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.