On the trajectories of bodies in non-inertial reference frames

This paper considers a trajectory of the body moving under the influence of the force $\mathbf{F}$ in a non-inertial reference frame (NRF), which is "tied" to a given curve $y=y(x)$ and is described by a natural movable basis $\tau$-$\mathbf{n}$. For this NRF, a system of linear differential equations is obtained to simulate various types of trajectories resulting from the action of certain forces. The common Cartesian coordinate system is chosen as a fixed basis $\mathbf{i}$-$\mathbf{j}$. Several examples of motion along the given trajectories $y=y(x)$ are considered with gravity as an acting force $\mathbf{F}$. For these specific cases, the analytic expressions for absolute (in the system $\mathbf{i}$-$\mathbf{j}$), relative (in the system $\tau$-$\mathbf{n}$), and translational accelerations are given. The corresponding trajectories of motion under free fall conditions in terms of NRF are constructed. The following trajectories $y=y(x)$ are studied: uneven motion along a straight line, a brachistochrone, and a circle. Using computer modeling tools, the results are presented as plots showing the qualitative difference between the trajectories of the same body in the inertial and non-inertial frames of reference. The considered limiting cases of motion confirm the validity of the obtained general system of equations in the NRF.