Welcome to the Home Page for Physics 573
Nonlinear Mechanics

This link gives an introduction, a syllabus, and comments on textbooks.
The listing of other books on nonlinear mechanics is found at this link.

Some lecture notes, or handouts to supplement the texts; parts of the course will follow these closely.

1. Description of the basic geometry needed for Hamiltonian mechanics.
2. Description of canonical transformations
3. Description of Hamilton-Jacobi equation, and action-angle variables
4. Some brief notes concerning elliptic functions .
5. Beginning studies of perturbation theory, using the pendulum as an example: naïve perturbation approach, secular perturbation theory, canonical perturbation theory.
   Some more detailed Maple calculations for the motion of a charged particle in a constant B field and an oscillating E field.
6. Various examples are being put here. At the moment we have notes on the Toda 3-particle molecule, the Hénon-Heiles potential, and the Ford and Walker 2-2 resonances. Also there are Maple worksheets showing 3-dimensional orbits and associated Poincaré sections, both for the HH potential, and for the Walker-Ford 2-2 resonance.
7. Notes beginning more serious calculations in perturbation theory, beginning with adiabatic expansions, and leading to understanding of motions near (primary) resonances.
8. Some notes on flows, based on Steve's notes, and connecting them with some parts of the texts.
9. Notes on Lie perturbation theory
10. A copy of Deprit's original paper on Lie perturbation theory.
11. Some notes on the use of maps to better understand the structure of Hamiltonian trajectories in phasespace:
    • The first set.
12. Soliton equations:
    • a list of some physically-interesting, soliton-bearing equations,
          and some of their symmetry characteristics
    • A derivation of the KdV equation from basic hydrodynamics, for incompressible, irrotational water flow along a narrow channel for waves of wavelength long with respect to the depth of the channel.
    • Some useful properties of the Korteweg-de Vries equation, for shallow, one-dimensional, unidirectional water waves
      • A scanned copy of a printed version of a photograph of a KP wave in the Andaman Sea
      • At the link is a discussion and some examples of solutions to the KdV equation.
    • Some extensive notes on methods to determine local- and non-local symmetries for pde's
    • At this link are some brief notes concerning Kac-Moody algebras, and examples

Homework Assignments, Solutions, and related Things:
either in Acrobat mode or Maple mode

The Final Exam will be in the form of a project/term paper.

Term Papers turned in by 2 students are now online, and may be read by everyone. Begin by clicking on this link. This project turned out rather well, I believe, with interesting additional material chosen and studied by the students, although in the end only 2 students did such papers.

• the Chaos Study Group at the University of Maryland.
• Non-Linear Physics Group at the Naval Research Laboratory.
• Chaos on the Web, a course at CalTech.
• A link to an online numerical resolver for an arbitrary ode, first- or second-order is at this website, and seems to function quite well. It is Java-based, on top of MATLAB code, and written by John Polking, at Rice University.
• If you find other interesting links, please let me know.

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