Dark Matter: The Problem of Motion.

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.