Undergraduate

modules

Modules

MAS3809 : Variational Methods

Semesters
Semester 2 Credit Value: 10
ECTS Credits: 5.0

Aims

To present basic ideas and techniques of variational calculus, including relevant applications.

Module Summary
What is the shortest route between two places on the Earth's surface (the answer isn't a straight line, because the Earth isn't flat!)? What is the optimum roller coaster design that provides the minimum descent time? How do you curve a string of fixed length such that it encloses the maximal area (a problem solved, according to legend, by Queen Dido of Carthage). To answer questions such as these, we need a way to find the path between two points which minimises some quantity (such as length or time). The calculus of variations is a very elegant and powerful way of doing this, and consequently has wide application to real-world problems. The ideas also provide the basis for the mathematical formulation of general principles that underpin various fundamental governing equations of mathematical physics e.g. mechanics.

Outline Of Syllabus

Review of standard results from the differential calculus. Differentiation under the integral sign. Definition of and method for calculating extremals (minima/maxima) of functionals. The Euler-Lagrange equation. Classical examples from everyday life. The Legendre condition for a minimum. The Euler-Poisson Equation. Fixed and variable end conditions. Integral constraints. Many dependent variables.

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture31:003:00Problem classes
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures
Scheduled Learning And Teaching ActivitiesLecture251:0025:00Formal lectures
Guided Independent StudyAssessment preparation and completion16:006:00Revision for class test
Guided Independent StudyAssessment preparation and completion113:0013:00Revision for unseen exam
Guided Independent StudyAssessment preparation and completion12:002:00Unseen exam
Scheduled Learning And Teaching ActivitiesLecture11:001:00Class test
Guided Independent StudyIndependent study127:0027:00Studying, practising, and gaining understanding of course material
Guided Independent StudyIndependent study33:009:00Review of coursework assignments and course test
Guided Independent StudyIndependent study26:0012:00Preparation for coursework assignments
Total100:00
Jointly Taught With
Code Title
PHY3029Variational Methods
MAS8809Variational Methods
Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems. Tutorials are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. In addition, office hours (two per week) will provide an opportunity for more direct contact between individual students and the lecturer.

Assessment Methods

The format of resits will be determined by the Board of Examiners

Exams
Description Length Semester When Set Percentage Comment
Written Examination1202A90N/A
Exam Pairings
Module Code Module Title Semester Comment
PHY3029Variational Methods2N/A
MAS8809Variational Methods2N/A
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises2M5Coursework assignments
Prob solv exercises2M5Course test
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The written exercises are expected to consist of two assignments of equal weight: the exact nature of assessment will be explained at the start of the module. The coursework assignments and the (in class) coursework test allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these are thus formative as well as summative assessments.

Reading Lists

Timetable