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Conserved Quantities from the Equations of Motion (with applications to natural and gauge natural theories of gravitation)

M. Ferraris·M. Francaviglia·M. Raiteri
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Abstract

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.

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M. Ferraris, M. Francaviglia, M. Raiteri. 2003-05-13. Conserved Quantities from the Equations of Motion (with applications to natural and gauge natural theories of gravitation). https://doi.org/10.1088/0264-9381%2F20%2F18%2F312

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