# CALCULUS OF VARIATIONS - 2019/0

Module code: MATM059

Module Overview

The objective of this module is to introduce students to some of the classical and modern methods in the Calculus of Variations. There will be many examples to illustrate the theory in action.

Module provider

Mathematics

BEVAN Jonathan (Maths)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 7

JACs code: G110

Module cap (Maximum number of students): N/A

Module Availability

Semester 2

Prerequisites / Co-requisites

MAT1032 Real Analysis 1 and MAT2004 Real Analysis 2, or equivalent.

Module content

• The Euler-Lagrange equations in one and several variables

• Conservation laws and constrained variational problems

• Weak derivatives

• The fundamental lemma of the Calculus of Variations.

• Necessary conditions for the existence of a minimizer: the Euler-Lagrange equation, the Legendre condition; convexity as a condition for the uniqueness of solutions to the Euler-Lagrange equation. Hamilton's equations.

• How to solve the Euler-Lagrange equation. Examples to include the Brachistochrone, minimal surfaces of revolution, the catenary (=cable on a suspension bridge). Fermat's principle.

• The direct method: Tonelli's theorem for the existence of a minimizer. Examples of problems with no minimizer,

• Weak and strong local minimizers: sufficient conditions for a solution of the Euler-Lagrange equation to be a minimizer.

Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-Semester Test 50 minutes 20
Examination Examination 80

Alternative Assessment

N/A

Assessment Strategy

The strategy of assessment via Examination and In-Semester Test is designed to provide:

• A detailed introduction to the classical and modern methods in the Calculus of Variations

• Experience of the methods used to interpret, understand and solve applicable problems using such techniques.

Module aims

• Equip students with some of the basic tools of the Calculus of Variations
• Develop the students’ appreciation of where the Calculus of Variations is useful and how it fits in with other mathematical disciplines

Learning outcomes

Attributes Developed
001 Accurately identify conditions under which the Euler-Lagrange equation can be derived, and to derive it from first principles under these conditions
002 Recognize and determine appropriate function spaces in which to set given variational problems, and to apply previous knowledge of ordinary differential equations in order to solve them
003 Understand and apply theorems concerning weak and strong local minimizers to given variational problems

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

Independent Study Hours: 106

Lecture Hours: 33

Tutorial Hours: 11

Methods of Teaching / Learning

The learning and teaching methods include:

• 3 x 1 hour lectures per week x 11 weeks interspersed with tutorials when appropriate for 11 weeks.

• Formative feedback will be provided both on submitted problem sheets and in the In-Semester test, to help students develop their understanding as the course progresses.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.