# Calculus of Variations

## Official Course Description

## Description

Course upgraded to FMAN25 starting 2014/15, see http://www.maths.lth.se/course/newcalcvar/

In connection with investigations of the law of gravity 1637 Galileo posed the following question:

If a pearl on which gravity is acting slides without friction along a curve in a vertical plane from a point A to a lower point B , will the travel time be shorter if the curve is a circular arc than if it is a straight line segment?

In 1696, a more general question was posed by the Swiss mathematician Johann Bernoulli who challenged his colleagues to find the BRACHISTOCHRONE, i.e. the curve from A to B along which the travel time is minimal. Correct solutions were provided by Newton, Leibniz, l'Hôpital, Tschirnhaus and Johann's brother Jakob Bernoulli. This was the foundation of a new branch of mathematics: the CALCULUS OF VARIATIONS.

The main problem of the calculus of variations is how to optimize a functional (a definite integral involving a function and its derivatives) with respect to the occuring function. During the 18th century it was shown that many interesting problems in geometriy and mechanics could be formulated as variational problem. As an example Euler in 1734 published a solution to the problem of finding the surface of revolution between to given circles which has the least area. Towards the end of the nineteenth century, however, the theoretical basis of the calculus of variations was called into question by among others Weierstrass, who reformulated the theory making it both simpler and more rigorous at the same time. It is essentially this formulation which is used today.

During the twentieth century the so called DIRECT METHODS appeared. They have played an important role in the development of geometry and the theory for nonlinear partial differential equations. Modern applications of the calculus of variations occur within e.g. physics, control theory, financial economics, biology and image analysis.

The course starts with some classical examples. Then one considers the definition of the variation of a functional and Euler's differential equations. These methods are used on problems with different constraints. Thereafter Legendre's, Jacobi's and Weierstrass' conditions for local maxima and minima are studied. We will use software packages, for example MATLAB to solve some problems.

The course is intended for students who wish to fördjupa deepen their knowledge of mathematical analysis and to see applications of the content of the calculus courses. It is also suitable for post graduate students of mechanics, control theory, computer vision and solid mechanics.