  The Eulerian variational formulation is presented to obtain governing
equations of the electromagnetic turbulent gyrokinetic system. A local momentum
balance in the system is derived from the invariance of the Lagrangian of the
system under an arbitrary spatial coordinate transformation by extending the
previous work [H. Sugama et al., Phys. Plasmas 28, 022312 (2021)]. Polarization
and magnetization due to finite gyroradii and electromagnetic microturbulence
are correctly described by the gyrokinetic Poisson equation and Amp\`{e}re's
law which are derived from the variational principle. Also shown is how the
momentum balance is influenced by including collisions and external sources.
Momentum transport due to collisions and turbulence is represented by a
symmetric pressure tensor which originates in a variational derivative of the
Lagrangian with respect to the metric tensor. The relations of the axisymmetry
and quasi-axisymmetry of the toroidal background magnetic field to a
conservation form of the local momentum balance equation are clarified. In
addition, an ensemble-averaged total momentum balance equation is shown to take
the conservation form even in the background field with no symmetry when a
constraint condition representing the macroscopic Amp\`{e}re's law is imposed
on the background field. Using the WKB representation, the ensemble-averaged
pressure tensor due to the microturbulence is expressed in detail and it is
verified to reproduce the toroidal momentum transport derived in previous works
for axisymmetric systems. The local momentum balance equation and the pressure
tensor obtained in this work present a useful reference for elaborate
gyrokinetic simulation studies of momentum transport processes.
