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Minimal curves of constant torsion
Author(s):
Thomas
A.
Ivey
Journal:
Proc. Amer. Math. Soc.
128
(2000),
2095-2103.
MSC (2000):
Primary 49K15, 53A04;
Secondary 58A17, 58A30, 73C02
Posted:
March 2, 2000
MathSciNet review:
1694865
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Additional information
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:
- 1.
- R. Bishop, There is more than one way to frame a curve, Am. Math. Monthly 82 (1975), 246-251. MR 51:6604
- 2.
- K. Brakke, The Surface Evolver, Experiment. Math. 1 (1992), #2, 141-165. MR 93k:53006
- 3.
- R. Bryant, P. Griffiths, Reduction for constrained variational problems and
 , Amer. J. Math. 108 (1986) 525-570. MR 88a:58044 - 4.
- P. Byrd, M. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer, 1953. MR 15:702a
- 5.
- A. Calini, T. Ivey, Bäcklund transformations and knots of constant torsion, to appear in J. Knot Theory and its Ramifications 7 (1998), 719-746.
- 6.
- C. Carathéodory, Untersuchungen über das Denaunaysche Problem der Variationsrechnung, Abh. Math. Sem. Hamburg 8 (1930), 32-55.
- 7.
- W.L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Annalen 117 (1939), 98-105. MR 1:313d
- 8.
- G. Darboux, Leçons sur la Théorie Générale des Surfaces, Gauthier-Villars, 1917.
- 9.
- P.A. Griffiths, Exterior Differential Systems and the Calculus of Variations, Birkhäuser, 1983. MR 84h:58007
- 10.
- L. Hsu, Calculus of Variations via the Griffiths formalism, J. Diff. Geom. 36 (1992), 551-589. MR 94a:58003
- 11.
- J. Langer, D. Singer, Lagrangian Aspects of the Kirchhoff Elastic Rod, SIAM Reviews 38 (1996), 605-618. MR 97h:73050
- 12.
- L.S. Pontrjagin, V.G. Boltyanskii, R.V.Gamkrelidze, E.F. Mischenko, The mathematical theory of optimal processes, Wiley-Interscience, 1962. MR 29:3316b
- 13.
- R. Strichartz, Sub-Riemannian Geometry, J. Diff. Geom. 24 (1983), 221-263. MR 88b:53055
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MSC (2000):
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MSC (2000):
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58A17, 58A30, 73C02
Additional Information:
Thomas
A.
Ivey
Affiliation:
Department of Mathematical Sciences, Ball State University, Muncie, Indiana 47306
Email:
tivey@math.bsu.edu
DOI:
10.1090/S0002-9939-00-05526-X
PII:
S 0002-9939(00)05526-X
Keywords:
Curves of constant torsion,
Griffiths formalism,
elastic rods
Received by editor(s):
September 2, 1998
Posted:
March 2, 2000
Communicated by:
Christopher Croke
Copyright of article:
Copyright
2000,
American Mathematical Society
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