Results in Mathematics

, Volume 27, Issue 3–4, pp 328–332 | Cite as

Hamiltonian flows on the space of star-shaped curves

  • Ulrich Pinkall


We provide a geometric interpretation of the KdV equation as an evolution equation on the space of closed curves in the centroaffine plane. There is a natural symplectic structure on this space and the KdV-flow is generated by a Hamiltonian given by the total centroaffine curvature. In this way we obtain another example for a soliton equation coming naturally from a differential geometric problem [1]. Furthermore, we present a group action of the diffeomorphism group of the circle on the space of closed centroaffine curves.

1991 Mathematics Subject Classification

35Q53 53A15 

Key words and phrase

KdV equation affine curves 


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  1. [1]
    Bobenko, A., Surfaces in terms of 2 by 2 matrices. Old and new integrable cases, In: Fordy A., Wood J. (eds) “Harmonic Maps and Integrable Systems”, Vieweg (1994)Google Scholar
  2. [2]
    Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M., Method for solving the Korteweg — de Vries equation. — Phys. Rev. Lett., 1967, 1095-1097.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 1995

Authors and Affiliations

  1. 1.Fachbereich MathematikBerlin

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