We consider an extremal problem for the functionalwhere X is a Banach spase, μ is a smooth measure onX, Ais a.map from a functional spaceB1(X) to a functional spaceB2(X), and extend the main results of classical calculus of variations to the case under consideration. The infinite-dimensional analogs of the Euler-Lagrange equation, the Noether theorem, the canonical Hamilton system are obtained. The illustrations of these results for(wherez(x) is a vector field) are given. The example related to the stochastic optimal control theory is considered