Spring
Semester
2011

## Course: MAT 213- Complex function
theory

All information is found at MY
SPACE.

## Course: MAT 331- Advanced
topics in analysis

This course has two parts:

I. Introduction to generalized functions and Sobolev spaces (Irina
Markina)

II. Integrable systems (Alexander Vasiliev)

# Exam: May 6, 2011, 09:00

# Topics for exam (Part II. Integrable systems)

Calculus of variations

Examples of extremal problems

Brachistochrone problem

Euler-Lagrange equations

Boundary conditions

Variational derivatives

Lagrangian mechanics

Lagrange equation of classical mechanics

Motion in central field

Legendre transform

Hamilton equations, equivalence between Hamilton and Lagrange equations

Cyclic coordinates

Phase space and Hamiltonian flow

Liouville and Poincare theorems

Holonomic constrains

Differentiable manifolds and tangent space

Hamiltonian mechanics

Exterior forms

Differential forms

Exterior derivatives

Integration over chains and Stokes theorem

Closed and exact forms, cohomology group

Cycles and boundaries, homology group

Symplectic structure

Cotangent space

Hamiltonian vector field

Liouville theorem for manifolds

Lie algebra of vector fields

Poisson structure and Lie algebra of Hamiltonian functions

Symplectic geometry

Integrable systems

Poincare-Cartan invariant

Jacobi-Hamilton method of integration of Hamiltonian systems

Generating functions

Liouville theorem

Action-angle variables

KdV and infinite dimensional integrable systems

# Syllabus:

VENUE:

Room 510, Johannes Brunsgate 12:

Wednesday- 10:15-12:00

Exam: May 6, 2011, 09:00

INSTRUCTOR: Prof. Alexander Vasiliev

Language of instruction:
English

EXAM: oral exam

Howeworks: there will be
no obligatory homeworks.

TEXT: V.I.Arnold
"Mathematical methods of classical mechanics", 2nd edition, Springer.