2.6 Variational Problems Depending on Several Functions of One Variable
141
Proof When the curves y(x) and z(x) change into y + δy 、 z + δz, the variation
of the corresponding functional is
δ J =
x 1
x 0
(F y δy + F y δy
+ F z δz + F z δz
)dx
(2.6.4)
Do integration by parts to the second term and the forth term of the right side of
the above expression respectively, we obtain
δ J = (F y δy + F z δz)
x 1
x 0
+
x 1
x 0
F y −
d
dx
F y
δydx +
x 1
x 0
F z −
d
dx
F z
δzdx
According to the necessary conditions of the extremum of a functional δ J = 0,
and δy, δz are equal to zero at x = x 0 and x = x 1 , we obtain
δ J =
x 1
x 0
F y −
d
dx
F y
δydx +
x 1
x 0
F z −
d
dx
F z
δzdx
Moreover, due to the arbitrariness of δy, δz in the interval (x 0 , x 1 ), according to
the fundamental lemma of the calculus of variations Lemma 1.5.2, there must be
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
F y −
d
dx
F y = 0
F z −
d
dx
F z = 0
Quod erat demonstrandum.
Corollary 2.6.1 Let the function with n unknown functions y 1 (x), y 2 (x), . . . , y n (x)
J [y 1 , y 2 , · · · , y n ] =
x 1
x 0
F(x, y 1 , y 2 , · · · , y n , y
1 , y
2 , · · · , y
n )dx
(2.6.5)
obtain extremum and satisfy the boundary conditions
y i (x 0 ) = y i0 , y i (x 1 ) = y i1 (i = 1, 2, · · · , n)
(2.6.6)
then the extremal curves y i = y i (x)(i = 1, 2, · · · , n) must satisfy the Euler
equations
F y i −
d
dx
F y
i
= 0 (i = 1, 2, · · · , n)
(2.6.7)
Generally speaking, the Eq. (2.6.7) can determine a family of the integral curves
with 2n parameters, which can be determined by the boundary conditions. The family
of the integral curves is a family of extremal curves of the variational problem.