
4
Higher order derivatives
The fundamental problem of the calculus of variations involved the first deriva-
tive of the unknown function. In this chapter we will allow the presence of
higher order derivatives.
4.1 The Euler-Poisson equation
First let us consider the variational problem of a functional with a single func-
tion, but containing its higher derivatives:
I(y)=
x
1
x
0
f(x, y, y
,...,y
(m)
)dx.
Accordingly, boundary conditions for all derivatives will also be given as
y(x
0
)=y
0
,y(x
1
)=y
1
,
y
(x
0
)=y
0
,y
(x
1
)=y
1
,
y
(x
0
)=y
0
,y
(x
1
)=y
1
,
and so on until
y
(m−1)
(x
0
)=y
(m−1)
0
,y
(m−1)
(x
1
)=y
(m−1)
1
.
As in the past chapters, we introduce an alternative solution