Course detail

Calculus of Variations

FSI-S1MAcad. year: 2025/2026

The calculus of variations. The classical theory of the variational calculus: the first and the second variations, conjugate points, generalizations for a vector function, higher order problems, relative maxima and minima and isoperimaterical problems, integraks with variable end points, geodesics, minimal surfaces. Applications in mechanics and optics.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

doc. RNDr. Miroslav Kureš, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

The calculus in the conventional ammount, boundary value problems of ODE and PDE.

Rules for evaluation and completion of the course

Classified seminar credit: the attendance, the brief paper, the semestral work

Seminars: required
Lectures: recommended

Aims

Students will be made familiar with fundaments of variational calculus. They will be able to apply it in various engineering tasks.
The variational calculus makes access to mastering in a wide range
of classical results of variational calculus. Students get up apply results
in technical problem solutions.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Elsgolc., L., Calculus of Variations, Dover Publications 2007 (EN)
Fox, Charles: Introduction to the Calculus of Variations, New York: Dover, 1988 (EN)
Kureš, Miroslav, Variační počet, PC-DIR Real, Brno 2000 (CS)
Wasserman. R., Tensors And Manifolds: With Applications to Physics, 2nd ed., Oxford University Press 2009 (EN)

Recommended reading

Kureš, Miroslav, Variační počet, PC-DIR Real Brno 2000 (CS)

Classification of course in study plans

  • Programme N-MAI-P Master's 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Miroslav Kureš, Ph.D.

Syllabus

1. Introduction. Instrumental results.
2. The fundamental lemma. First variation. Euler equation.
3. Second variation.
4. Classical applications.
5. Generalizations of the elementary problem.
6. Methods of solving of first order partial differential equations.
7. Canonical equations and Hamilton-Jacobi equation.
8. Problems with restrictive conditions.
9. Isoperimetrical problems.
10. Geodesics.
11. Minimal surfaces.
12. n-bodies problem.
13. Solvability in more general function spaces.

Exercise

13 hod., compulsory

Teacher / Lecturer

doc. RNDr. Miroslav Kureš, Ph.D.

Syllabus

Seminars related to the lectures in the previous week.