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Course, academic year 2023/2024
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Variational methods - NBCM174
Title: Variační metody
Guaranteed by: Department of Chemical Physics and Optics (32-KCHFO)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
Semester: summer
E-Credits: 5
Hours per week, examination: summer s.:2/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. Ing. Lucie Augustovičová, Ph.D.
Annotation -
Solutions of problems of classical variational calculus, Finding and Examination of Extreme of functionals. Formulation of a variational problem and determination of their properties. Modern variational calculus. Application of variational methods to the solution of boundary value problems. Applications in problems of mathematical physics.
Last update: Kapsa Vojtěch, RNDr., CSc. (02.05.2018)
Aim of the course -

The aim of this course is to deepen and broaden knowledge of variational methods with applications in physics.

Last update: Augustovičová Lucie, doc. Ing., Ph.D. (02.05.2018)
Course completion requirements -

The course credit is awarded for participation and exercise activity. Lack of participation can not be compensated by other means.

Course credit is a condition of admission to the exam.

The exam is oral and the requirements correspond to the syllabus of the subject in the range that was presented at the lecture.

Last update: Augustovičová Lucie, doc. Ing., Ph.D. (02.05.2018)
Literature -
  • K. Rektorys, Variační metody v inženýrských problémech a v problémech matematické fyziky, Academia, Praha 1999.
  • L. E. Elsgolc, Variační počet, SNTL, Praha 1965.
  • K. W. Cassel, Variational Methods with Applications in Science and Engineering, Cambridge University Press, 2013.
  • S. V. Fomin, R. A. Silverman, Calculus of variations, Courier Dover Publications, Dover 2000.
  • B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press, London 2004.

Last update: Kapsa Vojtěch, RNDr., CSc. (02.05.2018)
Teaching methods -

lecture and excercise

Last update: Kapsa Vojtěch, RNDr., CSc. (02.05.2018)
Syllabus -

1. Introduction and motivational examples

2. Fundamental lemma of variational calculus

3. Extreme of functional, Euler-Lagrange equations

4. Conditions for the existence of extreme of functional

5. Sturm-Liouville problem and quadratic functional

6. Sobolev spaces

7. Weak solution of boundary value problems for elliptic equations

8. Lax-Milgram theorem

9. Rayleigh-Ritz method

10. Hamilton's principle for discrete systems

11. Hamilton's principle for continuous systems

12. Stability of dynamical systems

The exercises include the solution of specific tasks of the variational calculus - e.g. the problem of the shortest line, the brachistochrone problem, the shape of a liquid drop, the soap film between two coaxial circular rings, the rod deflection, the static deflection of an elastic string, applications of Hamilton’s principle

Last update: Augustovičová Lucie, doc. Ing., Ph.D. (02.05.2018)
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