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Type: Journal article
Title: Geometric Poisson brackets on Grassmannians and conformal spheres
Author: Eastwood, M.
Beffa, G.
Citation: Proceedings of the Royal Society of Edinburgh Section A Mathematics, 2012; 142(3):525-561
Publisher: Cambridge University Press on behalf of The Royal Society of Edinburgh
Issue Date: 2012
ISSN: 0308-2105
Statement of
M. Eastwood, G. Marí Beffa
Abstract: We relate the geometric Poisson brackets on the 2-Grassmannian in R⁴ and on the (2, 2) Möbius sphere. We show that, when written in terms of local moving frames, the geometric Poisson bracket on the M¨obius sphere does not restrict to the space of differential invariants of Schwarzian type. But when the concept of conformal natural frame is transported from the conformal sphere into the Grassmannian, and the Poisson bracket is written in terms of the Grassmannian natural frame, it restricts and results in either a decoupled system or a complexly coupled system of Korteweg–de Vries (KdV) equations, depending on the character of the invariants. We also show that the bi-Hamiltonian Grassmannian geometric brackets are equivalent to the non-commutative KdV bi-Hamiltonian structure. Both integrable systems and Hamiltonian structure can be brought back to the conformal sphere.
Rights: © 2012 The Royal Society of Edinburgh
DOI: 10.1017/S0308210510001071
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Mathematical Sciences publications

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