Minimal curves of constant torsion

Ivey, Thomas

doi:10.1090/S0002-9939-00-05526-X

Minimal curves of constant torsion
HTML articles powered by AMS MathViewer

by Thomas A. Ivey PDF

Proc. Amer. Math. Soc. 128 (2000), 2095-2103 Request permission

Abstract:

The Griffiths formalism is applied to find constant torsion curves which are extremal for arclength with respect to variations preserving torsion, fixing the endpoints and the binormals at the endpoints. The critical curves are elastic rods of constant torsion, which are shown to not realize certain boundary conditions.

References

Richard L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly 82 (1975), 246–251. MR 370377, DOI 10.2307/2319846
Kenneth A. Brakke, The surface evolver, Experiment. Math. 1 (1992), no. 2, 141–165. MR 1203871
Robert Bryant and Phillip Griffiths, Reduction for constrained variational problems and $\int {1\over 2}k^2\,ds$, Amer. J. Math. 108 (1986), no. 3, 525–570. MR 844630, DOI 10.2307/2374654
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
A. Calini, T. Ivey, Bäcklund transformations and knots of constant torsion, to appear in J. Knot Theory and its Ramifications 7 (1998), 719–746.
C. Carathéodory, Untersuchungen über das Denaunaysche Problem der Variationsrechnung, Abh. Math. Sem. Hamburg 8 (1930), 32-55.
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
G. Darboux, Leçons sur la Théorie Générale des Surfaces, Gauthier-Villars, 1917.
Phillip A. Griffiths, Exterior differential systems and the calculus of variations, Progress in Mathematics, vol. 25, Birkhäuser, Boston, Mass., 1983. MR 684663
Lucas Hsu, Calculus of variations via the Griffiths formalism, J. Differential Geom. 36 (1992), no. 3, 551–589. MR 1189496
Joel Langer and David A. Singer, Lagrangian aspects of the Kirchhoff elastic rod, SIAM Rev. 38 (1996), no. 4, 605–618. MR 1420839, DOI 10.1137/S0036144593253290
L. S. Pontrjagin, V. G. Boltjanskiĭ, R. V. Gamkrelidze, and E. F. Miščenko, Matematicheskaya teoriya optimal′nykh protsessov, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961 (Russian). MR 0166036
Robert S. Strichartz, Sub-Riemannian geometry, J. Differential Geom. 24 (1986), no. 2, 221–263. MR 862049