In differential geometry, the theory of curves has an important place. The concept of moving frames defined on curves is an important part of this theory. Recently, Ozen and Tosun have introduced a new moving frame for the trajectories with non-vanishing angular momentum in 3-dimensional Euclidean space (J. Math. Sci. Model. 4(1), 2021). This frame is called positional adapted frame. In the present study, we investigate the special trajectories generated by Smarandache curves according to positional adapted frame in E3 and we calculate the Serret-Frenet apparatus of these trajectories. Later, we consider a specific curve and obtain the parametric equations of the aforesaid special trajectories for this curve. Finally, we give the graphics of these obtained special trajectories which were drawn with the mathematica program. The results obtained here are new contributions to the field. We expect that these results will be useful in some specific applications of differential geometry and particle kinematics in the future.