
1
The foundations of calculus of variations
The problem of the calculus of variations evolves from the analysis of func-
tions. 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 func-
tion 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 relation-
ship between these sets is now called a functional. The topic of the calculus
of variations is to find extrema of functionals, most commonly ...