This free course concerns the calculus of variations. Section 1 introduces some key ingredients by solving a seemingly simple problem – finding the shortest distance between two points in a plane. The section also introduces the notions of a functional and of a stationary path. Section 2 describes basic problems that can be formulated in terms of functionals. Section 3 looks at partial and total derivatives. Section 4 contains a derivation of the Euler-Lagrange equation. In Section 5 the Euler-Lagrange equation is used to solve some of the earlier problems, as well as one arising from a new topic, Fermat’s principle.
Course learning outcomes
After studying this course, you should be able to:
understand what functionals are, and have some appreciation of their applications
apply the formula that determines stationary paths of a functional to deduce the differential equations for stationary paths in simple cases
use the Euler-Lagrange equation or its first integral to find differential equations for stationary paths
solve differential equations for stationary paths, subject to boundary conditions, in straightforward cases.
Very bad no mathematical techniques given, complete dissapointment