Precise solution of the brachistochrone curve problem taking into account the dry friction force

Background. Currently, there is quite a lot of competition among scientific groups dealing with issues related to brachistochrone, so our study, based on the idea of a mobile basis, is quite relevant to the present time, especially since the purpose of the study is to find an accurate solution to dynamic equations taking into account dry friction on the trough. Materials and methods. The research method is based on the theory of the mobile basis, repeatedly tested on a set of problems relating to curvilinear motion. Results. Using the movable basis method, the equations of motion of the ball along the chute are obtained without taking into account slippage, but taking into account the force of dry friction. The exact solution of the problem in parametric form for the case was found $\mu \neq 0$. Conclusions. The proposed approach made it possible to find an accurate analytical solution to the brachistochrone problem, taking into account the forces of dry friction.