Vol. 10, No. 5, 2017

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Weak and strong solutions to the inverse-square brachistochrone problem on circular and annular domains

Christopher Grimm and John A. Gemmer

Vol. 10 (2017), No. 5, 833–856

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.

brachistochrone problem, calculus of variations of one independent variable, eikonal equation, geometric optics
Mathematical Subject Classification 2010
Primary: 49K05, 49K30, 49S05
Received: 5 May 2016
Accepted: 24 July 2016
Published: 14 May 2017

Communicated by John Baxley
Christopher Grimm
Department of Computer Science
Brown University
Providence, RI 02912
United States
John A. Gemmer
Department of Mathematics
Wake Forest University
Winston Salem, NC 27109
United States