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.

# Simplectic geometry and integrability

• 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:

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.

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.

Lectures, exercises, homework, consultations

written exam amd homeworks

oral exam

grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

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