Stress–energy–momentum tensors in higher order variational calculus

Abstract Given a variational problem defined by a natural Lagrangian density L ω on the k -jet extension J k ( Y / X ) of a natural bundle p : Y → X over an n -dimensional manifold X , oriented by a volume element ω , a stress–energy–momentum tensor T ( s ) is constructed for each section s ∈ Γ ( X , Y ) from the multimomentum map μ Θ :Γ(X,Y)→ Hom R ( X (X),Ω n−1 (X)) associated to any Poincare–Cartan form Θ and to the natural lifting of vector fields X (X) to the bundle Y → X . The characterization made for T ( s ) gives an intrinsic expression of this tensor as well as a generalization of the classical Belinfante–Rosenfeld formula. This tensor satisfies the typical properties of a stress–energy–momentum tensor: Diff( X )-covariance, Hilbert formula, conservation law, etc. Furthermore, it plays the expected role in the theory of minimal gravitational interactions.