Conserved quantities from the equations of motion: with applications to natural and gauge natural theories of gravitation

We present an alternative field theoretical approach to the definition of conserved quantities, based directly on the field equation 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-step algorithmical procedure can thence be 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 into the Kosmann lift in gauge natural theories, which arises when trying to restore naturality in a gauge natural variational Lagrangian.