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

Grimm, Christopher; Gemmer, John A.

doi:10.2140/involve.2017.10.833

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-4184 (online)

ISSN 1944-4176 (print)

Author index

To appear

Other MSP journals

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

DOI: 10.2140/involve.2017.10.833

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.

Keywords

brachistochrone problem, calculus of variations of one independent variable, eikonal equation, geometric optics

Mathematical Subject Classification 2010

Primary: 49K05, 49K30, 49S05

Milestones

Received: 5 May 2016

Accepted: 24 July 2016

Published: 14 May 2017

Communicated by John Baxley

Authors

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

Vol. 10, No. 5, 2017