In this paper we study the brachistochrone problem in an inverse-square
gravitational field on the unit disk. We show that the time-optimal solutions
consist of either smooth strong solutions to the Euler–Lagrange equation or
weak solutions formed by appropriately patched together strong solutions.
This combination of weak and strong solutions completely foliates the unit
disk. We also consider the problem on annular domains and show that the
time-optimal paths foliate the annulus. These foliations on the annular domains
converge to the foliation on the unit disk in the limit of vanishing inner
radius.
Keywords
brachistochrone problem, calculus of variations of one
independent variable, eikonal equation, geometric optics
