Textbook: 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
SYLLABUS
1. Variational approach
Calculus of variations, Euler-Lagrange equations.
Hamilton’s Principal Function ("Action"),
Hamilton's Principle.
Cyclic
coordinates, invariance to transformations and conservation laws
Electromagnetic fields and Lagrangian.
Lagrange’s multipliers and forces of constraints
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 (subject to omission)
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 (subject to revision)
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-Jacobi equation as a formulation of classical mechanics
Hamilton’s characteristic function and quantum-mechanical phase
Action-angle variablesReturn to: Course Descriptions