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
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.