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Advances in Mathematical Physics
Volume 2016, Article ID 9679460, 15 pages
Review Article

On the Definition of Energy for a Continuum, Its Conservation Laws, and the Energy-Momentum Tensor

Laboratory “Soils, Solids, Structures, Risks”, 3SR, Grenoble Alpes University and CNRS, Domaine Universitaire, BP 53, 38041 Grenoble Cedex 9, France

Received 11 May 2016; Accepted 15 June 2016

Academic Editor: Manuel De León

Copyright © 2016 Mayeul Arminjon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We review the energy concept in the case of a continuum or a system of fields. First, we analyze the emergence of a true local conservation equation for the energy of a continuous medium, taking the example of an isentropic continuum in Newtonian gravity. Next, we consider a continuum or a system of fields in special relativity: we recall that the conservation of the energy-momentum tensor contains two local conservation equations of the same kind as before. We show that both of these equations depend on the reference frame and that, however, they can be given a rigorous meaning. Then, we review the definitions of the canonical and Hilbert energy-momentum tensors from a Lagrangian through the principle of stationary action in general space-time. Using relatively elementary mathematics, we prove precise results regarding the definition of the Hilbert tensor field, its uniqueness, and its tensoriality. We recall the meaning of its covariant conservation equation. We end with a proof of uniqueness of the energy density and flux, when both depend polynomially on the fields.