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). We hope that the course will
be helpful for mathematicians to develop intuition about ODE's and PDE's
and for Physicists and Engineers to get a taste of the Mathematical analysis
of models. We will try to accomodate the different backgrounds of the students.
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.)
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Review.
Differential forms.
Some background in analysis and PDE.
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Mathematical formulations of mechanics.
Newtonian formulation.
Lagrangian formulation.
Hamiltonian formulation.
Variational problems.
Aubry-Mather theory.
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Examples.
Kepler problem.
Oscillators.
Integrable systems (Toda Lattice, Calogero system).
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Perturbation theory.
Lindstedt series.
Canonical perturbation theory.
Bifurcation theory.
KAM theorem.
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Classical field theory.
Electromagnetic fields
Hyperbolic equations
Radiation, difraction, Huygens principle.
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Introduction to thermodynamics and statistical mechanics.
Basics of thermodynamics
Convex analysis
Equilibrium statistical mechanics
Non-equilibrium statistical mechanics (transport, relaxation)
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Conservation laws.
Riemann's method
Shocks
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Continuum mechanics (solids).
The Cauchy model of Elastic solids
The equations of equilibrium and elliptic regularity
Bifurcation theory in equilibrium elasticity
Elastic waves
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Models of fluids (Navier Stokes equations).
Stationary solutions
Existence and uniqueness in 2-D
Textbook(s):
We recommend:
Also very good:
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W. Thirring: A course in Mathematical Physics Vol I, II
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C.C. Lin & L. A. Segel: Mathematics applied to deterministic models
in the Natural Sciences.