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Generalized elastic curves in the Lorentz flat space L 4

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Abstract

In this paper, we study the extremals of the curvature energy actions on non-null curves in the four-dimensional Lorentz-Minkowski space. We derive the motion equations and find three Killing fields along the generalized elastic curves. We also construct a cylindrical coordinate system using these Killing fields and express the generalized elastic curves by means of quadratures.

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References

  1. Goldstine, H. A History of the Calculus of Variations from the 17th Through the 19th Century, Springer-Verlag, New York (1980)

    Google Scholar 

  2. Langer, J. and Singer, D. A. The total squared curvature of closed curves. Journal of Differential Geometry 20(1), 1–22 (1984)

    MATH  MathSciNet  Google Scholar 

  3. Linnér, A. Explicit elastic curves. Annals of Global Analysis and Geometry 16(5), 445–475 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Shi, Y. and Hearst, J. The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling. Journal of Chemical Physics 101(6), 5186–5200 (1994)

    Article  Google Scholar 

  5. Langer, J. and Singer, D. A. Curve-straightening in Riemannian manifolds. Ann. Global Anal. Geom. 5(2), 133–150 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  6. Huang, R. A note on the p-elastica in a constant sectional curvature manifold. Journal of Geometry and Physics 49(3–4), 343–349 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gürbüz, N. p-elastic in the 3-dimensional Lorentzian space forms. Turkish Journal of Mathematics 30(1), 33–41 (2006)

    MATH  MathSciNet  Google Scholar 

  8. Ferrández, A., Guerrero, J., Javaloyes, M., and Lucas, P. Particles with curvature and torsion in three-dimensional pseudo-Riemannian space forms. Journal of Gemoetry and Physics 56(9), 1666–1687 (2006)

    Article  MATH  Google Scholar 

  9. O’Neill, B. Semi-Riemannian Geometry, Academic Press, New York (1983)

    MATH  Google Scholar 

  10. Langer, J. and Singer, D. A. Lagrangian aspects of the Kirchhoff elastic rod. SIAM Review 38(4), 605–618 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, East China Normal University, Shanghai, 200241, P. R. China

    Rong-pei Huang  (黄荣培) & Dong-hu Shang  (商东虎)

Authors
  1. Rong-pei Huang  (黄荣培)
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  2. Dong-hu Shang  (商东虎)
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Corresponding author

Correspondence to Rong-pei Huang  (黄荣培).

Additional information

Communicated by Li-qun CHEN

Project supported by the National Natural Science Foundation of China (No. 10671066) and the Shanghai Leading Academic Discipline Project (No. B407)

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Huang, Rp., Shang, Dh. Generalized elastic curves in the Lorentz flat space L 4 . Appl. Math. Mech.-Engl. Ed. 30, 1193–1200 (2009). https://doi.org/10.1007/s10483-009-0914-x

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  • DOI: https://doi.org/10.1007/s10483-009-0914-x

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