Calculus of Variations (FSI-S1M)

Academic year 2018/2019
Supervisor: doc. RNDr. Miroslav Kureš, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
Students will be made familiar with fundaments of variational calculus. They will be able to apply it in various engineering tasks.
Learning outcomes and competences:
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.
Prerequisites:
The calculus in the conventional ammount, boundary value problems of ODE and PDE.
Course contents:
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.
Teaching methods and criteria:
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
Controlled participation in lessons:
Seminars: required
Lectures: recommended
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture 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 Seminars related to the lectures in the previous week.
Literature - fundamental:
1. Fox, Charles: Introduction to the Calculus of Variations, New York: Dover, 1988
2. Kureš, Miroslav, Variační počet, PC-DIR Real, Brno 2000
Literature - recommended:
1. Kureš, Miroslav, Variační počet, PC-DIR Real Brno 2000
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
M2A-P full-time study M-MAI Mathematical Engineering -- GCr 3 Compulsory 2 1 S