  We present an alternative field theoretical approach to the definition of
conserved quantities, based directly on the field equations content of a
Lagrangian theory (in the standard framework of the Calculus of Variations in
jet bundles). The contraction of the Euler-Lagrange equations with Lie
derivatives of the dynamical fields allows one to derive a variational
Lagrangian for any given set of Lagrangian equations. A two steps algorithmical
procedure can be thence applied to the variational Lagrangian in order to
produce a general expression for the variation of all quantities which are
(covariantly) conserved along the given dynamics. As a concrete example we test
this new formalism on Einstein's equations: well known and widely accepted
formulae for the variation of the Hamiltonian and the variation of Energy for
General Relativity are recovered. We also consider the Einstein-Cartan
(Sciama-Kibble) theory in tetrad formalism and as a by-product we gain some new
insight on the Kosmann lift in gauge natural theories, which arises when trying
to restore naturality in a gauge natural variational Lagrangian.
