A brief note on field equations in generalized theories of gravity

In the work [1](arXiv:1109.3846 [gr-qc]), to obtain a simple and economic formulation of field equations of generalised theories of gravity described by the Lagrangian $\sqrt{-g}L\big(g^{\alpha\beta},R_{\mu\nu\rho\sigma}\big)$, the equality $\big(\partial L/\partial g^{\mu\nu}\big)_{R_{\alpha\beta\kappa\omega}} =2R_{\mu}^{~\lambda\rho\sigma}P_{\nu\lambda\rho\sigma}$ was derived. In this short note, it is demonstrated that such an equality can be directly derived from an off-shell Noether current associated with an arbitrary vector. As a byproduct, a generalized Bianchi identity related to the divergence for the expression of field equations is obtained. The results reveal that using the Noether current to determine field equations even can avoid calculating the derivative of the Lagrangian density with respect to the metric. Besides, the analysis is extended to the Lagrangian including terms of the covariant derivative of the Riemann tensor.