On the brachistochrone shape under the Magnus effect

This paper studies the effect of the rotational motion of a body on the trajectory of its fastest descent into the gravity field. The body is considered as a ball rotating about its instantaneous axis, which is perpendicular to the pattern, with a variable angular frequency. The rotation of the ball creates a vortex flow that induces the highest pressure at the top of the ball and the least pressure at the bottom. Thus, the Magnus force (down-force), which is opposed to the reaction force of a trough, occurs. It provides an “antilifting” effect resulting in strong changes in the brachistochrone shape. Based on the fundamental principle of dynamics, a general vector equation of motion is obtained in the form of projections on a moving basis represented as unit vectors of the tangent and normal to the trajectory of the motion. A parametric solution to the equations describing the shape of the trough in Cartesian coordinates is obtained in the absence of dissipative forces. It follows from the resulting solution that the Magnus effect is most noticeable only for massive bodies of long radius. Using the numerical integration methods, various shapes of the deformed brachistochrone are presented as a result of the Magnus effect.