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Motion of test particles along rotating curved trajectories is considered. The problem is studied both in the laboratory and the rotating frames of reference. It is assumed that the system rotates with the constant angular velocity ω = const. The solutions are found and analyzed for the case when the form of the trajectory is given by an Archimedes spiral. It is found that particles can reach infinity while they move along these trajectories and the physical interpretation of their behaviour is given. The analogy of this idealized study with the motion of particles along the curved rotating magnetic field lines in the pulsar magnetosphere is pointed out. We discuss further physical development (the conserved total energy case, when ω ≠ const) and astrophysical applications (the acceleration of particles in active galactic nuclei) of this theory.