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2023/24 Undergraduate Module Catalogue

MATH2650 Calculus of Variations

10 creditsClass Size: 95

Module manager: Professor Alastair Rucklidge

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2023/24

Pre-requisite qualifications

MATH1005 or (MATH1010 and MATH1012) or (MATH1050 and MATH1400) or (PHYS1290 and PHYS1300) or equivalent.

This module is not approved as a discovery module

Module summary

The calculus of variations concerns problems in which one wishes to find the extrema (usually the minima) of some quantity over a system that has functional degrees of freedom. Many important problems arise in this way across pure and applied mathematics and physics.In this course it is shown that such variational problems give rise to a system of differential equations, the Euler-Lagrange equations. These equations, which have far reaching applications, and the techniques for their solution, will be studied in detail.


Students will:
- learn how to formulate and analyse variational problems;
- be able to apply the Calculus of Variations to a range of minimisation problems in physics and mechanics.


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.

Teaching methods

Delivery typeNumberLength hoursStudent hours
Private study hours68.00
Total Contact hours32.00
Total hours (100hr per 10 credits)100.00

Private study

Studying and revising of course material.
Completing of assignments and assessments.

Opportunities for Formative Feedback

Regular examples sheets and quizzes.

Methods of assessment

Assessment typeNotes% of formal assessment
Total percentage (Assessment Coursework)15.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exam typeExam duration% of formal assessment
Standard exam (closed essays, MCQs etc)2 hr 00 mins85.00
Total percentage (Assessment Exams)85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 28/04/2023 14:54:41


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