Calculus of Variations

Calculus of Variations

Kursstart: Fredag 19/1, kl 10

Course content

Variational techniques is one of the most powerful way to solve complicated differential equations. The area goes back to the Bernoullis, Newton, Leibniz and the history of variational techniques runns through Euler, Lagrange, Jacobi, Hilbert and many other of the most reknown mathematicians of forlorn days.

Calculus of variations is concerned with finding the minimal value of some function, in general a function from some infinite dimensional space to the real numbers. This appears to be a very narrow problem – but nothing could be further from the truth. Almost all of classical physics, and even much of modern physics, can be formulated in terms of variational problems. There are also many applications of variational techniques in pure mathematics.

In this course, we will study Calculus of Variations and its connection to differential equations. We will treat several classical problems that can be formulated in terms of calculus of variations and also discuss how to use variational methods to understand partial differential equations.

Content: (A more detailed but still preliminary plan can be found here)

Convex functions
The Dirichlet principle
Elliptic operators
The Brachistocrone problem
The Harmonic oscillator
Euler-Lagrange and Hamilton-Jacobi equations of classical mechanics
The Isoperimetric Inequality
Sobolev Spaces
The Direct Method in the Calculus of Variations
Regularity theory

Lecturer

Erik Lindgren, eriklin@kth.se

Course literature

Other relevant literature:

Examination