MU:MU03265 Variational Analysis on Manif.

MU03265 Variational Analysis on Manifolds

Mathematical Institute in Opava
Summer 2012

Extent and Intensity

2/2/0. 6 credit(s). Type of Completion: zk (examination).

Guaranteed by

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

Course objectives

Methods for finding extrema of functionals on manifolds with suitable properties. Modern mthods in calculus of variations.

Syllabus

- Jets of differentiable mappings, fiber bundles and their prolongations, manifolds of contact elements
- The Lagrange structures (horizontal and contact forms, Lepagean forms, the first variation formula, the Euler-Lagrange equations, the Hamilton equations)
- Symmetries of the Lagrange structures (invariance transformations of the Lagrange structure, generalized symmetries, the first Noether theorem, the natural Lagrange structures, the second Noether theorem)
- The field of extremals and the Hamilton-Jacobi equations
- Foundations of the theory of bundles, variational sequence.

Literature

recommended literature

D. Krupka. Jets and Contact Elements. Proc. of the Seminar on Differential Geometry, M, 2000. info
D. Krupka. The Geometry of Lagrange Structures II. - Elementary Sheaf Theory. Silesian University, Opava, 1998. info
D. Krupka. The Geometry of Lagrange Structures. Silesian University, Opava, 1997. info
P.J. Olver. Applications of Lie groups to differential equations. 1993. info
I. M. Gelfand, S. V. Fomin. Calculus of Variations. Englewood Cliffs, Prentice-Hall, 1963. URL info

Language of instruction

Czech

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.

Teacher's information

Oral exam; further requirements to be specified in the course of the semester.