Abstract
In accordance with the Kirchhoff analogy, the equilibrium equations of an elastic thread on a plane are equivalent to the equations of motion of a simple pendulum. This analogy is generalized to the case when the thread is situated on a smooth curved surface. The equilibrium equations for the threads in the general case and in the particular cases of flat, cylindrical, and spherical surfaces are derived. For these surfaces the Kirchhoff analogy is generalized to the case of a simple pendulum in an additional force field. There are also considered the electromagnetic and nonholonomic analogies for the equilibrium equations of an elastic thread.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Cambridge Univ. Press, Cambridge, 1927).
V. A. Svetlitskii, Mechanics of Flexible Rods and Threads (Mashinostroenie, Moscow, 1978) [in Russian].
D. R. Merkin, Introduction to the Mechanics of Flexible Threads (Nauka, Moscow, 1980) [in Russian].
J. Langer and D. Singer, “The Total Squared Curvature of Closed Curves,” J. Diff. Geometry 20, 1–22 (1984).
V. A. Svetlitskii, “Steady Motion of an Elastic Thread on a Surface,” Mashinostroenie. Priborostroenie 2, 104–109 (1959).
J. Langer and D. Singer, “Curve-Straightening in Riemannian Manifolds,” Ann. Global Anal. Geom. 5, 133–150 (1987).
Author information
Authors and Affiliations
Additional information
Original Russian Text © I.E. Glagolev. 2013. published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2013. Vol. 68, No. 6, pp. 31–36.
About this article
Cite this article
Glagolev, I.E. Various analogies for the equilibrium shapes of an elastic thread on two-dimensional surfaces. Moscow Univ. Mech. Bull. 68, 133–138 (2013). https://doi.org/10.3103/S0027133013060010
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0027133013060010