FB210 Mathematical foundations of the variational theories in physics

Faculty of Science
Autumn 2010 - only for the accreditation
Extent and Intensity
2/1. 2 credit(s) (plus extra credits for completion). Type of Completion: k (colloquium).
Teacher(s)
prof. RNDr. Jana Musilová, CSc. (lecturer)
Guaranteed by
prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Prerequisites
differential and integral calculus of functions of one and many variables, fundamental problems of multilinear algebra (tensor calculus), differential forms on euclidean spaces
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
As a basic undelying geometrical structures of variational theories fibered manifolds are considered. For mechanics the base of a fibered manifold is one-dimensional, for field theories it is m-dimensional (m>1). The presentation of fundamental concepts and theorems concerning the geometry of fibered manifolds, differential forms on fibered manifolds, formulation of variational problems and proofs of basic formulas of variational theories allows students to obtain a basic knowledge of correct approaches to mathematical problems of variational theories. The presentation includes also problems of variational sequences and their meaning for understanding such aspects of theories as the variationality of equations of motion and the trivial variational problem.

Absolving the discipline students obtain following abilities and skills:

* Understanding of fundamental problems of calculus of variations on fibered manifolds and the connection of the general mathematical theory with physical variational theories.
* Skill of the practical calculus of jets, fibered manifolds and their jet prolongations, fibered charts, associated charts, etc.
* Skills of the practical calculus of vector fields, their prolongations, calculus of differential forms on fibered manifolds.
* Skill of practical formulation of variational problems, especially those with physical meaning, derivation of equations of motion.
* Practical use of the variational sequence for solving problems o variational triviality of a lagrangian or problems of variationality of equations of motion.
Syllabus
  • 1. Fundaments of the jet theory.
  • 2. Fibered manifolds and their jet prolongations, sections on fibered manifolds.
  • 3. Vector fields and differential forms of fibered manifolds and on their prolongations.
  • 4. Horizontal and contact forms.
  • 5. Basic operations with differential forms.
  • 6. Lagrangian, variational integral and variational formulas.
  • 7. Equations of motion, Euler-Lagrange form.
  • 8. Variational sequences on fibered manifolds.
  • 9. Representations of variational sequences, Euler-Lagrange and Helmholtz-Sonin mappings.
  • 10. Lepage forms, Lepage equivalents.
  • 11. Trivial variational problem.
  • 12. Variationality of equations of motion.
  • 13. Aplications, examples, mechanics.
Literature
  • KRUPKA, Demeter and David SAUNDERS. Handbook of Global Analysis. 1st ed. Nizozemí: Elsevier, 2008, 1244 pp. ISBN 04-4452-833-4. info
  • Bude průběžně doporučována.
Assessment methods
Teaching: lectures, consultative exercises
Exam: Colloquial discussion.
Current requirements: Elaborating of two examples with satisfactory complexity or two proofs of theorems. The presence in during exercises is obligatory (75 %)
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
General note: S.
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2010, Autumn 2011 - acreditation, Autumn 2024.