Stress–energy–momentum tensors for natural constrained variational problems

Abstract Under certain parameterization conditions for the “infinitesimal admissible variations”, we propose a theory for constrained variational problems on arbitrary bundles, which allows us to introduce in a very general way the concept of multi-momentum map associated to the infinitesimal symmetries of the problem. For natural problems with natural parameterization, a stress–energy–momentum tensor is constructed for each “admissible section” from the multi-momentum map associated to the natural lifting of vector fields on the base manifold. This tensor satisfies the typical properties of a stress–energy–momentum tensor (Diff( X )-covariance, Belinfante–Rosenfeld type formulas, etc.), and also satisfies corresponding conservation and Hilbert type formulas for natural problems depending on a metric. The theory is illustrated with several examples of geometrical and physical interest.