A4 7 Brachistochrone Problem in a Central Force Field

Abstract

The brachistochrone problem, which seeks the curve of fastest descent between two points under gravity, is a foundational problem in the calculus of variations. In this paper, we revisit this problem and extend the analysis to a non-uniform gravitational field governed by an inverse-square central force field. A corresponding time functional is constructed for this system and numerically evaluated for four candidate curves. The result identifies the optimal path among these candidates and provides a quantitative assessment of how the optimal trajectory deviates from the cycloid when the accelerating force varies with distance.