Curves in the Lorentz-Minkowski plane with curvature depending on their position
Metadata
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DE GRUYTER POLAND SP Z O O, BOGUMILA ZUGA 32A STR, 01-811 WARSAW, MAZOVIA, POLAND
Materia
Lorentzian curves Curvature Singer’s problem Sinusoidal spirals Elastic curves Grim-reaper curves
Date
2020-07Referencia bibliográfica
Castro, I., Castro-Infantes, I., & Castro-Infantes, J. (2020). Curves in the Lorentz-Minkowski plane with curvature depending on their position. Open Mathematics, 18(1), 749-770. [https://doi.org/10.1515/math-2020-0043]
Sponsorship
European Union (EU) Spanish Government MTM2017-89677-P; MECD FPU16/03096Abstract
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 the Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those
curves in 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 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 nondegenerate conics. We are able to get arc-length parametrizations of them and they are depicted graphically.