F4260 Calculus of variations and its applications

Faculty of Science
Spring 2009
Extent and Intensity
2/0/0. 3 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.
Timetable
Thu 14:00–15:50 F4,03017
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 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
there are 13 fields of study the course is directly associated with, display
Course objectives
Physical theories are often based on so called variational principle, which lies in searching the stationary conditions for an appropriate functional. For example, in mechanics such a functional assigns to admissible trajectories in the configuration space of a mechanical system a real number given by an appropriately defined integral (the definition itself arises from physics). The condition for stationary points then leads to equations of motion of the system. The situation is similar in field theories, where expressions of field quantities as functions of space-time coordinates are understood as "trajectories". However, the basic principle is the same. On the other hand, the problem of boundary conditions must be solved (e.g. problems with fixed or free end points, respectively). In physics there are also situations in which a studied system is subjected to certain constraint conditions. In such a case we have a so called constraint variational problem. All above mentioned problems, and many others, are from the point of view of mathematics solved in the discipline "Calculus of variations".

The aim of this course is to present to students the mathematical background of the calculus of variations, especially as for above exposed problems. Some applications of the mathematical theory to physical or technical problems are discussed as well.

Absolving the discipline student obtains following basic knowledge nad skills:

* Understanding of the concept of a variational problem, its formulation and solution.
* Understanding of differencies among variation problems with various types of boundary conditions (fixed ends, free ends).
* Practical calculation procedures in solving equations resulting from variational problems.
* Understanding of integrals of motion.
* Applications of the calculus of variations for solving problems resulting from physical variational theories.
Syllabus
  • I. Introduction.
  • I-1. Variational problems in geometry and physics (light propagation, brachistochrone problem, isoperimetric problem, problem of minimal rotational surface,....).
  • II. Elementary methods of solving stationary problem – functions of a single variable.
  • II-2. Functional, stacionarity condition, Euler equation and its derivation, special cases.
  • II-3. Applications (geometrical problems, problems in mechanics of one mass particle or in mechanics of a system of mass particles).
  • II-4. Approximate solutions of variational problems.
  • III. Method of variations – functions of a single variable.
  • III-5. Classification of stationary points.
  • III-6. Variation of a function, variation of a functional, theorems.
  • III-7. Euler equations, invariance.
  • IV. Functionals for multiple variable functions.
  • IV-8. Formulation of a problem, Euler equations.
  • IV-9. Applications - field theories.
  • V. Free end point problems.
  • V-10. Formulation of a problem, the problem of free end points in one dimensional space, applications.
  • V-11. The problem of free end points in threedimensional space, applications.
  • VI. Constrained variational problems.
  • VI-12. General formulation of a constrained problem, types of constraint conditions in physics, examples.
  • VI-13. Method of Lagrange multipliers.
  • VII. Introduction to calculus of variations on fibred spaces.
  • VII-14. Fibred euclidean spaces, sections and their prolongations, vector fields, differential forms.
  • VII-15. Variational problem on a fibred space, Lagrange structure, extremals, applications.
Literature
  • Průběžně zveřejňovaný text k přednášce
  • GEL'FAND, Izrail Moisejevič and Sergej Vasil'jevič FOMIN. Calculus of variations. Edited by Richard A. Silverman. Mineola, N. Y.: Dover Publications, 2000, vii, 232 s. ISBN 0-486-41448-5. info
Assessment methods
Teaching: lecture
Exam: colloquium - discussion based on prepared solution of problems
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Spring 2008, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, spring 2018, Spring 2021, Spring 2023, Spring 2025.
  • Enrolment Statistics (Spring 2009, recent)
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