Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Relaxed elastic line on a curved surface

Author: Gerald S. Manning
Journal: Quart. Appl. Math. 45 (1987), 515-527
MSC: Primary 53A04; Secondary 53A05, 58E10, 92A09, 92A40
DOI: https://doi.org/10.1090/qam/910458
MathSciNet review: 910458
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Abstract: In an effort to begin to understand the mechanics of various forms of biologically packaged DNA, we develop the Euler--Lagrange equations for the equilibrium path of an elastic line constrained to a surface but otherwise relaxed. We find, in contrast to a statement by Hilbert and Cohn-Vossen [1], that whether or not the solutions are geodesic curves of the surface depends on the boundary conditions and on the surface. Not surprisingly, the relaxed elastic line on a plane or a sphere is always a geodesic (straight line and great circle, respectively). On a cylinder and a ``pseudotorus,'' however, the relaxed line is a geodesic only if both ends are free. For example, a relaxed line on a cylinder, with fixed initial point and oblique tangent, does not wind on the corresponding geodesic (helix).

References [Enhancements On Off] (What's this?)

  • [1] D. Hilbert and S. Cohn-Vossen, Geometry and the imagination, Chelsea Publishing Company, New York, N. Y., 1952. Translated by P. Neményi. MR 0046650
  • [2] L. D. Landau and E. M. Lifshitz, Course of theoretical physics. Vol. 7, 3rd ed., Pergamon Press, Oxford, 1986. Theory of elasticity; Translated from the Russian by J. B. Sykes and W. H. Reid. MR 884707
  • [3] J. D. McGhee and G. Felsenfeld, Nucleosome structure, Ann. Rev. Biochem. 49, 1115-1156 (1980)
  • [4] T. J. Richmond, J. T. Finch, B. Rushton, D. Rhodes, and A. Klug, Structure of the nucleosome core particle at 7 A resolution, Nature 311, 532-537 (1984)
  • [5] J. D. McGhee, D. C. Rau, E. Charney, and G. Felsenfeld, Orientation of the nucleosome within the higher order structure of chromatin, Cell 22, 87-96 (1980)
  • [6] J. Widom and R. L. Baldwin, Monomolecular condensation of DNA induced by cobalt hexamine, Biopolymers 22, 1595-1620 (1983)
  • [7] Chuan Chih Hsiung, A first course in differential geometry, John Wiley & Sons, Inc., New York, 1981. Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 608028
  • [8] Robert Weinstock, Calculus of variations, Dover Publications, Inc., New York, 1974. With applications to physics and engineering; Reprint of the 1952 edition. MR 0443487

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DOI: https://doi.org/10.1090/qam/910458
Article copyright: © Copyright 1987 American Mathematical Society

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