The classical theory

Abstract

The basic problem in the subject that is referred to as the calculus of variations consists in minimizing an integral functional of the type

$$J(x) \: =\: \int_{a}^{\,b} \Lambda \big(t,\, x(t),\, x \,' (t)\big)\,dt $$

over a class of functions x defined on the interval [ a,b ], and which take prescribed values at a and b. The study of this problem (and its numerous variants) is over three centuries old, yet its interest has not waned. Its applications are numerous in geometry and differential equations, in mechanics and physics, and in areas as diverse as engineering, medicine, economics, and renewable resources. It is not surprising, then, that modeling and numerical analysis play a large role in the subject today. In the following chapters, however, we present a course in the calculus of variations which focuses on the core mathematical issues: necessary conditions, sufficient conditions, existence theory, regularity of solutions. This chapter deals with the case in which these variables are one-dimensional and all the data are smooth.

We will now discuss in a little more detail the Struggle for Existence.

Charles Darwin (The Origin of Species)

Life is grey, but the golden tree of theory is always green.

Goethe (Journey by Moonlight)

It’s the question that drives us, Neo.

Morpheus (The Matrix)