A New Family of Buckled Rings on the Unit Sphere

David A. Singer ()
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David A. Singer: Department of Mathematics, Applied Math & Statistics, Case Western Reserve University, Cleveland, OH 44106, USA

Mathematics, 2025, vol. 13, issue 8, 1-7

Abstract: Buckled rings, also known as pressurized elastic circles, can be described as critical points for a variational problem, namely the integral of a quadratic polynomial in the geodesic curvature of a curve. Thus, they are a generalization of elastic curves, and they are solitary wave solutions to a flow in a (three-dimensional) filament hierarchy. An example of such a curve is the Kiepert Trefoil, which has three leaves meeting at a central singular point. Such a variational problem can be considered for curves in other surfaces. In particular, researchers have found many examples of such curves in a unit sphere. In this article, we consider a new family of such curves, having a discrete dihedral symmetry about a central singular point. That is, these are spherical analogues of the Kiepert curve. We determine such curves explicitly using the notion of a Killing field, which is a vector field along a curve that is the restriction of an isometry of the sphere. The curvature k of each such curve is given explicitly by an elliptic function. If the curve is centered at the south pole of the sphere and has minimum value ρ , then k − ρ is linear in the height above the pole.

Keywords: buckled ring; filament equation; soliton (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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