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.

## Instruction Details

Type of instruction: Lectures

## Examination Details

Examination formats: Oral exam, written assignments