On the Definition of Energy for a Continuum, Its Conservation Laws, and the Energy-Momentum Tensor
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...
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| Format: | Article |
| Language: | English |
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Wiley
2016-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2016/9679460 |
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| author | Mayeul Arminjon |
| author_facet | Mayeul Arminjon |
| author_sort | Mayeul Arminjon |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-5d9c3979cfa24b1c8065aa213f60741d |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-5d9c3979cfa24b1c8065aa213f60741d2025-02-03T01:00:14ZengWileyAdvances in Mathematical Physics1687-91201687-91392016-01-01201610.1155/2016/96794609679460On the Definition of Energy for a Continuum, Its Conservation Laws, and the Energy-Momentum TensorMayeul Arminjon0Laboratory “Soils, Solids, Structures, Risks”, 3SR, Grenoble Alpes University and CNRS, Domaine Universitaire, BP 53, 38041 Grenoble Cedex 9, FranceWe 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.http://dx.doi.org/10.1155/2016/9679460 |
| spellingShingle | Mayeul Arminjon On the Definition of Energy for a Continuum, Its Conservation Laws, and the Energy-Momentum Tensor Advances in Mathematical Physics |
| title | On the Definition of Energy for a Continuum, Its Conservation Laws, and the Energy-Momentum Tensor |
| title_full | On the Definition of Energy for a Continuum, Its Conservation Laws, and the Energy-Momentum Tensor |
| title_fullStr | On the Definition of Energy for a Continuum, Its Conservation Laws, and the Energy-Momentum Tensor |
| title_full_unstemmed | On the Definition of Energy for a Continuum, Its Conservation Laws, and the Energy-Momentum Tensor |
| title_short | On the Definition of Energy for a Continuum, Its Conservation Laws, and the Energy-Momentum Tensor |
| title_sort | on the definition of energy for a continuum its conservation laws and the energy momentum tensor |
| url | http://dx.doi.org/10.1155/2016/9679460 |
| work_keys_str_mv | AT mayeularminjon onthedefinitionofenergyforacontinuumitsconservationlawsandtheenergymomentumtensor |