Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers-Reference-Cited by-同舟云学术

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

Published:2018-01-01 Issue:1 Volume:16 Page:747-766
ISSN:2391-5455
Container-title:Open Mathematics
language:en
Short-container-title:

Author:

Castro Ildefonso¹,Castro-Infantes Ildefonso²,Castro-Infantes Jesús³

Affiliation:

1. Departamento de Matemáticas , Universidad de Jaén , 23071 Jaén , Spain and Instituto de Matemáticas (IEMath-GR)

2. Departamento de Geometría y Topología , Universidad de Granada , 18071 Granada , Spain and Instituto de Matemáticas (IEMath-GR)

3. Departamento de Geometría y Topología , Universidad de Granada , 18071 Granada , Spain

Abstract

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.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/math-2018-0069/pdf

Reference25 articles.

1. Singer D., Curves whose curvature depends on distance from the origin, Amer. Math. Monthly, 1999, 106, 835–841.

2. Singer D., Lectures on elastic curves and rods, Curvature and variational modelling in Physics and Biophysics, AIP Conf. Proc., 2008, 1002, 3–32.

3. Djondjorov P.A., Vassilev V-M., Mladenov I.M., Plane curves associated with integrable dynamical systems of the Frenet-Serret type, Proc. of the 9th International Workshop on Complex Structures, Integrability and Vector Fields, World Scientific, Singapore, 2009, pp. 57–63.

4. Vassilev V.M., Djondjorov P.A., Mladenov I.M., Integrable dynamical systems of the Frenet-Serret type, Proc. of the 9th International Workshop on Complex Structures, Integrability and Vector Fields, World Scientific, Singapore, 2009, pp. 234–244.

5. Mladenov I.M., Hadzhilazova M.T., Djondjorov P.A., Vassilev V.M., On the plane curves whose curvature depends on the distance from the origin, XXIX Workshop on Geometric Methods in Physics. AIP Conf. Proc., 2010, 1307, 112–118.

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