Introductory description

This module consists of a study of the mathematical techniques of variational methods, with applications to problems in physics and geometry. Critical point theory for functionals in finite dimensions is developed and extended to variational problems.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

This module is an introduction into mathematical techniques of variational methods, with applications to problems in Physics and Geometry. The basic problem in the calculus of variations is to minimise an integral which depends on a differentiable function and its derivatives. The module covers the following topics: a brief revision of critical points in finite dimension, the mathematical set up of a variational problem, Euler-Lagrange equations for functionals of different types (including a derivation of these equations), a discussion of appropriate boundary conditions, first integrals of the Euler Lagrange equations, applications of variational principles to classical mechanics (including the least action principle) and optics (Fermat's principle). Existence of minimizers by the direct method provides an important application of the analysis modules taught in the first two years. Constrained variational problems using Lagrange multipliers. Noether's theorem for Lagrangians that are symmetric with respect to a flow. The theory is illustrated by numerous examples.

Learning outcomes

By the end of the module, students should be able to:

• set up extremal problems and develop requisite analytical tools, particularly those needed for the application of the direct method
• derive and solve Euler-Lagrange equations in weak and classical forms
• solve extremal problems with and without constraints
• have a good understanding of how the laws of mechanics and geometrical problems involving least length and least area fit into the general framework
• understand the relation between symmetry of a Lagrangian and the associated conservation law

Indicative reading list

A useful and comprehensive introduction can be found in:
Bruce van Brunt, The Calculus of Variations, Springer 2004
H Kielhofer, Calculus of Variations, Springer, 2018

Other useful texts are:
IM Gelfand & SV Fomin. Calculus of Variations, Prentice Hall, 1963.
R Weinstock, Calculus of Variations with Applications to Physics and Engineering, Dover, 1974.
The module will not, however, closely follow the syllabus of any book.

Subject specific skills

At the conclusion of the module the student should be able to set up and solve various minimisation problems with and without constraints, to derive Euler-Lagrange equations and appreciate how the laws of mechanics and geometrical optics, as well as some geometrical problems involving least length and least area, fit into this framework. The student should also appreciate the mathematical difficulties encountered in variational problems, such as

• the lack of compactness in infinite dimensional spaces which may prevent the existence of a minimising function and
• the weak form of the Euler-Lagrange equations (and associated regularity theory if appropriate).
  The mathematical underpinning of the link between symmetry and conservation laws should also be understood.

Transferable skills

The student will learn how the methods of mathematical analysis studied in the first and second year can be applied to model some simple real world phenomena such as the motion of a mechanical object or the shape of a ray of light. The student will see how mathematical methods can be used to restate some fundamental laws of Physics. The students will also learn some basic features of optimisation problems.