  We establish a version of the first Noether Theorem, according to which the
(equivalence classes of) conserved quantities of given Euler-Lagrange equations
in several independent variables are in one-to-one correspondence with the
(equivalence classes of) vector fields satisfying an appropriate pair of
geometric conditions, namely: (a) they preserve the class of vector fields
tangent to holonomic submanifolds of a jet space; (b) they leave invariant the
action, from which the Euler-Lagrange equations are derived, modulo terms
identically vanishing along holonomic submanifolds. Such correspondence between
symmetries and conservation laws is built on an explicit linear map f_A from
the vector fields satisfying (a) and (b) into the conserved differential
operators, and not into their divergences as it occurs in other proofs of
Noether Theorem. This map f_A is not new: it is the map determined by
contracting symmetries with a form of Poincare'-Cartan type A and it is
essentially the same considered for instance in a paper by Kupershmidt. There
it was shown that f_A determines a bijection between symmetries and
conservation laws in a special form. Here we show that, if appropriate
regularity assumptions are satisfied, any conservation law is equivalent to one
that belongs to the image of f_A, proving that the corresponding induced map
F_A between equivalence classes of symmetries and equivalence classes of
conservation laws is actually a bijection. All results are given
coordinate-free formulations and rely just on basic differential geometric
properties of finite-dimensional manifolds.
