Course syllabus

Calculus of Variations

FMA200, 6 credits, A (Second Cycle)

Valid for: 2013/14
Decided by: Education Board B
Date of Decision: 2013-04-10

General Information

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.

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.


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.

Examination details

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

Reading list

Contact and other information

Course coordinator: Studierektor Anders Holst,