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)	-

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 Eudoxus. 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.