FMAN25 - Calculus of Variations
About the course
In connection with investigations of the law of gravity, Galileo asked himself the following question around 1637:
If a bead under the influence of gravity slides frictionlessly along a curve in a vertical plane from a point A to another, lower, point B, will the time of fall be shorter if the curve is shaped like a circular arc rather than a straight line?
In 1696, this question was brought to a head by the Swiss mathematician Johann Bernoulli challenging his colleagues to find the BRACHISTOCHRON, i.e. the curve between A and B along which the fall time becomes the shortest possible. Correct solutions were submitted by Newton, Leibniz, l'Hôpital, Tschirnhaus and the brother Jakob Bernoulli. Thus, the foundation was laid for a new subfield of mathematics: CALCULUS OF VARIATION.
Calculus of variations is about optimizing integral expressions (functionals) with respect to functions included in the integrand. During the 18th century, it turned out that a number of interesting problems in geometry and mechanics could be formulated as variational problems. For example, in 1734 Euler published the solution to the problem of determining the surface of rotation whose area is the smallest possible. Towards the end of the nineteenth century, however, the fundamentals of the calculus of variations were questioned by, among others, a. Weierstrass, who reformulated the theory so that it became both simpler and more rigorous at the same time. It is essentially this form that is used today.
During the twentieth century, the so-called DIRECT VARIATION METHODS were added, which played a major role in geometry and the theory of non-linear partial differential equations. Modern applications of the calculus of variations can be found in e.g. physics, control technology, financial economics, biology and image analysis.
The course begins with some classic examples. Then one moves on to the definition of the variation of a functional and Euler's differential equations. These methods are used in problems with various boundary and side conditions. Then Legendre's, Jacobi's and Weierstrass' conditions for local maxima and minima are treated. We will use computer aids, such as MATLAB, to solve some tasks.
The course is designed for students who want to deepen their knowledge of mathematical analysis and see applications of the material in the basic courses. It is also suitable for PhD students in mechanics, control theory, computer vision and strength theory.
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Syllabus - Study period - Course literature LTH.se (Länkar till en extern sida.)