Tri-hamiltonian vector fields, spectral curves and separation coordinates

Details Tri-hamiltonian vector fields, spectral curves and separation coordinates Tri-hamiltonian vector fields, spectral curves and separation coordinates

2001-08-23

arxiv.org/abs/nlin/0108041v2

Rev. Math. Phys. 14 (2002), 1115-1163

Nonlinear Sciences

Exactly Solvable and Integrable Systems

49 pages Section on reduction revisited

Scientific paper

10.1142/S0129055X0200151X

We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P_0, P_1, P_2), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalise those considered by E. Sklyanin in his algebro-geometric approach, are obtained from the knowledge of: (i) a common Casimir function for the two Poisson pencils (P_1 - \lambda P_0) and (P_2 - \mu P_0); (ii) a suitable set of vector fields, preserving P_0 but transversal to its symplectic leaves. The frameworks is applied to Lax equations with spectral parameter, for which not only it unifies the separation techniques of Sklyanin and of Magri, but also provides a more efficient ``inverse'' procedure not involving the extraction of roots.

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