Course detail
Calculus of Variations
FSI-S1M-A Acad. year: 2024/2025 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
4
Supervisor
Department
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.
The study programmes with the given course
Programme N-AIM-A: Applied and Interdisciplinary Mathematics, Master's, compulsory-optional
Programme C-AKR-P: , Lifelong learning
specialization CLS: , elective
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.