Simplectic geometry and integrability

2022/2023
Programme:
Physics, Second Cycle
Orientation:
Astrophysics
Year:
2. year
Semester:
first
Kind:
optional
ECTS:
5
Hours per week – 1. semester:
Lectures
2
Seminar
0
Tutorial
1
Lab
0
Prerequisites

Enrollment into the program.
Positive result from written exam is necessary to enter the oral exam.

Content (Syllabus outline)

Symplectic geometry: Symplectic and Poisson manifolds. Darboux theorem.
Lagrange submanifolds. Lie algebra of hamiltonian vector fields. Hamiltonian actions of Lie groups, momentum maps, symplectic reduction. Co-adjoint orbits, Kostant-Kirillov symplectic forms and their application in the theory of representations of Lie groups.
Integrability: Arnold-Liouville theorem and action-angle coordinates. Lax equation, spectral curve, linearization of an algebraically integrable system on the Jacobian torus associated to the spectral curve. Lax formalism fort he partial differential equations, zerocurvature
condition. Inverse scattering problem. Description of integrable systems by means of the loop Lie algebras and the R-matrix.

Readings

• V. Arnold, Mathematical Methods of Classical Mechanics, Springer, Berlin (1989)
• V. Guillemin, S. Sternberg: Symplectic techniques in physics, Cambridge University Press, Cambridge (1986)
• O. Babelon, D. Bernard, Introduction to classical integrable systems, Cambridge University Press, Cambridge (2003)
• A. I. Bobenko et al., Algebro-Geometrical Approach to Nonlinear Integrable Equations, Springer, Berlin (1994)
• T. Dauxois, M. Peyrard, Physics of Solitons, Cambridge University Press, (2006)

Objectives and competences

Objectives:
In the first part of the course students get
acquainted with the basic symplectogeometric
notions which are needed for
global descriptions of Hamiltonian systems.
In the second part the fundamental concepts
of integrability are studied.
Competences:
Students learn the global treatment of
certain dynamical systems whose phase
spaces are manifolds with nontrivial
topology and geometry. Students learn to
apply some techniques, related to the theory
of symmetries, in finding analytical
solutions of certain Hamiltonian systems.

Intended learning outcomes

Knowledge and understanding
Students learn certain geometric
techniques which are necessary fort he
global formulation of Hamiltonian
formalism. Then they learn the
fundamentals of integrability.
Application
Students learn some approaches to the
study of physical symmetries.
Knowledge about the relevant
symmetries can enhance the
understanding of physical systems in an
essential way. Integrability provides
means for analytic solutions of certain
physical systems from various areas of
physics.
Reflection
Students enhance their understanding
of the notion of symmetry as one of the
key fenomena in mathematical physics.
Transferable skills
Methods studied in this course can be
applied in the treatment of various
systems, found in mechanics, statistical
physics, optics and other disciplines.

Learning and teaching methods

Lectures, exercises, homework, consultations

Assessment

written exam amd homeworks
oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

prof. dr. Pavle Saksida

1.)P. Saksida, On the nonlinear Fourier transform associated with periodic AKNS-ZS systems
and its inverse, J. Phys. A: Math. Theor., 46 (2013) 22pp
2.) P. Saksida, Lattices of Neumann oscillators and Maxwell-Bloch equations, Nonlinearity, 19 (2006) 22pp
3.) P. Saksida, Maxwell-Bloch equations, C. Neumann systems and Kaluza-Klein theory, Journal of Physics A: Mathematical and Theoretical, 38 (2005) 23pp
4.) P. Saksida, Integrable anharmonic oscillators on spheres and on hyperbolic spaces, Nonlinearity, 14 (2001) 18pp
5.) P. Saksida, Nahm's equations and generalizations of Neumann systems, Proceedings of London Mathematical Society, 78 (1999) 19pp