Physics 305 -
Course Content
(based on
textbook by Goldstein,
and additional handouts)
Review of kinematics and
dynamics in classical mechanics: Coordinate systems.
Transformation of coordinate systems.
Principle of least
action and Lagrange equations: Constrained motion. Generalized
coordinates. Principle of least action, variational
calculus. Galilei's relativity principle and Galilei invariance. Construction of Lagrange function for
free particle from basic spacetime properties, Lagrangian for N interacting particles.
Spacetime symmetries --> conservation laws: Homogeneity
of time --> energy conservation. Homogeneity of space --> linear momentum
conservation. Total linear momentum for N particles and center-of-mass motion,
isotropy of space --> angular momentum conservation
Poisson brackets: Classical
analogue of quantum mechanical commutators, quantum mech. eqns
of motion in the "Heisenberg picture"
Canonical
transformations, Hamilton-Jacobi equation: Point
transformations and phase-space transformations, canonical transformations and
generating functions, relation to Poisson brackets, Hamilton-Jacobi equation..
Two-body problems: 2-body
interactions. Reduction of two-body problem to two independent one-particle
problems. Motion in a central field (Lagrange formulation). Effective potential
with applications. Kepler problem (planetary motion).
Conservation of "Runge-Lenz vector" for 1/r
potentials. Bertrand's theorem for closed orbits.
Theory of elastic scattering: Laboratory and cms reference frames. Deflection function and diff.
scattering cross section in cms. Examples: (1) hard
sphere, (2)finite potential well, (3) arbitrary central field, with application
to
scattering.
Small oscillations: 1-D harmonic oscillator with
friction and external driving force. Resonance behavior. Coupled problems.
Normal modes. Discrete and continuous systems.
Rigid body motion: 6
degrees of freedom (3 for translation, 3 for rotation), kinetic energy of
translation and rotation, moment of inertia tensor and diagonalization
--> principal axes system, Lagrange function and equations of motion in
external field, Euler's equations and application to torque-free rotation of
symmetric rigid body, precession and nutation of
Earth, stability / instability of asymmetric rotor
Accelerated
(non-inertial) reference frames: Most general treatment of
rotational and translational accelerations for one point particle: centrifugal
and Coriolis force, motion on surface of rotating
Earth and free fall. Connection with Einstein's equivalence
principle (of gravitation and inertia)
Introduction to fluid
dynamics: Importance in applied and fundamental sciences (e.g.
climate modeling, aircraft design, car engine, supernova explosions),