Course detail
Calculus of Variations
FSI-S1M-A Acad. year: 2021/2022 Summer semester
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
English
Number of ECTS credits
3
Supervisor
Department
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.
Prerequisites
The calculus in the conventional ammount, boundary value problems of ODE and PDE.
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
Aims
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
The study programmes with the given course
Programme N-MAI-A: Mathematical Engineering, Master's, compulsory
Type of course unit
Lecture
26 hours, optionally
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 hours, compulsory
Syllabus
Seminars related to the lectures in the previous week.