ABSTRACT

The problem of the calculus of variations evolves from the analysis of functions. In the analysis of functions the focus is on the relation between two sets of numbers, the independent (x) and the dependent (y) set. The function f creates a one-to-one correspondence between these two sets, denoted as y=f(x). The generalization of this concept is based on allowing the two sets not to be restricted to being real numbers and to be functions themselves. The relationship between these sets is now called a functional. The topic of the calculus of variations is to find extrema of functionals, most commonly formulated in the form of an integral. The Euler-Lagrange differential equation just introduced represents a necessary, but not sufficient, condition for the solution of the fundamental variational problem.