MAS3809 : Variational Methods
- Offered for Year: 2017/18
- Module Leader(s): Dr James Waldron
- Owning School: Mathematics, Statistics and Physics
- Teaching Location: Newcastle City Campus
Semester 2 Credit Value:
To present basic ideas and techniques of variational calculus, including relevant applications.
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.
|Guided Independent Study||Assessment preparation and completion||1||2:00||2:00||Unseen exam|
|Scheduled Learning And Teaching Activities||Lecture||1||1:00||1:00||Class test|
|Scheduled Learning And Teaching Activities||Lecture||3||1:00||3:00||Problem classes|
|Scheduled Learning And Teaching Activities||Lecture||2||1:00||2:00||Revision lectures|
|Scheduled Learning And Teaching Activities||Lecture||25||1:00||25:00||Formal lectures|
|Guided Independent Study||Assessment preparation and completion||1||6:00||6:00||Revision for class test|
|Guided Independent Study||Assessment preparation and completion||1||13:00||13:00||Revision for unseen exam|
|Guided Independent Study||Independent study||1||27:00||27:00||Studying, practising, and gaining understanding of course material|
|Guided Independent Study||Independent study||3||3:00||9:00||Review of coursework assignments and course test|
|Guided Independent Study||Independent study||2||6:00||12:00||Preparation for coursework assignments|
Jointly Taught With
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.
The format of resits will be determined by the Board of Examiners
|Prob solv exercises||2||M||5||Coursework assignments|
|Prob solv exercises||2||M||5||Course 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.