  Equations of non-geodesic and non-geodesic deviations for different particles
are obtained, using a specific type of classes of the Bazanski Lagrangian. Such
type of paths has been found to describe the problem of variable mass in the
presence of Riemannian geometry. This may give rise to detect the effect of
dark matter which reveals the mystery of motion of celestial objects that are
not responding neither to Newtonian nor Einsteinian gravity. An important link
between non-geodesic equations and the dipolar particle or fluids has been
introduced to apply the concept of geometization of physics. This concept has
been already extended to represent the hydrodynamic equations in a geometric
way. Such an approach, demands to seek for an appropriate theory of gravity
able to describe different regions, eligible for detecting dark matter. Using
different versions of bi-metric theory of gravity, to examine their associate
non-geodesic paths. Due to implementing the geometrization concept, the
stability problem of non-geodesic equations are essential to be studied for
detecting the behavior of those objects in the presence of dark matter.
