Applied Calculus of Variations for Engineers, 2nd Edition

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