Graduate course MAT 1347F, Fall 2011
Instructor: Prof. Boris Khesin
Time/location: Wed. 11am-1pm at RW 141 and Fri. 11am-1pm at SS 1080.
- Billiards, symplectic structure on lines, Crofton's formula
- Integrability of the geodesic flow on an ellipsoid
- Hamiltonian systems, first integrals, the Arnold-Liouville theorem, tori
- Lie algebras, the Lie-Poisson bracket; example: a rigid body
- Bihamiltonian systems, the Lenard-Magri scheme, the Lax form; example: the Toda lattice
- Two Hamiltonian structures of the Korteweg-de Vries (KdV) equation
- The NLS and filament equations
- Introduction to the inverse scattering; example: KdV
- Introduction to solitons
- Near-integrability and non-integrability: glimps of the KAM and Arnold's diffusion
- Topics and generalizations (time permitted):
- the Kadomtsev-Petviashvili (KP) hierarchy on PDE's
- spectral parameter and spectral curve
- contact integrability and null geodesics
- S. Tabachnikov "Geometry and billiards" (Chapters 3 and 4),
Amer. Math. Soc., 2005.
- S. Tabachnikov "Introduction to symplectic topology" (Chapter 2),
Lecture notes (PennState Univ.).
- B. Khesin and R. Wendt "Geometry of infinite-dimensional groups" (Chapters I and II),
- R.S. Palais
"The symmetries of solitons"
Bull. Am. Math. Soc., New Ser. 34, No.4, 339-403 (1997).
- G. Segal, in the book "Integrable Systems."
Oxford Sci. Publ. 1999
- J. Moser "Various aspects of integrable Hamiltonian systems."
Progr. Math. 8, Birkhauser, Boston, Mass.,1980, pp. 233-289.
Some familiarity with the main notions of classical mechanics or symplectic geometry will be useful, but not required
Topics for mini-papers