Course syllabus

# Variationskalkyl

Calculus of Variations

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

## General Information

## Aim

## Learning outcomes

## Contents

## Examination details

## Admission

## Reading list

## Contact and other information

Calculus of Variations

Valid for: 2021/22

Faculty: Faculty of Engineering, LTH

Decided by: PLED F/Pi

Date of Decision: 2021-04-23

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

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 approaches 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.

- Variational problems without and with constraints. Euler's equations with and without 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.

Grading scale: TH - (U,3,4,5) - (Fail, Three, Four, Five)

Assessment: Written assignments and oral exam.

The examiner, in consultation with Disability Support Services, may deviate from the regular form of examination in order to provide a permanently disabled student with a form of examination equivalent to that of a student without a disability.

Assumed prior knowledge: FMAB30 Calculus in Several Variables.

The number of participants is limited to: No

The course overlaps following course/s: FMA200, MATC25

- Kot, M: A First Course in the Calculus of Variations. American Mathematical Society, 2014, ISBN: 978-1-4704-1495-5.

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

Course administrator: Studerandeexpeditionen, expedition@math.lth.se

Teacher: Niels Christian Overgaard, niels_christian.overgaard@math.lth.se

Course homepage: http://www.maths.lth.se/course/newcalcvar/