  In MOND (modified Newtonian dynamics)-based theories the strong equivalence
principle is generically broken in an idiosyncratic manner, manifested in the
action of an "external field effect (EFE)". The internal dynamics in a
self-gravitating system is affected even by a constant external field. In disk
galaxies the EFE can induce warps and modify the rotational speeds. Due to the
non-linearity of MOND, it is difficult to derive analytic expressions of this
important effect in a disk. Here we study numerically the EFE in two
non-relativistic Lagrangian theories of MOND: the `Aquadratic-Lagrangian'
theory (AQUAL) and `Quasilinear MOND' (QUMOND). For AQUAL we consider only the
axisymmetric field configurations with the external field along the disk axis,
or a spherical galaxy with test-particle orbits inclined to the external field.
For the more manageable QUMOND we calculate also the three-dimensional field
configurations, with the external field inclined to the disk axis. We
investigate particularly to what degree an external field modifies the
quasi-flat part of rotation curves. While our QUMOND results agree well with
published numerical results in QUMOND, we find that AQUAL predicts weaker EFE
than published AQUAL results. However, AQUAL still predicts stronger EFE than
QUMOND, which demonstrates current theoretical uncertainties. We also
illustrate how the MOND prediction on the rising part of the rotation curve, in
the inner parts, depends largely on disk thickness but only weakly on a
plausible external field for a fixed galaxy model. Finally, we summarize our
results for the outer parts as an improved, approximate analytic expression.
