Elastic curves with constant curvature at rest in the hyperbolic plane

JOURNAL OF GEOMETRY AND PHYSICS

卷:61

Article

Arroyo, J.¹  Garay, O. J.¹  Mencia, J. J.¹ 

[1] Univ Basque Country, Dept Matemat, Fac Ciencia & Tecnol, E-48080 Bilbao, Spain

关键词: Euler-Lagrange equations;    Generalized elastic curves;    Energy functionals;   

DOI  :  10.1016/j.geomphys.2011.04.006

来源: Elsevier

【 摘 要 】

We extend the classical variational model for elastic curves that are circular at rest to the hyperbolic plane H-2(-1). For simplicity, we call them lambda-elastica. We show that there are three types of critical curves according to their symmetries: rotational, translational and horocyclical type curves. By explicitly solving the Euler-Lagrange equation and giving a closedness criterion in each case, we can show that there exists a 2-parameter family of closed rotational lambda-elastica and that there exists an eight-shaped example of closed translational lambda-elastica in H-2 (-1). However, we prove that there are no examples of closed lambda-elastica of horocyclical type. The second variation formula is applied to study the stability of the constant curvature solution multiple covers. As an application, we combine these results with a Lorentzian version of the Hopf map to construct examples of closed elastic membranes in the anti de Sitter 3-space. A numerical approach is used to gain insight into the space of closed lambda-elastica. One plausible consequence of this numerics is that the eight-shaped critical curve mentioned before appears to be the only closed translational lambda-elastica and that it is also a candidate for a local minimum of elastic energy. (C) 2011 Elsevier B.V. All rights reserved.

【 授权许可】

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