Syllabus academic year 2011/2012
(Created 2011-09-01.)
CALCULUS OF VARIATIONSFMA200
Credits: 6. Grading scale: TH. Cycle: A (Second Cycle). Main field: Technology. Language of instruction: The course might be given in English. Optional for: D4, E4, F4, F4bs, Pi4, Pi4fm. Course coordinator: Director of Studies Anders Holst, Anders.Holst@math.lth.se, Mathematics. 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.

Aim
The course aims at a presentation of basic theory and applications of the calculus of variations, i.e., optimization problems for "functions of functions". A classical example is the isoperimetric problem, to find which closed curve of a given length encloses maximal area. Many physical laws can be formulated as variational principles, i.e. the law of 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 demonstrate an ability 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. Matematikcentrum.