Introduction to Hamiltonian Chaos and Statistical Mechanics

Outline

1. Hamiltonian dynamics

   1. Hamilton's principle
   2. Hamiltonian formulation
   3. General properties of Hamiltonian dynamics
2. Transformation theory

   1. Canonical transformations
   2. Hamilton-Jacobi equation. Action-angle variables
   3. Integrable systems: tori

      1. Symmetries and conserved quantities
3. Maps I
4. Perturbation theory

   1. Regular perturbation theory for ODE
   2. Canonical perturbation theory: N = 1
   3. Canonical perturbation theory: N > 1
5. KAM-theorem

   1. Example of a superconvergent perturbation theory
   2. How irrational is an irrational number
6. Maps II

   1. Twist maps
   2. Poincaré-Birkhoff fixed-point theorem
   3. The fate of heteroclinic orbits
7. Kneading of dough as chaotic system
   
8. Reviewing sketch of thermodynamics
9. Statistical ensembles

   1. Microcanonical ensemble
   2. Canonical ensemble
   3. Phase transitions

      1. Magnetism and Ising model
      2. Ising model in d = 1
      3. Mean-field theory

Recommended books:
M. Tabor, Chaos and Integrability in Nonlinear Dynamics
I. Percival and D. Richards, Introduction to Dynamics
J.B. Marion, Classical Dynamics of Particles and Systems
H. Goldstein, Classical Mechanics
A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

These books will be on reserve.

There will be homework assignments which will be graded. There will be no midterm or final.

Office Hours:
M 3:30-4, 5-5:30
We 3:30-5:30
in M458

e-mail: h-riecke@northwestern.edu