Department of Physics and Astronomy
PHYS
8010 – ADVANCED CLASSICAL MECHANICS
SPRING
SEMESTER 2008
Monday, Wednesday
Instructor: Mark Stockman
Office: 455 Science Annex
Phone: 678-457-4739 (mobil)
E-mail: mstockman@gsu.edu
Internet: http://www.phy-astr.gsu.edu/stockman/
Final Exam: Monday,
May 5 from
5:00 to 7:00 pm in the regular classroom
Text: Herbert Goldstein, Classical Mechanics,
Addison-Wesley,
Reading, MA, 1980
Supplementary text: L.D. Landau and E.M. Lifshitz, Mechanics,
Pergamon, New York, 1976.
Supplementary text: Alexander L. Fetter and John Dirk Walecka, Theoretical
Mechanics of Particles and Continua, McGraw-Hill, New York, 1980
Grading: midterm exam: 30%, final exam: 70%. The grading is mostly relative to the class.
Basic Information
The rules of the class room are designed to allow students to get the maximum benefit for their time and money spent. The physical attendance of lectures is not required but strongly recommended. If you happen to be late, enter class, do not apologize, quietly take your seat and start working. If you need to leave for any reason, do so also as quietly as possible, do not ask permission. The home assignments will not be graded; their solutions will distributed approximately in one week after handing out. Solving the home problems is an integer part of the course and is absolutely essential for the passing exams and ultimately mastering the material. Do not talk in class even in a low voice since it is disruptive (asking a fellow student a brief question is admissible, but should be kept to a minimum). Do not hesitate to interrupt the lecturer with any questions or comments, since it is beneficial for the class. (Do not assume that your question is too trivial to ask -- it may well be not so trivial. Many students may have a similar problem. No questions, answers, olr comments in class will affect your grades in any way.)
At the exams, you may not use any notes or books, except for
mathematical tables
(integrals and special functions), unless specifically allowed. You may
briefly
(for not more than five minutes) leave the class room after one hour of
work
without asking permission; any longer absence will terminate your exam
at that
point. You should bring your calculator (no data banks) and pen, or
pencils.
The paper needed will be given to you. The date and time of the final
exam
cannot be changed.
SYLLABUS
1. Variational approach
Calculus of variations, Euler-Lagrange equations.
Hamilton’s
Functional ("Action") and Hamilton's Principle.
Cyclic
coordinates,
invariance to transformations and conservation laws
Electromagnetic
fields
interacting with particles and their Lagrangian.
Lagrange’s
multipliers and forces of constraints (time permitting)
2. Newton Laws and Conservation Laws
Newton laws. Inertial
systems
Center of mass and conservation of momentum.
Angular momentum. Conservation of angular momentum in the c.o.m. system
Conservation of energy
Mechanics of car driving.
3. Central Forces
Reduction to one-body problem in the general case. Effective mass
Conservation of
angular
momentum and areal velocity
Effective
potential and
reduction to one-dimensional motion.
Gravitational
(Coulomb)
field and Keppler's laws.
Quasi-periodic
motion
and Bertrand theorem.
4. Infinite motion and scattering problem
Scattering cross section
Infinite motion
in a
gravitational (Coulomb) potential: Hyperbolic orbits
Rutherford
scattering.
Scattering by a
hard
sphere
Small angle
scattering.
5. Non-inertial coordinate systems (time permitting)
Rotating systems, infinitesimal rotations
Accelerations
Coriolis and
centrifugal
forces
Motion on the
surface of
the Earth
6. From Newton’s Equations back to Lagrangian Formalism
Constraints and generalized coordinates
Virtual
displacements
and d'Alembert principle
Lagrangian
equations
Generalized
momenta and
integrals of motion.
7. Small oscillations
Coupled problem
Linear equations
(review)
General solution
Matrix formalism
Normal modes.
Many-body
problems:
Springs and beads (optional)
8. Rigid Bodies
Rotation about a fixed center and free rotation
Inertia tensor.
Rotational and translational motion.
Fixed-body frame: Euler equations
Rotating bodies:
Some
applications
Compound
pendulum
Torque-free
symmetric
top (a model of molecule rotation and deformed nuclei)
Asymmetric top
Euler angles
Symmetric top
with a
fixed point. Precession and nutation.
11. Hamiltonian dynamics (Canonical formalism)
Hamiltonian and Hamilton’s equations
Modified Hamilton's principle
Canonical transformations
Poisson brackets as canonical invariants.
Transition
to quantum mechanics
Infinitesimal canonical transformations and
conservation laws
Hamilton's principal function and Hamilton-Jacobi
equation
as a formulation of classical mechanics
Hamilton’s characteristic function and
quantum-mechanical phase
Action-angle variables