Field equations and Noether potentials for higher-order theories of gravity with Lagrangians involving ⁱ R, ⁱ Rμν and ⁱ Rμνρσ

In this paper, we aim to perform a systematical investigation on the field equations and Noether potentials for the higher-order gravity theories endowed with Lagrangians depending on the metric and the Riemann curvature tensor, together with ith (i=1,2,···) powers of the Beltrami-d'Alembertian operator acting on the latter. We start with a detailed derivation of the field equations and the Noether potential corresponding to the Lagrangian √-gLR(R, R,···,ᵐ R) through the direct variation of the Lagrangian and a method based upon the conserved current. Next the parallel analysis is extended to a more generic Lagrangian √-gLRic(gμν, Rμν, Rμν, ···,ᵐ Rμν), as well as to the generalization of the Lagrangian √-gLRic, which depends on the metric gμν, the Riemann tensor Rμνρσ and ⁱ Rμνρσs. Finally, all the results associated to the three types of Lagrangians are extended to the Lagrangian relying on an arbitrary tensor and the variables via ⁱ acting on such a tensor. In particular, we take into consideration of equations of motion and Noether potentials for nonlocal gravity models. For Lagrangians involving the variables ⁱ R, ⁱ Rμν and ⁱ Rμνρσ, our investigation provides their concrete Noether potentials and the field equations without the derivative of the Lagrangian density with respect to the metric. Besides, the Iyer-Wald potentials associated to these Lagrangians are also presented.