El problema de la Braquistócrona en el cilindro S1 × R con varias vueltas

Abstract

We briefly present the brachistochrone problem in a vertical plane. Next, we present the parametric formulation of the brachistochrone problem on the surface of a vertical cylinder of radius R, and we find the curve that solves this problem. We immediately formulate the problem of the tautochronous in the cylinder, and we demonstrate that the brachistochrone curve found previously has tautochronous behavior, that is, two loose particles from the rest of the points other than the brachistochrone curve, reach the lowest point of the trajectory simultaneously. It is also verified that the brachistochrone curve in a vertical plane (inverted cycloid) is the limit of the brachistochrone curve found on the cylindrical surface when the radius of the cylinder tends to infinity. Later we analyze the brachistochronous problem between two fixed points A and B on the cylindrical surface with the additional condition that the particle before reaching the end point B must give a certain number of turns previously defined. We find the curve that solves this problem and additionally, we find a mathematical relationship that determines how many turns can be maximum if we set the values of the coordinates of the starting point (A), end (B), the radius of the cylinder and g (the acceleration due to gravity)