Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2025/26
| MATH1000 | Core Mathematics |
| MATH2650 | Calculus of Variations |
MATH2650 Calculus of Variations
This module is not approved as a discovery module
The calculus of variations concerns problems in which one wishes to find the extrema of some quantity over a system that has functional degrees of freedom. Many important problems arise in this way across pure and applied mathematics. In this course, you will meet the system of differential equations arising from such variational problems: the Euler-Lagrange equations. These equations and the techniques for their solution, will be studied in detail.
This module aims to provide an introduction to the calculus of variations. Students will learn how to approach minimisation problems where the outcome depends on unknown functions.
On successful completion of the module students will have demonstrated the following learning outcomes:
1. Learn how to formulate and analyse variational problems;
2. Apply the calculus of variations to a range of minimisation problems.
Skills Learning Outcomes
On successful completion of the module students will have demonstrated the following skills learning outcomes:
a. Communicate through written work technical information and reasoning.
b. Apply analytical thinking and technical knowledge to solve problems.
c. Write in a clear, concise, and focused way.
d. Manage workloads, deadlines, and workplace pressure through prioritisation and productivity skills.
1. Introduction to the general ideas of the Calculus of Variations. Derivation of the Euler equation.
2. Simple problems involving one independent variable and one dependent variable; catenary (the hanging chain problem), brachistochrone (the sliding particle problem), the shape of soap films.
3. Extensions to cases with several dependent variables, higher derivatives and natural boundary conditions. Lagrange's necessary condition.
4. Finding extrema with constraints. Lagrange multipliers. Variable end points.
5. Rayleigh-Ritz method and eigenvalue problems for Sturm-Liouville equations.
6. Hamilton's principle for mechanics. Application to the double pendulum.
Additional topics that build on these may be covered as time allows. Further details of possible topics will be delivered closer to the time that the module runs.
Methods of assessment
The assessment details for this module will be provided at the start of the academic year
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 22 | 1 | 22 |
| Seminar | 5 | 1 | 5 |
| Private study hours | 73 | ||
| Total Contact hours | 27 | ||
| Total hours (100hr per 10 credits) | 100 | ||
The reading list is available from the Library website
Last updated: 28/05/2025
Errors, omissions, failed links etc should be notified to the Catalogue Team