FB210 Mathematical foundations of the variational theories in physics

Faculty of Science
Autumn 2024
Extent and Intensity
2/1/0. 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. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
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.
Learning outcomes
By finishing the course the students will acquire the following skills

* The understanding of basic problems of the calculus of variations on fibered manifolds and the connection with variational theories in physics
* Practical calculations with jets, fibered manifolds and their prolongations, fibered coordinate systems and other associated systems
* Practical calculations with vector fields, their prolongations and with differential forms on fibered manifolds
* Practical formulation of variational problems, especially physical ones, derivation of equations of motion
* Practical use of the variational sequence for the solution of the trivial Lagrangian problem and the problem 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
    required literature
  • MUSILOVÁ, Jana and Pavla MUSILOVÁ. Matematika : pro porozumění i praxi : netradiční výklad tradičních témat vysokoškolské matematiky. První vydání. Brno: VUTIUM, 2017, ix, 367-79. ISBN 9788021455030. info
    recommended literature
  • KRUPKA, Demeter and David SAUNDERS. Handbook of Global Analysis. 1st ed. Nizozemí: Elsevier, 2008, 1244 pp. ISBN 04-4452-833-4. info
Teaching methods
lectures and theoretical exercices
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 once in two years.
The course is taught: every week.
General note: L.
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 2010 - only for the accreditation, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2010, Autumn 2011 - acreditation.
  • Enrolment Statistics (recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2024/FB210