Math 467: Dynamical systems theory

Basic Information

• Instructor: Eugene Lerman
• e-mail:lerman@math.uiuc.edu
• Homepage: http://www.math.uiuc.edu/~lerman
• Course page: http://www.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

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

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