  In the work (arXiv:1109.3846 [gr-qc]), to obtain a simple and economic
formulation of field equations for generalised theories of gravity described by
the Lagrangian $\sqrt{-g}L\big(g^{\alpha\beta},R_{\mu\nu\rho\sigma}\big)$, the
key equality $\big(\partial L/\partial
g^{\mu\nu}\big)_{R_{\alpha\beta\kappa\omega}}
=2P_{\mu}^{~\lambda\rho\sigma}R_{\nu\lambda\rho\sigma}$ was derived. In this
note, it is demonstrated that such an equality can be directly derived from an
off-shell Noether current associated with an arbitrary vector field. As
byproducts, a generalized Bianchi identity related to the divergence for the
expression of field equations, together with the Noether potential, is
obtained. On the basis of the above, we further propose a systematic procedure
to derive the equations of motion from the Noether current, and then this
procedure is extended to more general higher-order gravities endowed with the
Lagrangian encompassing additional terms of the covariant derivatives of the
Riemann tensor. To our knowledge, both the detailed expressions for field
equations and the Noether potential associated with such theories are first
given at a general level. All the results reveal that using the Noether current
to determine field equations establishes a straightforward connection between
the symmetry of the Lagrangian and the equations of motion and such a remedy
even can avoid calculating the derivative of the Lagrangian density with
respect to the metric.
