Gaudin Models and Bending Flows: a Geometrical Point of View

Details Gaudin Models and Bending Flows: a Geometrical Point of View Gaudin Models and Bending Flows: a Geometrical Point of View

2003-06-04

arxiv.org/abs/nlin/0306005v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

27 pages, LaTeX with amsmath and amssymb

Scientific paper

10.1088/0305-4470/36/46/009

In this paper we discuss the bihamiltonian formulation of the (rational XXX) Gaudin models of spin-spin interaction, generalized to the case of sl(r)-valued spins. In particular, we focus on the homogeneous models. We find a pencil of Poisson brackets that recursively define a complete set of integrals of the motion, alternative to the set of integrals associated with the 'standard' Lax representation of the Gaudin model. These integrals, in the case of su(2), coincide wih the Hamiltonians of the 'bending flows' in the moduli space of polygons in Euclidean space introduced by Kapovich and Millson. We finally address the problem of separability of these flows and explicitly find separation coordinates and separation relations for the r=2 case.

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