The Calculus of Variations, which plays an important role in both pure and applied mathematics, dates from the time of Newton. Development of the subject started mainly with the work of Euler and Lagrange in the eighteenth century and still continues. This course, which for most students begins the MSc in Mathematics programme, develops the theory of the Calculus of Variations. It also introduces other topics including the calculus of functions of several real variables; and ideas of convergence, particularly of sequences of functions and normed vector spaces – a working knowledge of which is required both in this and other courses in the MSc in Mathematics programme.

Problems such as the determination of the shortest curve between two points on a given smooth surface and the shapes of soap films, are most easily formulated using ideas from the Calculus of Variations. The Calculus of Variations also provides useful methods of approximating solutions of linear differential equations; furthermore, variational principles also provide the theoretical underpinning for the coordinate-free formulations of many laws of nature.

M820 provides an introduction to the central ideas of variational problems, as well as some of the mathematical background necessary for the subject. Many of the simple applications of Calculus of Variations are described and, where possible, the historical context of these problems is discussed.

The course also contains some more advanced material, such as an analysis of the second variation and of discontinuous solutions; it ends with a discussion of the general properties of the solutions of an important class of linear differential equations, namely Sturm-Liouville systems. Throughout, the emphasis is on the mathematical ideas and one aim is to illustrate the need for mathematical rigour. Applications will be discussed but you are not expected to have a detailed understanding of the underlying physical ideas.