Brachistochrone of off-centered cylinders

TL;DR

This paper investigates the shortest transit time paths for off-centered cylinders rolling under gravity, revealing new trajectory types and their dependence on initial conditions using numerical methods.

Contribution

It introduces a numerical approach to solve the brachistochrone problem for off-centered cylinders, uncovering novel trajectories and their critical dependence on initial parameters.

Findings

Discovery of distinct brachistochrone trajectories for off-centered cylinders

Critical dependence of paths on initial position and orientation

Enhanced understanding of brachistochrone solutions for rigid bodies

Abstract

We consider the problem of finding paths of shortest transit time between two points (popularly known as Brachistochrone) for cylinders with off-centered center of mass, rolling down without slip, subject solely to the force of gravity. This problem is set up using principles of classical rigid body dynamics and the desired path function is solved for numerically using the method of discrete calculus of variations. We discover a distinct array of brachistochrone trajectories for off-centered cylinders, demonstrate a critical dependence of such paths on the initial location and orientation of cylinders' centers of mass and bring new insights into the family of brachistochrone problems and solutions.