On the shape of the brachistichrone rotating in a vertical plane

The paper aims to study the influence of the brachistochrone rotating in its own plane on the gutter shape along which a body moves. The problem is solved with a moving basis, which allows one to account for all forces exerted on the body. Introduction of the moving basis yields a compact system of dynamical equations, whose validity was proven in previous author's papers. In limiting cases, such an approach is used to solve analytically the obtained equations of motion and to determine the shape of curves depending on the parameters in the equations by tabular integration. The latter is illustrated in the figures presented. According to the energy conservation law, which accounts for the rotation of the entire system as a whole, the resulting equations also include the angular frequency of rotation as an additional parameter. In this paper, the case of steady rotation is studied. Such conditions have a significant impact on the brachistochrone. In the limiting case of low rotational speeds, the curve, as it should be, degenerates smoothly into a classical brachistochrone, which is justified by the numerical methods used.