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 formulated in
the form of an integral.
1.1 The fundamental problem and lemma of calculus of
The fundamental problem of the calculus of variations is to ﬁnd the extremum
(maximum or minimum) of the functional
f(x, y, y
where the solution satisﬁes the boundary conditions
This problem may be generalized to the cases when higher derivatives or multi-
ple functions are given and will be discussed in Chapters 3 and 4, respectively.
These problems may also be extended with constraints, the topic of Chapter 2.
A solution process may be arrived at with the following logic. Let us as-
sume that there exists such a solution y(x) for the above problem that satisﬁes