Calculus of Variations 34

Unit aims

To introduce students to calculus of variations and use it to solve basic problems arising in physics, mathematics and materials science.

Unit description

Calculus of Variations is an important branch of optimization that deals with finding extrema of the functionals in certain functional spaces.  It has deep relation with various fields in 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 investigation of the problems involving free energy. The aim of this course is to present the basics of the calculus of variations, including 1D theory and its application to various problems arising in natural sciences.

Learning objectives

After taking this unit, students will:

  1. Understand the basics of the calculus of variations
  2. Will be able to analyze and solve various variational problems arising in physics


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. Hamilton-Jacobi theory. Geometric Optics. Eikonal. Hamilton-Jacobi equations (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: MATHM0015
Level of study: M/7
Credit points: 10
Teaching block (weeks): 2 (19-24)
Lecturer: Dr Yves Tourigny


MATH20901 Multivariable Calculus and MATH20101 Ordinary Differential Equations 2



Methods of teaching

Lectures & homeworks.

Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

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

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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