Department of
Physics
Classical Mechanics (Phys. 461/661)
Tuesday, Thursday 4:406:55 pm 227SC Winter 1998
Instructor 
Office 
Phone 
Homepage 
Prof. Mark I. Stockman 
455 SA 
(404)6513221 
www.phyastr.gsu.edu/stockman 
Textbook:
G.F.Fowles and G.L.Cassiday, Analytical Mechanics, Harcourt Brace,
Fort Worth, Texas, 1993.
H.Goldstein, Classical Mechanics, AddisonWesley, Reading, Massachusetts,
1980
Additional Texts:
L.D.Landau and E.M.Lifshitz, Mechanics, AddisonWesley, Reading,
Massachusetts, 1960.
A.L.Fetter and J.D.Walecka, Theoretical Mechanics of Particles and
Continua, McGrawHill, New York,1980
Grading:
Homework: 40%, Midterm Examination: 20%, Final Examination: 40%.
Outline of the Course

Newton's Laws and Conservation Laws (Chs.2 and 7,8)

Three Newton's laws in their contemporary understanding. The First Law
as a definition of inertial reference systems

Forces, masses and the Second and Third Laws.

Examples of forces (Coulomb, gravitational, friction, etc.).

Third Newton's Law and systems of many particles.

Linear momentum and its conservation. Center of mass (c.o.m.) (7.1).

Reaction forces and the equation of rocket motion. (7.6)

Central forces and conservation of angular momentum. Torque and the angularmomentum
theorem in the lab system (7.2)

Noncentral forces and unavoidability of internal angular
momentum (spin).

Angular momentum theorem in the c.o.m. system (8.5)

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 rollingover. How to corner: where
to break or accelerate, the mechanics of car racing. Mechanical basis of
car and road design. Advantages and disadvantages of forwardand rearwheel
drives. Why a formula 1 car is designed this way?

Newton's Mechanics of Rectilinear Motion (Ch.2)

Launches of a projectile near the earth surface.

Effects of friction, maximum shooting distance as the function of the inclination
angle (greater or less then 45 degrees?).

Rocket motion with and without friction. Why do we need multistage rockets?

Notion of impulse and suddenforce approximation.

Forces that depend on position. Potential, kinetic and total energy.
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

Oscillations (Ch.3)

Harmonic (linear) oscillator.

Energy for linear oscillator. Free motion without friction. The underdamped,
critically damped and overdamped harmonic oscillator. Oscillation frequency,
amplitude and damping rate.

Forced motion of a damped linear oscillator. The general solution, and
transient and stationary regimes.

The response function and resonance. Freeoscillation and resonance frequencies
(they are different). Lorentz contour and the resonance quality factor.

Electrical analogy and selectivity of a radio receiver.

Production of heat by an oscillator and the imaginary part
of the response function.

Harmonic oscillator driven by a periodic nonharmonic force.
The general solution as a Fourier series. Harmonics, higherorder resonances
and filters.

Exact solutions for sudden forces (kicked oscillators). Electronic
transitions in a molecule as a sudden perturbation of a molecular oscillator
(FranckCondon principle).

The eigenvalue equation and normal modes

Lagrangian Formulation

Holonomic and nonholonomic systems. Generalized coordinates and velocities.

The Lagrangian function and equations.
Examples: Onedimensional motion. Threedimensional motion in
a central field. Pulleys, blocks and cables.

Ignorable variables, transformation invariance and conservation laws.

Hamilton's principal function and the least action principle. EulerLagrange
equations for the variational problem as an alternative basis for mechanics.

One Particle in Three Dimensions (Ch.4)

The del operator, the scalar and vector potentials.

The conditions of potentiality of a force field.

The scalar potential as a solution of the Poisson equation
and as a contour integral, equivalence of the both.

Examples of potential and nonpotential forces.

Central forces and conservation of angular momentum revisited.

Harmonic oscillator in three dimensions: the direct solution of the Newton's
equations. An anisotropic oscillator, nonclosed and closed orbits, multiple
frequencies and Lissajous figures.

The isotropic oscillator, the amplitude and phase of motion.

General Solution for One Particle in a Central Potential
Field

Angular momentum conservation and areal velocity.

Energy equations of motion for a particle in a threedimensional central
field. Plane trajectories and the polar coordinates.

The general integral solution for radial motion and the form of a trajectory.

Angular solution. Closed and nonclosed trajectories.

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.

Elements Of Celestial Mechanics And Classical Atomic Mechanics

Finite, infinite and special trajectories.

The exact integration of the energy equations for the Coulomb (gravitational)
field. Elliptical, hyperbolic, and parabolic trajectories.

Keppler's laws. Comparison to the principle of mechanical similarity.

Infinite trajectories and scattering.

Scattering cross section and the Rutherford formula. The discovery of the
atomic nucleus. Scattering from a rigid sphere.

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.

Mechanics of Two Interacting Particles

Orbiting and scattering of two particles. The reduced mass and the reduction
of the twobody problem to the onebody problem.

Coorbiting and the discovery of double stars.

The number of the degrees of freedom and general integrals
of motion. The principal impossibility to find the general solution for
the three and manybody problems.

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).

Energy and momentum transfer in the laboratory system.
Applications: Nuclear decays and the discovery of neutrino.
The transfer of momentum and energy, and collisions of cars. Which cars
are safer?

Rotation of Rigid Bodies

Forced rotation about a given axis. The angular velocity, angular momentum
and the inertia tensor.

Moments and products of inertia

Inequalities for the moments of inertia and the case of laminas.

The principal axes and principal moments of inertia.

Axiallysymmetric bodies and the diagonalization of the inertia tensor.
Moments of inertia of some molecules.

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.)

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.

Euler equations. Free rotation of a body in the rotating
system. The space and body cones. Gyroscopic precession and nutations.

Hamiltonian Formulation

The Legendre transformation, Hamiltonian and the canonical equations.

Poisson brackets and the conservation of the Hamiltonian
as energy. Connections to quantum mechanics.

Canonical transformations
Final Exam: Thursday, March 12, 6:00 pm