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?)

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

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