# MA209 Variational Principles

Lecturer: Vassili Gelfreich

Term(s): Term 3

Status for Mathematics students: List A for Maths

NOTE: To avoid clashes with April exams this module starts in the 2nd week of Term 3 and is lectured 4 times a week. It overlaps with the 3rd/4th year examination periods in April and May so these students should be aware that they may miss examinable material.

Commitment: 15 lectures

Assessment: One-hour examination

Prerequisites: MA131 Analysis, a module on solving ordinary differential equations (and it is probably a good idea to revise at least separation of variables and linear constant coefficient ODEs) and MA259 Multivariable Calculus is also helpful.

Leads To: MA4L3 Large Deviation Theory.

Content: 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. The basic problem in the calculus of variations for continuous systems is to minimise an integral of the form

$$I(y)=\int_a^b f(x,y,y_x)\,dx$$

on a suitable set of differentiable functions $y\colon[a,b]\to\mathbb{R}$ where $y_x$ denotes the derivative of $y$ with respect to $x$. The Euler-Lagrange theory for this problem is developed and applied to dynamical systems (Hamiltonian mechanics and the least action principle), shortest time (path of light rays and Fermat's principle), shortest length and smallest area problems in geometry. The theory is extended to constrained variational problems using Lagrange multipliers.

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.

Objectives: At the conclusion of the course you should be able to set up and solve minimisation problems with and without constraints, to derive Euler-Lagrange equations and appreciate how the laws of mechanics and geometrical problems involving least length and least area fit into this framework.

Books:

A useful and comprehensive introduction is:

R Weinstock, Calculus of Variations with Applications to Physics and Engineering, Dover, 1974.

Other useful texts are:

F Hildebrand, Methods of Applied Mathematics (2nd ed), Prentice Hall, 1965.

IM Gelfand & SV Fomin. Calculus of Variations, Prentice Hall, 1963.

The module will not, however, closely follow the syllabus of any book.