MAGIC095: Calculus of Variations

Course details

A specialist MAGIC course


Autumn 2019
Monday, October 7th to Friday, December 13th


Live lecture hours
Recorded lecture hours
Total advised study hours


14:05 - 14:55 (UK)


In undergraduate calculus we learn to maximise or minimise functions of one variable, finding optimal points. The Calculus of Variations is concerned with finding optimal functions and the properties of these optimisers. Famous examples include surfaces of minimal area or the shortest or quickest paths between given points. In the classical indirect method, the optimisers are found as solutions of the Euler-Lagrange differential equations. In the modern direct method, one uses abstract means to find optimisers, which often yields existence results for solutions of differential equations.


Mostly undergraduate calculus, but we will also use the implicit function theorem and some existence and uniqueness theory for ODEs. Knowledge of Lebesgue integration is helpful but not required.

Related courses


In this 10-lecture series, elements of both the classical and the modern theory are presented. A tentative plan is as follows:

    1. Introduction. One-dimensional variational problems: Fundamental lemma, Euler-Lagrange equations
    2.-3. Convexity and existence and regularity issues
    4.-5. Second variations and necessary conditions for optimality
    6. Variational problems with constraints
    7. Problems involving multiple integrals
    8. Direct method: coercivity and lower semicontinuity
    9. Sobolev spaces, weak convergence and compactness
    10. The direct method for integral functionals and existence of solutions for some nonlinear PDEs


  • MK

    Dr Matthias Kurzke

    University of Nottingham


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The assessment for this course will be released on Monday 6th January 2020 at 00:00 and is due in before Sunday 19th January 2020 at 23:59.

The assessment for this course will be via a single take-home paper in January with 2 weeks to complete and submit online. There will be 4 or 5 questions, each giving between 10 and 30 marks for a total of 100 marks, and you will need 50 marks to pass.

Please note that you are not registered for assessment on this course.


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