Course syllabus

# Variationskalkyl

Calculus of Variations

## FMA200, 6 credits, A (Second Cycle)

## General Information

## Aim

## Learning outcomes

## Contents

## Examination details

## Admission

## Reading list

## Contact and other information

Calculus of Variations

Valid for: 2013/14

Decided by: Education Board B

Date of Decision: 2013-04-10

Elective for: D4, E4, F4, F4-fm, Pi4, Pi4-bs, Pi4-fm

Language of instruction: The course will be given in English on demand

The aim of the course is to present the basic theory for, 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.

Competences and skills

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.

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 some applications.

Grading scale: TH

Assessment: Written and/or oral test, to be decided by the examiner. Some written assignments.

Required prior knowledge: Calculus in one and several variables (FMA410, FMA430). FMA420 Linear algebra.

The number of participants is limited to: No

- Mesterton-Gibbons, M.: A Primer on the Calculus of Variations and Optimal Control Theory. AMS, 2009, ISBN: 978-0-8218-4772-5. The literature may be changed.

Course coordinator: Studierektor Anders Holst, Studierektor@math.lth.se