The Effect of Gravitational Field on Brachistochrone Problem

Abstract

The equation of the minimum time trajectory (the brachistochrone) does not contain the gravity acceleration explicitly. The proof of the gravitational field effect on the trajectory is revealed by solving functional for the uniform gravitational field and nonuniform gravitational field for the central gravity in this article. The Findings revealed that the effect of constant gravity acceleration is inversely proportional on the arc length of the cycloid, except at g = 0 m/s2, which means that the trajectory could not be formed without gravity acceleration at a location where a particle are not affected by the gravitational field, whereas in nonuniform gravitational field, the particle’s trajectory is not a cycloid and lies in two quadrant. The curve in first quadrant is a mirror image of the curve in fourth quadrant and vice versa. The difference trajectory between uniform and nonuniform gravitational cases is the proof of the existence of the gravitational field effect.