Weak and Strong Solutions to the Inverse-Square Brachistochrone Problem on Circular and Annular Domains

TL;DR

This paper analyzes the inverse-square brachistochrone problem on circular and annular domains, revealing that optimal solutions are either smooth or weakly patched solutions, forming foliations that converge as the domain changes.

Contribution

It characterizes the structure of time optimal solutions as weak and strong solutions on circular and annular domains, extending classical brachistochrone results.

Findings

Optimal solutions form foliations of the domain.

Weak solutions are constructed by patching strong solutions.

Foliations on annuli converge to those on the disk.

Abstract

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.