  Models of "modified-inertia" formulation of MOND are described and applied to
nonrelativistic many-body systems. They involve time-nonlocal equations of
motion. Momentum, angular momentum, and energy are (nonlocally) defined, whose
total values are conserved for isolated systems. The models make all the
salient MOND predictions. Yet, they differ from existing "modified-gravity"
formulations in some second-tier predictions. The models describe correctly the
motion of a composite body in a low-acceleration field even when the internal
accelerations of its constituents are high. They exhibit a MOND external field
effect (EFE) that shows some important differences from what we have come to
expect from modified-gravity versions: In one, simple example of the models,
what determines the EFE, in the case of a dominant external field, is
$\mu(\theta\langle a_{ex}\rangle/a_0)$, where $\mu(x)$ is the MOND
`interpolating function' that describes rotation curves, compared with
$\mu(a_{ex}/a_0)$ for presently-known modified-gravity formulations. The two
main differences are that while $a_{ex}$ is the momentary value of the external
acceleration, $\langle a_{ex}\rangle$ is a certain time average of it, and that
$\theta>1$ is an extra factor that depends on the frequency ratio of the
external- and internal-field variations. Only ratios of frequencies enter, and
$a_0$ remains the only new dimensioned constant. For a system on a circular
orbit in a galaxy (such as the vertical dynamics in a disc galaxy), the first
difference disappears, since $\langle a_{ex}\rangle=a_{ex}$. But the $\theta$
factor can appreciably enhance the EFE in quenching MOND effects, over what is
deduced in modified gravity. Some exact solutions are also described, such as
for rotation curves, for an harmonic force, and the general, two-body problem,
which in the deep-MOND regime reduces to a single-body problem.
