Separation of variables, Lax-integrable systems and \(gl(2) \otimes gl(2)\)-valued classical \(r\)-matrices
type: Item
instance of: scholarly article
title: Separation of variables, Lax-integrable systems and \(gl(2) \otimes gl(2)\)-valued classical \(r\)-matrices
zbMATH Open document ID: 1450.37054
published in: Journal of Geometry and Physics
publication date: 2020-07-21T00:00:00Z
review text: This paper discusses a separation of variables problem for a certain class of Lax-integrable Hamiltonian systems. It addresses specific questions about the construction of separated variables in the context of integrable Hamiltonian systems. The separated variables \(x_i\) and \(p_j\) (\(i, j =1, \dots, d\)) are a set of canonical coordinates that satisfy a general system of equations for separation. That is, \(\Phi_i (x_i, p_i, I_1, ..., I_d, C_1, ..., C_r) = 0\) for \(i = 1, \dots, d\). The \(I_k\)'s are Poisson-commuting integrals of motion, the \(C_k\)'s are Casimir functions, and \(d\) is one-half the dimension of the symplectic leaves in the phase space. Separated variables are typically used to allow explicit integration of the equations of motion. Separation of variables is therefore of interest for integrable systems in both classical and quantum physics. The author is specifically interested in integrable Hamiltonian systems for which a Lax representation with spectral parameter is known. He calls Lax-integrable systems those systems for which the spectral invariants of the Lax matrix comprise a complete set of Poisson-commuting integrals of motion. A conjecture called ``magic recipe'', that goes back to \textit{E. K. Sklyanin} [J. Sov. Math. 47, No. 2, 2473--2488 (1989; Zbl 0692.35107); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 164, 151--169 (1987)], is recalled here. An open question about this ``magic recipe'' is when the variables constructed in this way are actually canonical. This was answered by \textit{B. Dubrovin} and \textit{T. Skrypnyk} [J. Math. Phys. 59, No. 9, 091405, 39 p. (2018; Zbl 1403.37067)], who described general separability criteria. In the current paper the author considers what is necessary to produce a set of canonical variables that is complete in the simplest non-trivial case. It would be advisable for a reader interested in this paper to review first at least the two references cited above. The curr…
reviewed by: William J. Satzer Jr.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37J37
Mathematics Subject Classification ID: 17B80
Mathematics Subject Classification ID: 82B23
zbMATH DE Number: 7223539
zbMATH Keywords: separation of variables
zbMATH Keywords: classical \(r\)-matrices
zbMATH Keywords: Gaudin-type models
author: T. Skrypnyk
MaRDI profile type: Publication
full work available at URL: j.geomphys.2020.103733
OpenAlex ID: W3024060134
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DOI: 10.1016/J.GEOMPHYS.2020.103733
description: scientific article; zbMATH DE number 7223539