Valid from: Autumn 2013
Decided by: FN1/Anders Gustafsson
Date of establishment: 2014-01-27
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course code: FMA200
Teaching language: English
The aim of the course is to present the basic theory for, and applications of, the calculus of variations, i.e., optimization problems for "functions of functions". A classical example is the isoperimetric problem, to find which closed curve of a given length encloses maximal area. Many physical laws can be formulated as variational principles, i.e. the law of refraction. The calculus of variations is also a corner stone in classical mechanics, and has many other technological applications e.g. in systems theory and optimal control.
Knowledge and Understanding
For a passing grade the doctoral student must be able to explain the basic parts of the theory in the context of an oral examination.
Competences and Skills
For a passing grade the doctoral student must
Variational problems without and with constraints. Euler's equations without and with constraints. Legendre's, Jacobi's and Weierstrass' necessary conditions for a local minimum. Hilbert's invariant integral and Weierstrass' sufficient conditions for a strong local minimum. Hamilton's principle and Hamilton's equations. Lagrange's och Mayer's problems.
Mesterton-Gibbons, M.: A Primer on the Calculus of Variations and Optimal Control Theory. American Mathematical Soc., 2009. ISBN 9780821847725.
Type of instruction: Lectures
Examination formats: Oral exam, written assignments
Grading scale: Failed, pass
Assumed prior knowledge: Calculus in one and several variables corresponding to the courses FMAA05 and FMA430, and linear algebra corresponding to the course FMA420.