
3
Multivariate functionals
3.1 Functionals with several functions
The variational problem of multiple dependent variables is posed as
I(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
with a pair of boundary conditions given for all functions:
y
i
(x
0
)=y
i,0
and
y
i
(x
1
)=y
i,1
for each i =1, 2,...,n. The alternative solutions are:
Y
i
(x)=y
i
(x)+
i
η
i
(x); i =1,...,n
with all the arbitrary auxiliary functions obeying the conditions:
η
i
(x
0
)=η
i
(x
1
)=0.
The variational problem becomes
I(
1
,...,
n
)=
x
1
x
0
f(x,...,y
i
+
i
η
i
,...,y
i
+
i
η
i
,...)dx,
whose derivative with respect to the auxiliary variables is
∂I
∂
i
=
x
1
x
0
∂f
∂
i
dx =0.
Applying the chain rule we get
∂f
∂
i
=
∂f
∂Y