the 1/r^2 force law;
24.  Oct 24: Kepler's 3 laws of planetary motion: law of equal areas from angular momentum conservation, law of ellipsis from from 2-D equations of motion in spherical coordinates, differential equation of the orbit and its solution, conic sections (ellipse, parabola and hyperbola), eliipse discussion;
25.  Oct 26: Kepler's 'Harmonic Law' from Newton's 2. law and gravitational law, Keplerian motion, prediction of Neptune, dark matter problem, definition of gravitational field strength and potential, energy equation for the orbit and rederivation of the orbit equation, relation between total energy and orbit type;
26.  Oct 29: comet motion, astronomical units, eccentricity and orbital period, centrifugal and effective radial potential, 1-D radial equation of motion, turning points, minimal energy, bound and open orbit solutions;
27.  Oct 31: Rutherford scattering: repulsive Coulomb potential, orbit equation, impact parameter and scattering angle and their relation, differential cross section, derivation of Rutherford scattering formula, remark on deviations from the Rutherford behavior; CHPT. 7: definition of center of mass position, momentum and velocity for an ensemble of particles;
28.  Nov 02: Equation of motion for center of mass motion in the presence of external and internal forces, vanishing of the internal part for central forces, total angular momentum and its decomposition in internal (spin) and external (orbital) contributions, total kinetic energy as a sum of relative and cm motion;
29.  Nov 05:  MIDTERM II
30.  Nov 07: Example for angular momentum and kinetic energy decomposition: swinging rod, interacting 2-body system: reduced mass, equation of relative motion, examples for bound state problem: corrections to Kepler's 3.law for planetary motion, binary star system: mass determination from revolution velocities and period;
31.  Nov 09: Two-body collisions, short-range forces, conservation laws (momentum and energy) and energy-loss/-gain term, 1-D (central) collisions and coefficient of restitution, 2-D scattering: kinematics and cm-coordinates, general goal of scattering experiments, relation between incoming lab- and center of mass-velocity, relation between lab- and cm-scattering angle, relation between in- and outgoing cm-velocity;
32.  Nov 12: 2-D elastic collision limit (Q=0), heavy target limit and Rutherford scattering, equal mass case: lab scatt.-angle is half the cm-angle, inelastic collision: relation between energy loss and restitution coefficient, motion with variable mass, derivation of Newton's 2.law including mass loss term, example: rocket motion in free space; CHPT. 8: definition of Lagrange function and action;
33.  Nov 14: Hamilton's variational principle: minimum principle of the action to define the dynamics, example of free fall: definition of variations in position and velocity, Lagrange function, derivation of Newton's 2. law for free fall from Hamilton's variational principle using partial integration;
34.  Nov 16: parametrization of a variation of the true solution for the free-fall case, verification that the true solution is indeed a minimum of the action, definition of generalized coordinates, examples: simnple pendulum and 3-D dunbbell, general procedure of determing the number of generalized coordinates using equations of (holonomic) constraints, example for determining Lagarnge function: pendulum with movable support;
35.  Nov 26:
36.  Nov 28:
37.  Nov 30:
38.  Dec 03:
39.  Dec 05:
40.  Dec 07:
41.  Dec 10:
42.  Dec 12:
43.  Dec 19:  FINAL EXAM