  General properties of the three-body problem in a model of modified dynamics
are investigated. It is shown that the three-body problem in this model shares
some characters with the similar problem in Newtonian dynamics. Moreover, the
planar restricted three-body problem is solved analytically for this type of
extended gravity and it is proved that under certain conditions, which
generally happen at galactic and extragalactic scales, the orbits around
$L_{4}$ and $L_{5}$ Lagrange points are stable. Furthermore, a code is provided
to compare the behavior of orbits in the restricted three-body problem under
Newtonian and modified dynamics. Orbit integrations based on this code show
contrasting orbital behavior under the two dynamics and specially exhibit in a
qualitative way that the rate of ejections is smaller in the modified dynamics
compared to Newtonian gravity. These results could help us to search for
observational signatures of extended gravities.
