Course syllabus

# Variationskalkyl Calculus of Variations

## FMAN25, 7,5 credits, A (Second Cycle)

Valid for: 2014/15
Decided by: Education Board B
Date of Decision: 2014-04-08

## General Information

Elective for: D4, E4, F4, F4-bg, Pi4-bs, Pi4-fm, Pi4-bg, Pi4-bem
Language of instruction: The course will be given in English on demand

## 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.

## Learning outcomes

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.

## 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 integral and Weierstrass' sufficient conditions for a strong local minimum.
• Hamilton's principle and Hamilton's equations. Lagrange's och Mayer's problems.

## Examination details

Assessment: Written assignments and oral exam.

Required prior knowledge: FMA430 Calculus in Several Variables.
The number of participants is limited to: No
The course overlaps following course/s: FMA200