Calculus of Variations
Syllabus, Bachelor's level, 1MA099
- Code
- 1MA099
- Education cycle
- First cycle
- Main field(s) of study and in-depth level
- Mathematics G1F
- Grading system
- Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
- Finalised by
- The Faculty Board of Science and Technology, 12 October 2023
- Responsible department
- Department of Mathematics
Entry requirements
10 credits in mathematics. Participation in Several Variable Calculus, Several Variable Calculus M or Geometry and Analysis III. Participation in Linear Algebra II.
Learning outcomes
In order to pass the course the student should be able to
- give an account of the foundations of calculus of variations and of its applications in mathematics and physics;
- formulate important results and theorems covered by the course;
- use the theory, methods and techniques of the course to solve simpler variation- and boundary value problems;
- present mathematical arguments to others.
Content
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) applications within physics and engineering science.
The Euler-Lagrange equation. The brachistochrone problem. The isoperimetric problem. Fermat's principle (geometric optics). Hamilton's principle (particle dynamics), Lagrange's and Hamilton's equations of motion, the Hamilton-Jacobi equation, the principle of least action. The Euler-Lagrange equation for several independent variables. Minimal surfaces. Vibrating strings and membranes, eigenfunction expansions and Sturm-Liouville theory. Quantum mechanics: the Schrödinger equation. Noether's theorem. The min-max principle.
Instruction
Lectures and group discussions.
Assessment
Oral examination at the end of the course.
If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.
Reading list
- Reading list valid from Autumn 2024
- Reading list valid from Autumn 2023
- Reading list valid from Autumn 2022, version 2
- Reading list valid from Autumn 2022, version 1
- Reading list valid from Autumn 2021
- Reading list valid from Spring 2020
- Reading list valid from Autumn 2019
- Reading list valid from Spring 2013
- Reading list valid from Autumn 2012
- Reading list valid from Spring 2007