Calculus of Variations
FSI-S1MAcad. year: 2020/2021
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.
Recommended optional programme components
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)
Elsgolc., L., Calculus of Variations, Dover Publications 2007 (EN)
Wasserman. R., Tensors And Manifolds: With Applications to Physics, 2nd ed., Oxford University Press 2009 (EN)
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
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
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.
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.
eLearning: currently opened course