(Created 2010-07-25.)

CALCULUS OF VARIATIONS | FMA200 |

**Aim**

The course aims at a presentation of basic theory 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 surrounds encloses maximal area. Many physical laws can be formulated as principles of variations, 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

*Skills and abilities*

For a passing grade the student must

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**

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 the maximum principle and some applications.

**Literature**

Sparr, A.: Föreläsningar i variationskalkyl. Matematikcentrum.