On the definition of classical energy and its conservation laws

Mayeul Arminjon (1)

1 3SR - Laboratoire sols, solides, structures - risques [Grenoble]

We analyze the emergence of a true local conservation equation for energy, taking the example of a continuous medium in Newtonian gravity. We show that the same kind of conservation equation holds true in special relativity, and that there is one such equation per reference frame. Then we review the definitions of the canonical and Hilbert tensors from a Lagrangian through the principle of stationary action in a general spacetime. 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 of the fields.

Mots clés: energy conservation; conservation equation; Newtonian gravity; flat spacetime; curved spacetime; Hilbert tensor; variational principle

Domaines: Relativité Générale et Cosmologie Quantique [gr-qc]; Physique mathématique [math-ph]; Physique [physics]

Mayeul Arminjon. On the definition of classical energy and its conservation laws. 2016. ⟨hal-01262577v1⟩