We deduce intrinsic equations for a generalized elastic line deformed on the nonnull surface by an external field in the semi-Euclidean space \( \mathbb{E} \) 31 and give some applications.
Similar content being viewed by others
References
B. O’Neill, Semi-Riemannian Geometry, Academic Press, New York (1983).
H. H. Uğurlu and A. Çalışkan, “On the geometry of space-like surfaces,” J. Inst. Sci. Technol., Gazi Univ. (1997).
R. Capovilla, C. Chryssomalakos, and Guven, “Hamiltonians for curves,” J. Phys. A: Math. Gen., 35, 6571–6587 (2002).
J. Langer and D. A. Singer, “The total squared curvature of closed curves,” J. Different. Geom., 20, 1–22 (1984).
J. Arroyo, O. J. Garay, and J. J. Mencia, “Closed generalized elastic curves in S 2(1),” J. Math. Phys., 48, 339–353 (2003).
O. J. Garay, “Extremals of the generalized Euler–Bernoulli energy and applications,” J. Geom. Symmetry Phys., 12, 27–61 (2008).
M. Barros, A. Ferrandez, M. A. Javaloyes, and P. Lucas, “Relativistic particles with rigidity and torsion in D = 3 spacetimes,” Classical Quantum Gravity, 22, 489–513 (2005).
N. Gürbüz, “Intrinsic formulation for elastic line deformed on a surface by external field in the Minkowski 3-space,” J. Math. Anal. Appl., 327, 1086–1094 (2007).
A. Hasimoto, “Soliton on a vortex filament,” J. Fluid Mech., 51 (1972).
J. Lamb, “Solitons on moving space curves,” J. Math. Phys., 18 (1977).
R. E. Goldstein and D. M. Petrich, “Solitons, Euler’s equation and vortex patch dynamic,” Phys. Rev. Lett., 69, 555–558 (1992).
R. Lakshamanan, Phys. Lett. A, 92 (1982).
R. Bryant and P. Griffiths, “Reduction for constrained variational problem,” Amer. J. Math., 108 (1986).
R. E. Goldstein and D. M. Petrich, “The Korteweg–de Vries hierarchy as dynamics of closed curves in the plane,” Phys. Rev. Lett., 67, 3203–3206 (1991).
J. Langer and R. Perline, “The planar filament equation,” J. Math. Phys., 35 (1994).
J. Langer, “Recursion in curve geometry,” J. Math., 5, 25–51 (1999).
R. Huang and D. Shang, “Generalized elastic curves in Lorentz flat space L 4,” Appl. Math. Mech., 30, 1193–1200 (2009).
C. Ekici and A. Görgülü, “Intrinsic equations for a generalized elastic line on an oriented surface in the Minkowski space E 31 ,” Turk. J. Math., 33, 397–407 (2009).
G. Manning, “Elastic line deformed on a surface by an external field: Intrinsic formulation and preliminary application to nucleosome energetics,” Phys. Rev. A, 38, 3073–3081 (1988).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 3, pp. 333–340, March, 2015.
Rights and permissions
About this article
Cite this article
Gürbüz, N., Görgülü, A. Generalized Elastic Line Deformed on a Nonnull Surface by an External Field in the 3-Dimensional Semi-Euclidean Space \( \mathbb{E} \) 31 . Ukr Math J 67, 381–389 (2015). https://doi.org/10.1007/s11253-015-1087-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-015-1087-z