  We investigate the possibility that the behavior of the rotational velocities
of test particles gravitating around galaxies can be explained in the framework
of modified gravity models with non-minimal matter-geometry coupling.
Generally, the dynamics of test particles around galaxies, as well as the
corresponding mass deficit, is explained by postulating the existence of dark
matter. The extra-terms in the gravitational field equations with
geometry-matter coupling modify the equations of motion of test particles, and
induce a supplementary gravitational interaction. Starting from the variational
principle describing the particle motion in the presence of the non-minimal
coupling, the expression of the tangential velocity of a test particle, moving
in the vacuum on a stable circular orbit in a spherically symmetric geometry,
is derived. The tangential velocity depends on the metric tensor components, as
well as of the coupling function between matter and geometry. The Doppler
velocity shifts are also obtained in terms of the coupling function. If the
tangential velocity profile is known, the coupling term between matter and
geometry can be obtained explicitly in an analytical form. The functional form
of this function is obtained in two cases, for a constant tangential velocity,
and for an empirical velocity profile obtained from astronomical observations,
respectively. Therefore, these results open the possibility of directly testing
the modified gravity models with non-minimal coupling between matter and
geometry by using direct astronomical and astrophysical observations at the
galactic or extra-galactic scale.
