Calculus of Variations

Credits

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/05 Mathematical Analysis.

Examination method

Oral exam.

Learning
objectives

After presenting the classical methods of the Calculus of Variations, the course introduces the direct methods of the Calculus of Variations and discusses numerous applications.

Syllabus

Classical methods of the calculus of variations: Euler-Lagrange equations and necessary and sufficient conditions for the existence of strong and weak minima for one-dimensional variational problems. Discussion of the Hamiltonian formulation and of the Hamilton-Jacobi equations for one-dimensional problems. After a review of absolutely continuous functions and Sobolev spaces, the direct methods are then discussed. Numerous applications are presented and studied during the course.

Expected learning
outcomes

By the end of the course, students must demonstrate that they

know and understand the issues relating to the search for extremal solutions of integral and/or differential equations;
are able to apply the knowledge acquired to the study and solution of concrete examples, making correct use of proof techniques;
are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
are able to identify the most appropriate methods to analyse and solve a problem relating to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Command of the knowledge acquired, clarity of presentation, rigour in the use of language, confidence in using the notions acquired.

Calculus of Variations