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Introduction to the calculus of variations (SoSe 26)
Inscription via LSF. (if you cannot register on LSF, you can do it on Moodle, key: "calvados")The course consists in 2 lectures per week and no exercises (WP 37). There will be some exercise sessions instead of some lectures.
Description:
The calculus of variation is, to put it simply, the study of critical points of functionals (functions of functions), which is often used in minimization problems. It connects many domains of mathematics such as PDEs, functional analysis, geometry,... and finds applications in a wide variety of physical problems such as in the determination of geodesics (minimizing distances), minimal surfaces, in classical and quantum mechanics (minimizing energies) etc. In this course, we will discuss both the theory and and the applications to some (and more) of the aforementioned problems.
Content:
(might change during the lecture)- The classical method: Euler-Lagrange equation, isoperimetric and holonomic constraints,
- Hamiltonian formulation, Legendre transformation
- Symmetries and conserved quantities: Noether's theorem
- Application to standard examples: geodesics, brachistochrone, catenary problems, Clairaut's invariant
- The direct method: proof of existence of critical points
- Crash course on functional analysis, Sobolev spaces, the Dirichlet energy and its generalization, regularity of extrema,
- Application to the non-linear Schrödinger equation
- Method of concentration compactness to study the loss of compactness in Sobolev spaces
Audience:
The course will be of interest for Master students of mathematics (WP 37), or motivated Bachelor students, who want to intensify their knowledge in analysis and partial differential equations.Credits:
6 ECTS.Schedule:
- Lectures: Tuesday 10:00-12:00 and Thursday 10:00-12:00 in Room B134
References:
- B. van Brunt, The Calculus of Variations, Universitext, Springer, 2004
- B. Dacorogna, Introduction to the Calculus of Variations. Imperial College Press, 2004
- P. Blanchard, E. Brüning, Variational Methods in Mathematical Physics: A unified approach, Springer, 1992
- L.C. Evans, Partial Differential Equations, AMS, 1998
- M. Lewin, Describing lack of compactness in Sobolev spaces, unpublished lecture notes "Variational Methods in Quantum Mechanics", 2010, Université de Cergy-Pontoise
- E. H. Lieb, M. Loss, Analysis, Graduate studies in mathematics, AMS, 2001
- D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order. Vol. 224. No. 2. Berlin: springer, 1977.
- V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1989
- D. Cioranescu, P. Donato, An introduction to Homogeneization, Oxford University Press, 2000