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Geometric Poisson brackets on Grassmannians and conformal spheres

Published online by Cambridge University Press:  07 June 2012

M. Eastwood  and
G. Marí Beffa
M. Eastwood
Affiliation:
Centre for Mathematics and Its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia ([email protected])
G. Marí Beffa
Affiliation:
Mathematics Department, University of Wisconsin, Madison, Wisconsin 53706, USA ([email protected])
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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 Möbius 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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 142 , Issue 3 , June 2012 , pp. 525 - 561
Copyright
Copyright © Royal Society of Edinburgh 2012

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