Integrable systems (Math 99r, Spring 2024)

Integrable systems are particularly nice differential equations which have a large collection of “conserved quantities”. The geometry of the space of solutions to such systems is often highly interesting. The goal of this course is to give an example-based introduction to this theory, starting with classic examples like the harmonic oscillator and the Euler top, describe the theory of Lax pairs, and use all of this to motivate the symplectic geometry perspective on integrable systems. There is a lot of rich history behind this story (for instance, the theory of integrable systems was one of Lie’s original motivations for the theory of Lie algebras!). The main focus in the second half of this course will be the Calogero-Moser and Toda integrable systems, and their construction due to Kazhdan-Kostant-Sternberg. A more detailed syllabus and schedule will be posted later.

I will be typing up notes for (some of) the lectures and posting them on this webpage. For the most part, I will follow Arnold’s “Mathematical Methods of Classical Mechanics” in the beginning of the course. I also like Libermann-Marle’s “Symplectic Geometry and Analytical Mechanics”, as well as Abraham-Marsden’s “Foundations of Mechanics”. I will add further references for later parts as we go along.

Most of your grade will be based off a final paper (probably around 80% of the grade); and a few problems which will be due towards the end of the semester (second week of April).

We will meet in SC 232 on Tuesdays, 4:30-6 pm and in SC 530 on Fridays, 2-3 pm.

Lectures

These notes were written hastily, so you should be sure to check what I have written and supplement it through other sources.

Lecture 0: Introduction (Jan 26). I have uploaded a recording of the lecture on Canvas.

Lecture 1: The Lagrangian formalism (Jan 30).

Lecture 2: The Legendre transform and Hamiltonian mechanics (Feb 2).

Lecture 3: Symplectic geometry (Feb 6).

No lecture on Feb 9 (CMSA lecture series).

No lecture on Feb 13 (snowstorm).

Lecture 4: Hamiltonian reduction (Feb 16). (This spills over into the next lecture, and I expect that I’ve started a domino effect…)

Lecture 5: Rigid body motion (Feb 20).

Lecture 6: Lax pairs (Feb 23). (Incomplete)

Lecture 7: Integrability (Feb 27).

Lecture 8: The Calogero-Moser system; see the first few chapters of Etingof’s notes.

Problems

Here are some problems. I will keep updating this file throughout the semester. Please try to do at least a third of the problems, and submit them to me by email by the second week of April.