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2 Variational Problems with Fixed Boundaries
Example 2.6.7 Prove that the Euler-Poisson equation of the functional J [y, z] =
x 1
x 0
F(x, y, y
, z, z
)dx has the following form
F x −
d
dx
(F − y
F y − z
F z ) = 0
( 1 )
and if the integrand does not explicitly contain x, then the Euler equation of the
functional has the first integral F − y
F y − z
F z = c.
Proof Find the various derivatives
dF
dx
= F x + F y y
+ F z z
+ F y y
+ F z z
(2)
d
dx
(y
F y ) = y
F y + y
d
dx
F y
(3)
d
dx
(z
F z ) = z
F z + z
d
dx
F z
(4)
Subtracting Eqs. (3) and (4) from Eq. (2), we obtain
d
dx
(F − y
F y − z
F z ) = F x + y
F y −
d
dx
F y
+ z
F z −
d
dx
F z
(5)
From Eq. (2.6.3), the values in the two groups of brackets on the right of Eq. (5)
are equal to zero respectively, thus there is
d
dx
(F − y
F y − z
F z ) = F x
(6)
Equation (6) is the very Eq. (1). When F does not contain x, F x = 0, namely the
right-hand side of Eq. (6). Integrating once, we obtain
F − y
F y − z
F z = c
(7)
Quod erat demonstrandum.
2.7 Variational Problems Depending on Higher Order
Derivatives
This section discusses the variational problems of the functional with higher
derivative. First to discuss the functional with the second derivative, that is