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Field equations and Noether potentials for higher-order theories of gravity with Lagrangians involving , and
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 th () 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 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 , as well as to the generalization of the
Lagrangian , which depends on the metric ,
the Riemann tensor and 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
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 , and , 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.Comment: 73 pages, No figure
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