  This work contains an exposition of foundations of the variational calculus
in fibered manifolds. The emphasis is laid on the geometric aspects of the
theory. Especially functionals defined by real functions (Lagrange functions)
or differential forms (Lagrangian forms) on the first jet prolongation of a
given fibered manifold are studied. Critical points (critical cross sections)
of the functionals are examined and the Euler equations for them are derived in
a completely invariant manner. The first variation formula is derived by means
of the so-called Lepagian forms. All variations appearing in the theory are
generated by vector fields. Jet prolongations of projectable vector fields are
defined. The Euler form, associated with a given Lagrange function (of
Lagrangian form) is introduced by means of the Euler equations of the calculus
of variations. Necessary and sufficient conditions for the vanishing of the
Euler form are stated in terms of differential forms and their exterior
differential. The corresponding conditions for a Lagrange function leading to
identically vanishing Euler equations are given. Some special Lepagian forms
are studied. Classes of symmetries of a variational problem are defined.
Invariant, generalized invariant, and symmetry transformations are
characterized in terms of the Lie derivatives. The variational problem with
prescribed symmetry transformations is formulated, and necessary and sufficient
conditions for its solutions are studied. The geometrical aspects of the
so-called generally covariant variational theories are studied. Definitions and
theorems are well adapted to the situation in physical field theories.
