Lecturer: Mario Micallef

Term(s): Term 2

Status for Mathematics students: List A (note that students who have taken MA209 Variational Principles cannot register for this module)

Commitment: 30 lectures

Assessment: 100% 3 hour examination

Formal registration prerequisites: None

Assumed knowledge:

• MA113 Differential Equations A
• MA259 Multivariable Calculus
• MA258 Mathematical Analysis III
• Equicontinuity and the Ascoli-Arzela Compactness Theorem from MA260: Norms, Metrics and Topologies

Useful background: Newton's laws of motion, scalar potential of electrostatic field or gravitational field. However, this is a mathematics module and a physics background will not be required.

Synergies: Variational problems arise whenever some quantity is to be optimised. This quantity can come from geometry (length, area), physics (energy), biology, economics. So all modules in which optimisation is considered are related to this module. Examples of such modules include:

• PX267 Hamiltonian Mechanics
• MA250 Introduction to Partial Differential Equations

Leads to: The following modules have this module listed as assumed knowledge or useful background:

• MA4H0 Applied Dynamical Systems
• MA4L3 Large Deviation Theory

Aims: To introduce the calculus of variations and to see how central it is to the formulation and understanding of physical laws and to problems in geometry.

Content: 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). The theory is extended to constrained variational problems using Lagrange multipliers. The theory is illustrated by numerous examples.

Objectives: 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.

Books:

The books most relevant to this module are:

• 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.

More books, and a commentary on them, can be found in the module's Moodle page. The module will not, however, closely follow the syllabus of any book.

Additional Resources