Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers

Abstract

This article is motivated by a problem posed by David A. Singer in 1999 and by the classical Euler elastic curves. We study spacelike and timelike curves in the Lorentz-Minkowski plane 𝕃2 whose curvature is expressed in terms of the Lorentzian pseudodistance to fixed geodesics. In this way, we get a complete description of all the elastic curves in 𝕃2 and provide the Lorentzian versions of catenaries and grim-reaper curves. We show several uniqueness results for them in terms of their geometric linear momentum. In addition, we are able to get arc-length parametrizations of all the aforementioned curves and they are depicted graphically.

Curves in the Lorentz-Minkowski plane with curvature depending on their position

“…av b a b , 0, , recovering in this new way the special elastic curves described in Section 3 of [9] with energy = / E σ 4 2 , σ being the tension of the elastica. Moreover, in Section 7 we classify the causal curves in…”

“…In fact, we can only mention the articles [7,8] in this line, both devoted to Sturmian spiral curves. The authors initiated in [9] the study of the spacelike and timelike curves in 2 satisfying = ( ) κ κ y or = ( ) κ κ x . Both conditions were interpreted geometrically as the curvature is expressed in terms of the Lorentzian pseudodistance to timelike or spacelike fixed geodesics.…”

“…Both conditions were interpreted geometrically as the curvature is expressed in terms of the Lorentzian pseudodistance to timelike or spacelike fixed geodesics. In this way, in [9] we get a complete description of almost all the elastic curves in 2 and provide the Lorentzian versions of catenaries and some grim-reaper curves.…”

Non-Null Curves With Constant Weighted Curvature in Lorentz-Minkowski Plane With Density

“…In [10], Lopez has studied the minimal surfaces in Euclidean 3-space with a log-linear density φ(x, y, z) = αx + βy + γz, where α, β and γ are real numbers not all-zero. Also, Belarbi et al have studied the surfaces in R 3 with density and they have given some results in a Riemannian manifold M with density in [1] and [2], respectively. Furthermore, ruled and translation minimal surfaces in R 3 with density e z ; helicoidal surfaces in R 3 with density e −x 2 −y 2 and weighted minimal affine translation surfaces in Euclidean space with density have been studied in [6,18,19], respectively.…”

“…Also, Belarbi et al have studied the surfaces in R 3 with density and they have given some results in a Riemannian manifold M with density in [1] and [2], respectively. Furthermore, ruled and translation minimal surfaces in R 3 with density e z ; helicoidal surfaces in R 3 with density e −x 2 −y 2 and weighted minimal affine translation surfaces in Euclidean space with density have been studied in [6,18,19], respectively. Also, some types of surfaces have been studied by geometers in other spaces such as Minkowski 3-space and Galilean 3-space with density.…”