Applied Calculus of Variations for Engineers, Third edition, 3rd Edition

3

 

Multivariate functionals

The subject of this chapter is further expansion of the classes of functions used in variational problems. Specifically, we discuss functionals with multiple functions, functions with multiple variables will be discussed in a variational context. Parametric functions and minimal surface applications will also be presented.

 

3.1Functionals with several functions

The variational problem of multiple dependent variables is posed as

I(y1,y2,…,yn)=∫x0x1f(x,y1,y2,…,yn,y1′,y2′,…,yn′)dx

with a pair of boundary conditions given for all functions:

yi(x0)=yi,0

and

yi(x1)=yi,1

for each i = 1, 2, …, n. The alternative solutions are:

Yi(x)=yi(x)+ϵiηi(x);i=1,2,…,n

with all the arbitrary auxiliary functions obeying the conditions: ...