We consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form
$H\left(\int_{a}^{b}f(t,x^{\sigma}(t),x^{\Delta}(t))\Delta t \right)$.
Cases of calculus of variations as these appear in practical applications but cannot be solved using the classical theory. Therefore, an extension of this theory is needed. Euler-Lagrange equations, natural conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail. |