Rational Mechanics  393C (54525 (Math) 60140 (CAM))
R. de la Llave
Fall 98

Brief description:

The goal of this course is to introduce the students to models used to
describe mechanical systems and their mathematical treatment
(variational methods, qualitative theory of ODE, and PDE). No
background in Physics will be assumed.

Roughly a little less than half of the course will be devoted to
discrete systems and the rest to continuum models.

Students will be encouraged to carry out a computer project
(e.g. solving some ODE's and PDE's that are particularly significant,
working out some asymptotic expansions, etc.)

ffl Review.

Differential forms. Some background in analysis and PDE. 


Mathematical formulations of mechanics:
Newtonian formulation. Lagrangian formulation. Hamiltonian formulation.
Variational problems. Aubry-Mather theory.

Examples:
Kepler problem. Oscillators. Integrable systems (Toda Lattice, Calogero system).


Perturbation theory:
Lindstedt series. Canonical perturbation theory. Bifurcation theory.
KAM theorem. 

Classical field theory:
Electromagnetic fields Hyperbolic equations Radiation, difraction,
Huygens principle. 

Introduction to thermodynamics and statistical mechanics
Basics of thermodynamics Convex analysi

Equilibrium statistical mechanics Non-equilibrium statistical mechanics
(transport, relaxation) 

Conservation laws:
Riemann's method. Shocks.

Solids:
The Cauchy model of Elastic solids.
The equations of equilibrium and elliptic regularity.
Bifurcation theory in equilibrium elasticity. Elastic waves

Models of fluids (Navier Stokes equations):
Stationary solutions. Existence and uniqueness in 2-D

Textbook:

We recommend:
Lin and Segal: Mathematics applied to deterministic problems in the natural
sciences (SIAM) 

Thirring: A course in Mathematical Physics Vol I, II
(Springer)