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

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.

Latex file, 31 pages

References in corpus (4)

• Relativistic conservation laws and integral constraints for large cosmological perturbations
• Thermodynamics of $(d+1)$-dimensional NUT-charged AdS Spacetimes
• A covariant formalism for Chern-Simons gravity
• Covariant Charges in Chern-Simons AdS_3 Gravity

Cited by in corpus (15)

• Invariant conserved currents in gravity theories with local Lorentz and diffeomorphism symmetry
• Augmented Variational Principles and Relative Conservation Laws in Classical Field Theory
• The Nontriviality of Trivial General Covariance: How Electrons Restrict 'Time' Coordinates, Spinors (Almost) Fit into Tensor Calculus, and 7/16 of a Tetrad Is Surplus Structure
• Charges and Energy in Chern-Simons Theories and Lovelock Gravity
• Invariant conserved currents in gravity theories: diffeomorphisms and local gauge symmetries
• Global Generalized Bianchi Identities for Invariant Variational Problems on Gauge-natural Bundles
• Invariant conserved currents in generalized gravity
• Superpotentials from variational derivatives rather than Lagrangians in relativistic theories of gravity
• Variationally equivalent problems and variations of Noether currents
• Lagrangian reductive structures on gauge-natural bundles
• Generalized Bianchi identities in gauge-natural field theories and the curvature of variational principles
• Symmetries of Helmholtz forms and globally variational dynamical forms
• The relative energy of homogeneous and isotropic universes from variational principles
• Boundary Conditions, Energies and Gravitational Heat in General Relativity (a Classical Analysis)
• The Jacobi morphism and the Hessian in higher order field theory; with applications to a Yang-Mills theory on a Minkowskian background