Introduction to 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.