Introduction to the calculus of variations

Introduction

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This free OpenLearn course, Introduction to the calculus of variations, is an extract from the Open University course MS327 Deterministic and stochastic dynamics [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] , a third level applied mathematics course which is designed to be studied as a first applied mathematics course at Open University level 3. It introduces core topics at this level and is structured around three books: Fundamental concepts of dynamics, Deterministic dynamics and Stochastic processes and diffusion. Understanding how to analyse dynamical processes is a central competence for using mathematics in science, engineering and economics. The materials in this course provide powerful tools which are used by practising applied mathematicians and give a deep insight into the properties of dynamical systems.

Introduction to the calculus of variations consists of material from MS327 Unit 5, Introduction to the calculus of variations, and has five sections in total. You should set aside about three to four hours to study each of the sections; the whole extract should take about 16 hours to study. The extract is a small part (around 8%) of a large course that is studied over eight months, and so can give only an approximate indication of the level and content of the full course.

The aim of this extract is to set up some important mathematical apparatus called the calculus of variations, a hugely important topic in the natural sciences. The calculus of variations is an active topic of study in its own right. It also has applications in subjects as diverse as statics, optics, differential geometry, approximate solutions of differential equations and control theory. The extract is relatively self-contained and should be reasonably easy to understand for someone with a sound knowledge of relevant mathematics, such as could be gained from Open University level 2 study of applied mathematics including differential equations and mechanics.

Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. However, the extract isn't entirely representative of the module materials, because there are no explicit references to use of the MS327 software or to video material (although please note that the PDF may contain references to other parts of MS327). In this extract, some illustrations have also been removed due to copyright restrictions.

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Section 1 introduces many of the key ingredients of the calculus of variations by solving a seemingly simple problem – finding the shortest distance between two points in a plane. In particular, this section introduces the notion of a functional and that of a stationary path.

Section 2 briefly describes a few basic problems that can be formulated in terms of functionals, in order to give you a feel for the range of problems that can be solved using the calculus of variations.

Section 3 is a short interlude about partial and total derivatives, which are used extensively throughout the rest of the extract.

Section 4 is the most important section. It contains a derivation of the Euler-Lagrange equation, which will be used throughout the rest of the extract.

In Section 5, we apply the Euler-Lagrange equation to solve some of the problems discussed in Section 2, as well as a problem arising from a new topic, called Fermat’s principle.