## MATH2650 Calculus of Variations

### 10 creditsClass Size: 150

Module manager: Dr R. Sturman
Email: R.Sturman@maths.leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2008/09

### Pre-requisite qualifications

(MATH1015 or MATH1060) and (MATH1050 or MATH1960), or equivalent.

Module replaces

MATH2610

This module is approved as an Elective

### 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.

### Objectives

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

### Syllabus

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 case with several dependent variables and then several independent variables.
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. Euler-Lagrange equation. Formulation of problems involving small oscillations in terms of normal coordinates. Analysis of normal modes and eigenfrequencies.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Lecture 22 1.00 22.00 Tutorial 11 1.00 11.00 Private study hours 67.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 100.00

### Opportunities for Formative Feedback

Regular examples sheets

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 100.00 Total percentage (Assessment Exams) 100.00

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