Calculus of Variations
Variationskalkyl

FMA201F, 7.5 högskolepoäng

Gäller från och med: Autumn 2013
Beslutad av: FN1/Anders Gustafsson
Datum för fastställande: 2014-01-27

Allmänna uppgifter

Avdelning: Mathematics
Kurstyp: Gemensam kurs, avancerad nivå och forskarnivå
Kursen ges även på avancerad nivå med kurskod: FMA200
Undervisningsspråk: English

Syfte

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.

Mål

Kunskap och förståelse

För godkänd kurs skall doktoranden be able to explain the basic parts of the theory in the context of an oral examination.

Färdighet och förmåga

För godkänd kurs skall doktoranden

• be able to demonstrate an ability to identify problems which can be modelled with the concepts introduced.
• be able to integrate methods and views from the different parts of the course in order to solve problems and answer questions within the framework of the course.
• in writing and orally, with clear logic and proper terminology be able to explain the solution to a mathematical problem within the course.

Kursinnehåll

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.

Kurslitteratur

Mesterton-Gibbons, M.: A Primer on the Calculus of Variations and Optimal Control Theory. American Mathematical Soc., 2009. ISBN 9780821847725.

Kursens undervisningsformer

Undervisningsform: Föreläsningar

Kursens examination

Examinationsformer: Muntlig tentamen, inlämningsuppgifter
Betygsskala: Underkänd, godkänd
Examinator:

Antagningsuppgifter

Förutsatta förkunskaper: Calculus in one and several variables corresponding to the courses FMAA05 and FMA430, and linear algebra corresponding to the course FMA420.

Kurstillfällesinformation

Kontaktinformation och övrigt

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