A new approach to solve the Brachistochrone problem by constructing a lattice unit cell

A simple and general new approach to solve the Brachistochrone problem is presented in this paper. The Brachistochrone problem is concerned with finding the shortest time trajectory of a particlesliding on a frictionless path under gravity. The problem is solved in this project using a solid-state physics mechanism of building a lattice by a unit cell of a suitable lattice parameter and a transformation operator. This problem was solved analytically centuries ago by many scientists. To the author's knowledge, the approach considered here was not used before. The method clearly shows that the Brachistchrone is just a two-dimensional lattice with a parameter and a transformation angle that depend on the size of the trajectory. It has been found that the shortest time track is a cycloid, which is a curve that lies between a straight line and a circle. Thefindings of this work were compared to the exact results found previously and were found to be within an infinitesimally negligible margin of error.