(Created 2008-07-17.)
 CALCULUS OF VARIATIONS FMA200

Higher education credits: 6. Grading scale: TH. Level: A (Second level). Language of instruction: The course might be given in English. Optional for: D4, E4, F3, F3tvb, Pi3, Pi3bs, Pi3fm. Course coordinator: Director of Studies, Lars-Christer Böiers, Lars_Christer.Boiers@math.lth.se, Matematik. Recommended prerequisits: Calculus in one and several variables (FMA410, FMA430). FMA420 Linear algebra. Assessment: Written and/or oral test, to be decided by the examiner. Some written assignments. Further information: The course is given every second year. Home page: http://www.maths.lth.se/matematiklth/vitahyllan/vitahyllan.html.

Aim
The course aims at a presentation of basic theory and applications of the calculus of variations, i.e., optimisation problems for "functions of functions". A classical example is the isoperimetric problem, to find that closed curve of a given length which surrounds a domain of maximal area. Many physical laws can be formulated as principles of variations, i.e. the law of light refraction. The calculus of variations is also a corner stone in classical mechanics, and has many other technological applications e.g. in systems theory and optimal control.

Knowledge and understanding
For a passing grade the student must

be able to explain the basic parts of the theory in the context of an oral examination.

Skills and abilities
For a passing grade the student must

be able to show capability to identify problems which can be modelled with the concepts introduced.

be able to integrate methods and views from the different parts of the course in order to solve problems and answer questions within the framework of the course.

in writing and orally, with clear logic and proper terminology be able to explain the solution to a mathematical problem within the course.

Contents
Euler's equations without and with constraints. Canonical form. The Legendre transform. Noether's theorem. Hamilton's principle. Second order conditions. Weierstrass' sufficient conditions. Furthermore, direct methods (Ritz, ...) are treated, as well as the maximum principle and some applications.

Literature
Sparr, A.: Föreläsningar i variationskalkyl. Mat. inst.