## Math 467: Dynamical systems theory

### Basic Information

**Instructor: **Eugene Lerman
**e-mail:**lerman@math.uiuc.edu
**Homepage:**
`https://math.uiuc.edu/~lerman`

** Course page:**
`https://math.uiuc.edu/~lerman/467/467syl.html`

**Office:** 334 Illini Hall
**Phone:** 244-9510
**Office hours:** MW 11-12 and by appointment
**Class meets:** MWF 10 am
in 159 Altgeld Hall

## Prerequisites

A course in differentiable manifolds such as
Mathematics 423.
If you have any questions or concerns, please contact me by e-mail.

## Course outline

- Symplectic linear algebra.
- Basic examples of symplectic manifolds.
- Review of vector bundles.
- Lagrangian embedding theorem, applications.
- Classical Hamiltonian systems.
- Hamilton's principle, Euler-Lagrange equations, Legendre
transform, examples.
- Poisson bracket, complete integrability.
- A fast introduction to Lie groups and group actions.
- Symplectic group actions, symmetries of Hamiltonian systems.
- Coadjoint representation, coadjoint orbits.
- Moment map and its properties.
- Fiber bundles, basic forms, coisotropic reduction.
- Symplectic quotients.
- Completely integrable systems revisited, action-angle variables.
- Monodromy of the period lattice of the spherical pendulum.
- Stratified spaces, singular quotients.
- Application: stability of a symmetric top.

## Texts

The official text is The structure of dynamical systems. A symplectic view of physics by J.-M. Souriau

Lecture notes of an old version of the course are available here pdf file

You may also wish to look at a set of notes by Ana Cannas
PostScript file
A new version of these notes should be available soon.

Recommended texts:

- Symplectic geometry and analytical mechanics
by P. Liebermann and C.-M. Marle
- Introduction to
Mechanics and Symmetry by J. E. Marsden and T. S. Ratiu
- Symplectic Techniques in Physics by V. Guillemin and
S. Sternberg

## Grades

The course grade will be based on homework.
Homework will be assigned weekly. One problem per homework will be
graded.

Last modified: Tue January 18 2000