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TRI-Hamiltonian vector fields, spectral curves and separation coordinates. (English) Zbl 1026.37051

Summary: We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets \((\text{P}_0, \text{P}_1, \text{P}_2)\), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalize those considered by E. K. Sklyanin [Prog. Theor. Phys., Suppl. 118, 35–80 (1995; Zbl 0868.35002)] in his algebro-geometric approach, are obtained from the knowledge of: (i) a common Casimir function for the two Poisson pencils \((\text{P}_1 - \lambda\text{P}_0)\) and \((\text{P}_2 - \mu\text{P}_0)\); (ii) a suitable set of vector fields, preserving \(\text{P}_0\) but transversal to its symplectic leaves. The framework is applied to Lax equations with spectral parameter, for which not only it establishes a theoretical link between the separation techniques of Sklyanin [loc. cit.] and of F. Magri [J. Math. Phys. 19, 1156–1162 (1978; Zbl 0383.35065)], but also provides a more efficient “inverse” procedure to obtain separation variables, not involving the extraction of roots.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
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