  The Eulerian variational formulation of the gyrokinetic system with
electrostatic turbulence is presented in general spatial coordinates by
extending our previous work [H. Sugama, {\it et al}., Phys.\ Plasmas {\bf 25},
102506 (2018)]. The invariance of the Lagrangian of the system under an
arbitrary spatial coordinate transformation is used to derive the local
momentum balance equation satisfied by the gyrocenter distribution functions
and the turbulent potential which are given as solutions of the governing
equations. This derivation is in contrast with the conventional method using
the spatial translation in which the asymmetric canonical pressure tensor
generally enters the momentum balance equation. In the present study, the
variation of the Lagrangian density with respect to the metric tensor is taken
to directly obtain the symmetric pressure tensor which includes the effect of
turbulence on the momentum transport. In addition, it is shown in this work how
the momentum balance is modified when the collision and/or external source
terms are added to the gyrokinetic equation. The results obtained here are
considered useful for global gyrokinetic simulations investigating both
neoclassical and turbulent transport processes even in general non-axisymmetric
toroidal systems.
