F4260 Calculus of variations and its applications

Faculty of Science
Spring 2025
Extent and Intensity
2/1/0. 3 credit(s) (plus extra credits for completion). Type of Completion: k (colloquium).
Teacher(s)
Mgr. Michael Krbek, Ph.D. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)
Guaranteed by
Mgr. Michael Krbek, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: Mgr. Michael Krbek, Ph.D.
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
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 8 fields of study the course is directly associated with, display
Course objectives
Basic physical and other theories are often based on the variational principle, which consists in searching for the stationarity and minimality conditions of a given functional, functionals being understood as functions of functions. 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 the physical consideration of symmetries). The condition for stationary points then leads to the equations of motion of the system. The situation is similar in field theory, 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). The above mentioned problems, and many others, are solved in the discipline "Calculus of variations" from the point of view of mathematics.


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

Learning outcomes
By succesfully taking the course the student obtains the following basic knowledge and skills:

* Understanding of the concept of a variational problem, its formulation and solution.
* Understanding of differences among variation problems with various types of boundary conditions (fixed ends, free ends).
* Understanding the distinction between stationarity and minimality of a functional
* 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
  • 1st Lecture

  • (a) Introduction: content and learning outcomes, recommended literature (Giaquinta-Hildebrandt, Hildebrandt-Tromba, Jost) (b) Requirements for course completion (c) History: Ancient Greece (Euklides, Dido), Newton, Bernoullis, Euler, Lagrange, Jacobi, Weierstrass, Riemann, Hilbert, Caratheodóry (d) Applications: Physics, Geometry, Image manipulation, Economics , all other natural sciences, Engineering (e) Defining functionals (f) Euler's method of finite differences

  • 2nd Lecture

  • (a) The problem of Queen Dido and its solution (Hurwitz   1900) by direct method (b) Disadvantages of direct methods, indirect methods

  • 3rd Lecture

  • (a) Function spaces and norms on them (b) Linear funccionals (c) Basic lemmas of the Calculus of variations (d) Variations of functionals (e) Stationary points and relative extrema: Euler's equations 2nd time round

  • 4th Lecture

  • (a) Special integrands, reduction to quadratures (b) Generalization for several dependent variables (c) Open end boundary conditions

  • 5th Lecture

  • (a) Variational derivative (b) Invariance of Euler-Lagrange equations with respect to point and other transformations (c) Use of invariance for solving equations

  • 6th Lecture

  • (a) Necessary and sufficient conditions for minima (b) Legendre condition (c) Conjugate points and the Jacobi condition

  • 7th Lecture

  • (a) Canonical form of Euler-Lagrange equations (b) Legendre transformation (c) Hamilton equations

  • 8th Lecture

  • (a) Canonical transformations (b) Hamilton-Jacobi theory

  • 9th Lecture

  • (a) The theorems of Noether (b) Conservation laws

  • 10th Lecture

  • (a) Parametric variational problems (b) Finsler geometry (c) Caratheodory's Royal Road to the Calculus of variations

  • 11th Lecture

  • (a) Variational problems with multiple integrals (b) Linear elasticity theory, spherical symmetry as a limit of cubic lattice (c) Conservation laws, Energy-momentum tensor field

  • 12th Lecture

  • (a) Aproximative (direct) methods in the Calculus of Variations (b) Ritz's variational method (c) Sturm-Liouville problem

Literature
    recommended literature
  • 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
    not specified
  • GIAQUINTA, Mariano and Stefan HILDEBRANDT. Calculus of variations. Berlin: Springer-Verlag, 1996, 474 s. ISBN 354050625X. info
  • GIAQUINTA, Mariano and Stefan HILDEBRANDT. Calculus of variations. Berlin: Springer-Verlag, 1996, xxix, 652. ISBN 3540579613. info
  • HILDEBRANDT, Stefan and Anthony TROMBA. The parsimonious universe : shape and form in the natural world. New York: Copernicus, 1996, XIII, 330. ISBN 0387979913. info
  • JOST, Jürgen and Xianqing LI-JOST. Calculus of variations. First published. Cambridge: Cambridge University Press, 1998, xvi, 323. ISBN 9780521057127. info
Teaching methods
Lectures: theoretical explanation with practical examples
Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems. Occasional homework.
Assessment methods
Teaching: lectures and practical classes


Final grade: colloquium - discussion based on home prepared solution of a problem). The problem will be assigned during the semester based also on the interests of the student.
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: S.
Teacher's information
Colloquim preparation and organization: the student chooses a problem from a given list (considered problem may be suggested by students), the solution is then presented during a seminar and discussed.
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 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, spring 2018, Spring 2021, Spring 2023.
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