Involve: A Journal of Mathematics

  • Involve
  • Volume 10, Number 5 (2017), 833-856.

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

Christopher Grimm and John A. Gemmer

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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.

Article information

Involve, Volume 10, Number 5 (2017), 833-856.

Received: 5 May 2016
Accepted: 24 July 2016
First available in Project Euclid: 19 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49K05: Free problems in one independent variable 49K30: Optimal solutions belonging to restricted classes 49S05: Variational principles of physics (should also be assigned at least one other classification number in section 49)

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


Grimm, Christopher; Gemmer, John A. Weak and strong solutions to the inverse-square brachistochrone problem on circular and annular domains. Involve 10 (2017), no. 5, 833--856. doi:10.2140/involve.2017.10.833.

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