General Information

Place and time: MW 10:00-11:20am, Mathematics 4-130

Professor: Mikhail Movshev

Course description: In this course, we delve into the integrability aspect of classical mechanics and field theory, primarily concentrating on classical examples. We begin by revisiting the fundamentals of Lagrangian mechanics, discussing relation of symmetries and c onservation laws via the Noether theorem. Our examination includes examples of mechanical systems such as the Calogero-Moser system, various spin-tops, and the nonlinear Schrödinger equations, with the latter serving as a field-theoretic example. The idea of system integrability will be explored and expressed using the framework of Hamiltonian formalism. This involves a concise review of the fundamentals of symplectic and Poisson geometry, leading to a discussion on one of the primary methodologies for constructing integrable systems – the Hamiltonian reduction. This technique will be specifically applied to analyze the Calogero-Moser system. Additionally, the course will cover Lax pairs and their generalizations, particularly in relation to spin-tops. For field-theoretic instances, such as the nonlinear Schrödinger equation, we will explore the r-matrix approach to integrability. The latter part of the course is dedicated to the Lie-theoretic aspects of the r-matrix methods. This will include an in-depth look at its relationship with both the reduction method and the method of Lax pairs, providing a (hopefully) good understanding of these concepts.

Textbooks:

• Freed, D. S., & Uhlenbeck, K. K. (Eds.). (1995). Geometry and Quantum Field Theory. American Mathematical Society; Institute for Advanced Study.
• Semenov-Tian-Shansky, M. (2008). Integrable Systems: the R-Matrix Approach. Research Institute for Mathematical Sciences, Kyoto University.
• Faddeev, L. D., & Takhtajan, L. A. (1987). Hamiltonian Methods in the Theory of Solitons. Springer.
• Etingof, P. (2007). Calogero-Moser Systems and Representation Theory (Zurich Lectures in Advanced Mathematics). American Mathematical Society; European Mathematical Society. ISBN: 9783037190340.
• Audin, M. (1996). Spinning Tops: A Course on Integrable Systems. Cambridge University Press.

Information for students with disabilities
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