Instructor: Laurent O. Jay, 225L MLH, ljay@math.uiowa.edu

Time & Location: 2:30-3:20 MWF in 105 MLH.

Prerequisites: 22M:142 or equivalent or consent of instructor.

This course is a natural continuation of 22M:142 (nonlinear dynamics with numerical methods). Though the course is mostly about the theory of ordinary differential equations (ODEs), we may consider this theory in conjunction with some theoretical and practical aspects of numerical methods for ODEs.

Textbooks (recommended):

1. Calculus of Variations by I. M. Gelfand and S. V. Fomin, 2000, 240 pages, Paperback, Dover Publications, ISBN-10: 0486414485, ISBN-13: 978-0486414485, list price: $10.95. Library references: Mathematical Sciences Library QA315 .G417. The book on amazon.com.
2. Mathematical Methods of Classical Mechanics by V.I. Arnol'd, Springer, Series: Graduate Texts in Mathematics, Vol. 60, 2nd edition 1989, Corr. 4th printing (September 5, 1997), 509 pages, Hardcover, ISBN-10: 0387968903, ISBN-13: 978-0387968902, list price: $69.95. Library references: Engineering Library QA805 .A6813 1989a, Physics Library QA805 .A6813 1989, and Physics Reserve QA805 .A6813 1989b. Table of contents. The book on amazon.com.

Course topics:

• Existence and uniqueness
• Lyapunov-stability
• Variational calculus
• Lagrange's and Hamilton's mechanics
• Noether's theorem
• Divergence free ODEs (Poincare recurrence theorem)
• Reversible systems
• Invariant (stable, unstable, center) manifolds
• Poisson systems
• Variational/symplectic integrators
• Differential inequalities
• KAM (Kolmogorov-Arnold-Moser) theory
• Normal forms
• Structural stability
• Perturbations
• Averaging
• Bifurcations
• Homoclinic orbits
• Melnikov's method
• Boundary value problems (Sturm-Liouville systems)
• etc.