*Course Syllabus for*
# Calculus of Variations

Variationskalkyl

## FMA201F, 7.5 credits

**Valid from:** Autumn 2013

**Decided by:** FN1/Anders Gustafsson

**Date of establishment:** 2014-01-27

## General Information

**Division:** Mathematics

**Course type:** Course given jointly for second and third cycle

**The course is also given at second-cycle level with course code:** FMA200

**Teaching language:** English

## Aim

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.

## Goals

*Knowledge and Understanding*

For a passing grade the doctoral 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 doctoral 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.

## Course Contents

Variational problems without and with constraints. Euler's equations without and with constraints. Legendre's, Jacobi's and Weierstrass' necessary conditions for a local minimum.
Hilbert's invariant integral and Weierstrass' sufficient conditions for a strong local minimum.
Hamilton's principle and Hamilton's equations. Lagrange's och Mayer's problems.

## Course Literature

Mesterton-Gibbons, M.: A Primer on the Calculus of Variations and Optimal Control Theory. American Mathematical Soc., 2009. ISBN 9780821847725.

**Type of instruction:** Lectures

**Examination formats:** Oral exam, written assignments

**Grading scale:** Failed, pass

**Examiner:**

## Admission Details

**Assumed prior knowledge:** Calculus in one and several variables corresponding to the courses FMAA05 and FMA430, and linear algebra corresponding to the course FMA420.

## Course Occasion Information

**Course coordinators:**