Calculus of Variations (MAE849)

Undergraduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE849
Semester	8
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	Greek
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.

Syllabus

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

Delivery

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. Additionally:

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.