MU03265 Variational Analysis on Manifolds

Mathematical Institute in Opava
Summer 2019
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
Guaranteed by
prof. RNDr. Artur Sergyeyev, Ph.D., DSc.
Mathematical Institute in Opava
Course Enrolment Limitations
The course is offered to students of any study field.
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.
The course is also listed under the following terms Summer 2000, Summer 2001, Summer 2002, Summer 2003, Summer 2004, Summer 2005, Summer 2006, Summer 2007, Summer 2008, Summer 2009, Summer 2010, Summer 2011, Summer 2012, Summer 2013, Summer 2014, Summer 2015, Summer 2016, Summer 2017, Summer 2018.
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