Calculus of Variations (MAE849)

Από Wiki Τμήματος Μαθηματικών



School of Science

Academic Unit

Department of Mathematics

Level of Studies


Course Code




Course Title

Calculus of Variations

Independent Teaching Activities

Lectures (Weekly Teaching Hours: 3, Credits: 6)

Course Type

Special Background

Prerequisite Courses

Classical Mechanics

Language of Instruction and Examinations


Is the Course Offered to Erasmus Students

Yes (in English)

Course Website (URL) See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Calculus of Variations deals with optimisation problems where the variables, instead of being finite dimensional as in ordinary calculus, are functions. This course treats the foundations of calculus of variations and gives examples on some (classical and modern) physical applications. After successfully completing the course, the students should be able to:

  • give an account of the foundations of calculus of variations and of its applications in mathematics and physics.
  • describe the brachistochrone problem mathematically and solve it.
  • solve isoperimetric problems of standard type.
  • solve simple initial and boundary value problems by using several variable calculus.
  • formulate maximum principles for various equations.
General Competences
  • Search for, analysis and synthesis of data and information, with the use of the necessary technology.
  • Adapting to new situations.
  • Decision-making.


The Euler-Lagrange equation. The brachistochrone problem. Minimal surfaces of revolution. The isoperimetric problem. Fermat's principle (geometric optics). Hamilton's principle. The principle of least action. The Euler-Lagrange equation for several independent variables. Applications: Minimal surfaces, vibrating strings and membranes, eigenfunction expansions, Quantum mechanics: the Schrödinger equation, Noether's theorem, Ritz optimization, the maximum principle.

Teaching and Learning Methods - Evaluation


Face to face

Use of Information and Communications Technology Yes
Teaching Methods
Activity Semester Workload
Lectures 39
Self study 78
Exercises 33
Course total 150
Student Performance Evaluation
  • Weekly homework
  • Final project
  • Final exam

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

  • Calculus of Variations, I. M. Gelfand and S. V. Fomin, Dover Publications, 2000.
  • Εφαρμοσμένα Μαθηματικά, D. J. Logan, Πανεπιστημιακές Εκδόσεις Κρήτης, 2010.
  • Θεωρητική Μηχανική, Π. Ιωάννου και Θ. Αποστολάτος, ΕΚΠΑ, 2007.