# MA209 Variational Principles

**Lecturer: **Mario Micallef

**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:** 100% 1 hour examination

**Formal registration prerequisites: **None

**Assumed knowledge: **Solving ODEs including separation of variables and linear constant coefficient ODEs (this material is covered in both MA133 Differential Equations and MA113 Differential Equations A), MA259 Multivariable Calculus (and, by implication, the prerequisites for this module), MA244 Analysis III or MA258 Mathematical Analysis III

**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
- MA4G6 Calculus of Variations
- MA4L9 Variational Analysis and Evolution Equations

**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.