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

TL;DR

This paper investigates curves in the Lorentz-Minkowski plane with curvature depending on their position, deriving new families of such curves and providing explicit formulas, uniqueness results, and graphical representations.

Contribution

It extends classical curvature-position problems to Lorentzian geometry, introducing new integrability results and explicit solutions for spacelike and timelike curves.

Findings

Derived new Lorentzian spiral, elastic, and grim-reaper curves.

Provided explicit elementary function equations for these curves.

Established uniqueness and arc-length parametrizations for specific Lorentzian curves.

Abstract

Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in $\mathbb{L}^2$ whose curvature depends on the Lorentzian pseudodistance from the origin, and those ones whose curvature depends on the Lorentzian pseudodistance through the horizontal or vertical geodesic to a fixed lightlike geodesic. Making use of the notions of geometric angular momentum (with respect to the origin) and geometric linear momentum (with respect to the fixed lightlike geodesic) respectively, we get two abstract integrability results to determine such curves through quadratures. In this way, we find out several new families of Lorentzian spiral, special elastic and grim-reaper curves whose intrinsic equations are expressed in terms of elementary functions. In addition, we provide uniqueness results for the generatrix curve of the Enneper's surface of second kind and for Lorentzian versions of some well known curves in the Euclidean setting, like the Bernoulli lemniscate, the cardioid, the sinusoidal spirals and some non-degenerate conics. We are able to get arc-length parametrizations of them and they are depicted graphically.