Department of Physics

Classical Mechanics (Phys. 461/661)

Tuesday, Thursday 4:40-6:55 pm 227-SC Winter 1998
Instructor Office Phone  Homepage
Prof. Mark I. Stockman 455 SA  (404)651-3221

G.F.Fowles and G.L.Cassiday, Analytical Mechanics, Harcourt Brace, Fort Worth, Texas, 1993.
H.Goldstein, Classical Mechanics, Addison-Wesley, Reading, Massachusetts, 1980

Additional Texts:

L.D.Landau and E.M.Lifshitz, Mechanics, Addison-Wesley, Reading, Massachusetts, 1960.
A.L.Fetter and J.D.Walecka, Theoretical Mechanics of Particles and Continua, McGraw-Hill, New York,1980


Homework: 40%, Midterm Examination: 20%, Final Examination: 40%.

Outline of the Course

  1. Newton's Laws and Conservation Laws (Chs.2 and 7,8)
    1. Three Newton's laws in their contemporary understanding. The First Law as a definition of inertial reference systems
    2. Forces, masses and the Second and Third Laws.
    3. Examples of forces (Coulomb, gravitational, friction, etc.).
    4. Third Newton's Law and systems of many particles.
    5. Linear momentum and its conservation. Center of mass (c.o.m.) (7.1).
    6. Reaction forces and the equation of rocket motion. (7.6)
    7. Central forces and conservation of angular momentum. Torque and the angular-momentum theorem in the lab system (7.2)
    8. Non-central forces and unavoidability of internal angular momentum (spin).
    9. Angular momentum theorem in the c.o.m. system (8.5)
    10. Necessary equilibrium conditions and static as a part of mechanics. Applications: Stability of a bike and the cone of friction. Stability of a turning car with respect to sliding, skidding, and rolling-over. How to corner: where to break or accelerate, the mechanics of car racing. Mechanical basis of car and road design. Advantages and disadvantages of forward-and rear-wheel drives. Why a formula 1 car is designed this way?
  2. Newton's Mechanics of Rectilinear Motion (Ch.2)
    1. Launches of a projectile near the earth surface.
    2. Effects of friction, maximum shooting distance as the function of the inclination angle (greater or less then 45 degrees?).
    3. Rocket motion with and without friction. Why do we need multistage rockets?
    4. Notion of impulse and sudden-force approximation.
    5. Forces that depend on position. Potential, kinetic and total energy.
      1. Applications: General solution for one dimensional potential motion. Vertical launches of a projectile into the space. The escape velocity and time of flight. Oscillator motion from the general solution. See notebook for Mathematica
  3. Oscillations (Ch.3)
    1. Harmonic (linear) oscillator.
    2. Energy for linear oscillator. Free motion without friction. The underdamped, critically damped and overdamped harmonic oscillator. Oscillation frequency, amplitude and damping rate.
    3. Forced motion of a damped linear oscillator. The general solution, and transient and stationary regimes.
    4. The response function and resonance. Free-oscillation and resonance frequencies (they are different). Lorentz contour and the resonance quality factor.
    5. Electrical analogy and selectivity of a radio receiver.
    6. Production of heat by an oscillator and the imaginary part of the response function.
    7. Harmonic oscillator driven by a periodic non-harmonic force. The general solution as a Fourier series. Harmonics, higher-order resonances and filters.
    8. Exact solutions for sudden forces (kicked oscillators). Electronic transitions in a molecule as a sudden perturbation of a molecular oscillator (Franck-Condon principle).
    9. The eigenvalue equation and normal modes
  4. Lagrangian Formulation
    1. Holonomic and non-holonomic systems. Generalized coordinates and velocities.
    2. The Lagrangian function and equations.
      1. Examples: One-dimensional motion. Three-dimensional motion in a central field. Pulleys, blocks and cables.
    3. Ignorable variables, transformation invariance and conservation laws.
    4. Hamilton's principal function and the least action principle. Euler-Lagrange equations for the variational problem as an alternative basis for mechanics.
  5. One Particle in Three Dimensions (Ch.4)
    1. The del operator, the scalar and vector potentials.
    2. The conditions of potentiality of a force field.
    3. The scalar potential as a solution of the Poisson equation and as a contour integral, equivalence of the both.
    4. Examples of potential and non-potential forces.
    5. Central forces and conservation of angular momentum revisited.
    6. Harmonic oscillator in three dimensions: the direct solution of the Newton's equations. An anisotropic oscillator, non-closed and closed orbits, multiple frequencies and Lissajous figures.
    7. The isotropic oscillator, the amplitude and phase of motion.
  6. General Solution for One Particle in a Central Potential Field 
    1. Angular momentum conservation and areal velocity.
    2. Energy equations of motion for a particle in a three-dimensional central field. Plane trajectories and the polar coordinates.
    3. The general integral solution for radial motion and the form of a trajectory.
    4. Angular solution. Closed and non-closed trajectories.
    5. The isotropic harmonic oscillator described by the general solution and comparison to the previously obtained direct solution. The energy and momentum versus the amplitude and phase.
  7. Elements Of Celestial Mechanics And Classical Atomic Mechanics
    1. Finite, infinite and special trajectories.
    2. The exact integration of the energy equations for the Coulomb (gravitational) field. Elliptical, hyperbolic, and parabolic trajectories.
    3. Keppler's laws. Comparison to the principle of mechanical similarity.
    4. Infinite trajectories and scattering.
    5. Scattering cross section and the Rutherford formula. The discovery of the atomic nucleus. Scattering from a rigid sphere.
    6. The exact solution for the perturbed Coulomb potential,  and the explicit formula for the precession of the perihelion. The origin of perihelion precessions in the solar system.
  8. Mechanics of Two Interacting Particles
    1. Orbiting and scattering of two particles. The reduced mass and the reduction of the two-body problem to the one-body problem.
    2. Co-orbiting and the discovery of double stars.
    3. The number of the degrees of freedom and general integrals of motion. The principal impossibility to find the general solution for the three- and many-body problems.
    4. Kinematics of decay and scattering of two particles. The circular diagrams. The energy and momentum conservation laws and the transformations between the c.o.m. and a laboratory system (Landau).
    5. Energy and momentum transfer in the laboratory system.
      1. Applications: Nuclear decays and the discovery of neutrino. The transfer of momentum and energy, and collisions of cars. Which cars are safer?
  9. Rotation of Rigid Bodies
    1. Forced rotation about a given axis. The angular velocity, angular momentum and the inertia tensor.
    2. Moments and products of inertia
    3. Inequalities for the moments of inertia and the case of laminas.
    4. The principal axes and principal moments of inertia.
    5. Axially-symmetric bodies and the diagonalization of the inertia tensor. Moments of inertia of some molecules.
    6. Forces and torques, precession of the angular momentum. The static and dynamic balancing of a wheel. (Don't believe a mechanic who tells you that this damn wheel can not be balanced-- it can always be.)
    7. Free rotation, the momental and Poinsot ellipsoids Stability of rotation about the axes of the minimum and maximum moments of inertia. Instability of rotation about the intermediate axis and the technique of diving.
    8. Euler equations. Free rotation of a body in the rotating system. The space and body cones. Gyroscopic precession and nutations.
  10. Hamiltonian Formulation
    1. The Legendre transformation, Hamiltonian and the canonical equations.
    2. Poisson brackets and the conservation of the Hamiltonian as energy. Connections to quantum mechanics.
    3. Canonical transformations
Final Exam: Thursday, March 12, 6:00 pm