A new flow on starlike curves in ℝ³

Huang, Rongpei; Singer, David

doi:10.1090/S0002-9939-02-06631-5

A new flow on starlike curves in $\mathbb {R}^3$
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by Rongpei Huang and David A. Singer PDF

Proc. Amer. Math. Soc. 130 (2002), 2725-2735 Request permission

Abstract:

In this note we find a new evolution equation for starlike curves in $\mathbb {R}^3$. We study the evolution of the subaffine curvature and subaffine torsion under the flow and show that it is completely integrable. The solutions to the evolution which move without changing affine shape are subaffine elastic curves. We integrate the subaffine elastica by quadratures.

References

G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR 1744638
George Boole, A treatise on differential equations, Chelsea Publishing Co., New York, 1959. 5th ed. MR 0107033
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
Eugenio Calabi, Peter J. Olver, and Allen Tannenbaum, Affine geometry, curve flows, and invariant numerical approximations, Adv. Math. 124 (1996), no. 1, 154–196. MR 1423202, DOI 10.1006/aima.1996.0081
Peter J. Giblin and Guillermo Sapiro, Affine-invariant distances, envelopes and symmetry sets, Geom. Dedicata 71 (1998), no. 3, 237–261. MR 1631679, DOI 10.1023/A:1005099011913
Herman H. Goldstine, A history of the calculus of variations from the 17th through the 19th century, Studies in the History of Mathematics and Physical Sciences, vol. 5, Springer-Verlag, New York-Berlin, 1980. MR 601774
H.Hasimoto, A soliton on a vortex filament, J. Fluid Mech. 51 (1972), 477-486.
R. Huang, Affine and Subaffine Elastic Curves in $\mathbb {R}^2$ and $\mathbb {R}^3$, thesis, Case Western Reserve University, 1999.
George L. Lamb Jr., Elements of soliton theory, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1980. MR 591458
Joel Langer and Ron Perline, Poisson geometry of the filament equation, J. Nonlinear Sci. 1 (1991), no. 1, 71–93. MR 1102831, DOI 10.1007/BF01209148
Joel Langer and Ron Perline, Local geometric invariants of integrable evolution equations, J. Math. Phys. 35 (1994), no. 4, 1732–1737. MR 1267918, DOI 10.1063/1.530567
Joel Langer and David A. Singer, The total squared curvature of closed curves, J. Differential Geom. 20 (1984), no. 1, 1–22. MR 772124
Joel Langer and David A. Singer, Knotted elastic curves in $\textbf {R}^3$, J. London Math. Soc. (2) 30 (1984), no. 3, 512–520. MR 810960, DOI 10.1112/jlms/s2-30.3.512
J. Langer and D. Singer, Liouville integrability of geometric variational problems, Comment. Math. Helv. 69 (1994), no. 2, 272–280. MR 1282371, DOI 10.1007/BF02564486
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
Jerrold Marsden and Alan Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D 7 (1983), no. 1-3, 305–323. Order in chaos (Los Alamos, N.M., 1982). MR 719058, DOI 10.1016/0167-2789(83)90134-3
Katsumi Nomizu and Takeshi Sasaki, Affine differential geometry, Cambridge Tracts in Mathematics, vol. 111, Cambridge University Press, Cambridge, 1994. Geometry of affine immersions. MR 1311248
S. P. Novikov, Solitons and geometry, Lezioni Fermiane. [Fermi Lectures], Published for the Scuola Normale Superiore, Pisa; by Cambridge University Press, Cambridge, 1994. MR 1272686
Ulrich Pinkall, Hamiltonian flows on the space of star-shaped curves, Results Math. 27 (1995), no. 3-4, 328–332. MR 1331105, DOI 10.1007/BF03322836
Bu Chin Su, Affine differential geometry, Science Press Beijing, Beijing; Gordon & Breach Science Publishers, New York, 1983. MR 724783