Calculus of Variations 3

Unit aims

To introduce students to the calculus of variation, and to illustrate its use in the solution of some elementary problems arising in mathematics and in physics.

Unit description

Calculus of Variations is an important branch of optimisation in which the quantity (the functional) to be minimised depends on infinite-dimensional vectors that may for instance represent curves or surfaces. The subject has deep connections with various fields in the natural sciences, including differential geometry, ordinary and partial differential equations, materials science, mathematical biology, etc. It is one of the oldest and yet one of the most used tools for the investigation of problems involving the concept of "free energy". The aims of this course are (1) to cover the basics of the calculus of variations, including the one-variable case, and (2) to illustrate the theory by considering various applications arising in the natural sciences.

Relation to other units

The unit builds on Calculus 1, Linear Algebra and Geometry 1, and Multivariable Calculus 2. In terms of the mathematics involved, it is also closely connected with other units that pertain to differential and partial differential equations

Learning objectives

After taking this unit, students will:

  1. know the basic techniques and results of the calculus of variations
  2. be able to apply these techniques to solve some problems arising in other areas of science that can be formulated in terms of the minimisation of some functional.

Transferable skills

Increased understanding of the relationship between mathematics and the problems arising from the physical sciences.

Development of problem-solving and analytical skills.


1. Basic concepts of the calculus of variations: Definitions: functionals, extremum, variations, function spaces. Necessary conditions for an extremum. Euler-Lagrange equations. Convexity and it's role in minimization. Minimization under constraints. Existence and nonexistence of minimizers. Basic examples: Brachistochrone problem, Isoperimetric problem, Geodesics on the surface.

2. Generalizations: Higher derivatives. Functions of several variables. Least action principle. Basic examples: vibrating rod, vibrating membrane.

3. Second variation and local minimality/stability: Second variation. Legendre Condition. Relation between local stability and local minimality.

4. Direct methods in the calculus of variations. Minimizing sequences. Ritz method and method of finite differences.

5. Hamiltonian and duality principle. Relation between Hamiltonian systems and variational problems. Legendre transforms.  Examples from classical mechanics (if time permits)

Reading and References

I M Gelfand and S V Fomin, Calculus of Variations, Prentice-Hall Bruce van

Brunt, The Calculus of Variations, Dover

Unit code: MATH30005
Level of study: H/6
Credit points: 10
Teaching block (weeks): 2 (19-24)
Lecturer: Dr Yves Tourigny


MATH20901 Multivariable Calculus and MATH20101 Ordinary Differential Equations



Methods of teaching

Lectures and homeworks.

Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

  • 100% from a 1 hour 30 minute exam in May/June*

* Level 6 and 7 will have different exam papers

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.

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