Field equations and Noether potentials for higher-order theories of gravity with Lagrangians involving ${\mathrm{\square }}^{i}R$, ${\mathrm{\square }}^i{R}_{\mu \nu }$ and ${\mathrm{\square }}^i{R}_{\mu \nu \rho \sigma }$

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 $i$th ($i=1,2,\cdot \cdot \cdot$) powers of the Beltrami-d’Alembertian operator $\mathrm{\square }$ acting on the latter. We start with a detailed derivation of the field equations and the Noether potential corresponding to the Lagrangian $\sqrt{-g}{L}_R(R,\mathrm{\square }R,\cdot \cdot \cdot ,{\mathrm{\square }}^{m}R)$ 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 $\sqrt{-g}{L}_{\text{Ric}}(g^{\mu \nu },R_{\mu \nu },\mathrm{\square }{R}_{\mu \nu },\cdot \cdot \cdot ,{\mathrm{\square }}^m{R}_{\mu \nu })$, as well as to the generalization of the Lagrangian $\sqrt{-g}{L}_{\text{Ric}}$, which depends on the metric $g^{\mu \nu }$, the Riemann tensor $R_{\mu \nu \rho \sigma }$ and ${\mathrm{\square }}^i{R}_{\mu \nu \rho \sigma }$s. 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 ${\mathrm{\square }}^i$ 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 ${\mathrm{\square }}^{i}R$, ${\mathrm{\square }}^i{R}_{\mu \nu }$ and ${\mathrm{\square }}^i{R}_{\mu \nu \rho \sigma }$, 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.