motion of particles under pseudo-deformation

in this short article, we observe that the path of particle of mass $m$ moving along $mathbf{r}= mathbf{r}(t)$ under pseudo-force $mathbf{a}(t)$, $t$ denotes the time, is given by $mathbf{r}_d= int(frac{dmathbf{r}}{dt} mathbf{a}(t)) dt +mathbf{c}$. we also observe that the effective force $mathbf{f}_e$ on that particle due to pseudo-force $mathbf{a}(t)$, is given by $ mathbf{f}_e= mathbf{f} mathbf{a}(t)+ mathbf{l} dmathbf{a}(t)/dt$, where $mathbf{f}= m d^2mathbf{r}/dt^2 $ and $mathbf{l}= m dmathbf{r}/dt$. we have discussed stream lines under pseudo-force.