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| Module Availability |
| Spring |
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| Assessment Pattern |
Assessment Pattern
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Unit(s) of Assessment
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Weighting Towards Module Mark( %)
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Examination
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75
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Coursework
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25
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Qualifying Condition(s)
An overall aggregate mark of 50% for the module is required to pass the module.
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| Module Overview |
The objective of this module is to introduce students to some of the classical and modern methods in the Calculus of Variations. There will be many examples to illustrate the theory in action. |
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| Prerequisites/Co-requisites |
| Real Analysis |
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| Module Aims |
The course aims to:
• equip students with some of the basic tools of the Calculus of Variations
• develop the students’ appreciation of where the Calculus of Variations is useful and how it fits in with other mathematical disciplines
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| Learning Outcomes |
By the end of the module students will be expected to be able to:
• accurately identify conditions under which the Euler-Lagrange equation can be derived, and to derive it from first principles under these conditions
• recognize and determine appropriate function spaces in which to set given variational problems, and to apply previous knowledge of ordinary differential equations in order to solve them
• understand and apply theorems concerning weak and strong local minimizers to given variational problems
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| Module Content |
Weak derivatives
• The fundamental lemma of the Calculus of Variations.
• Necessary conditions for the existence of a minimizer: the Euler-Lagrange equation, the Legendre condition; convexity as a condition for the uniqueness of solutions to the Euler-Lagrange equation. Hamilton's equations.
• How to solve the Euler-Lagrange equation. Examples to include the Brachistochrone, minimal surfaces of revolution, the catenary (=cable on a suspension bridge). Fermat's principle.
• The direct method: Tonelli's theorem for the existence of a minimizer. Examples of problems with no minimizer,
• Weak and strong local minimizers: sufficient conditions for a solution of the Euler-Lagrange equation to be a minimizer.
• Variational problems with constraints: Lagrange multipliers, endpoint conditions.
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| Methods of Teaching/Learning |
| Learning will take place through a mixture of lectures and problem solving classes. Formative feedback will be provided both on submitted problem sheets and on the assessed coursework element to help students develop their understanding as the course progresses. |
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| Selected Texts/Journals |
Introduction to the Calculus of Variations by B. Dacorogna, published by Imperial College Press, 2004. |
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| Last Updated |
| March 2011 |
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