Brachistochrone curvs for particles in viscous fluid

The Brachistochrone is a classical problem in which a cycloid is determined to be the fastest path of descent for a body rolling under the influence of gravity. In this study, we examine the rolling motion of a particle under the combined effects of gravity, buoyancy, and the rolling motion of a particle possessing mass and a moment of inertia, investigating how these physical parameters modify the curve. Using the Euler-Lagrange equations, we derive an equation for the curves that minimize time and the curve that minimizes energy loss, which are notably different from the path that minimizes descent time. Our analyses provide insights into the competing effects at play during the gravity-driven rolling of bodies within a fluid.