Course detail

Calculus of Variations

FSI-S1MAcad. year: 2019/2020

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.

Learning outcomes of the course unit

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.


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


Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Fox, Charles: Introduction to the Calculus of Variations, New York: Dover, 1988 (EN)
Kureš, Miroslav, Variační počet, PC-DIR Real Brno 2000 (CS)
Kureš, Miroslav, Variační počet, PC-DIR Real, Brno 2000 (CS)

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

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

Language of instruction


Work placements

Not applicable.


Students will be made familiar with fundaments of variational calculus. They will be able to apply it in various engineering tasks.

Specification of controlled education, way of implementation and compensation for absences

Seminars: required
Lectures: recommended

Classification of course in study plans

  • Programme M2A-P Master's

    branch M-MAI , 1. year of study, summer semester, 3 credits, compulsory

Type of course unit



26 hours, optionally

Teacher / Lecturer


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.


13 hours, compulsory

Teacher / Lecturer


Seminars related to the lectures in the previous week.