On rational integrability of Euler equations on Lie algebra \(\mathrm{so}(4, \mathbb C)\)
type: Item
instance of: scholarly article
title: On rational integrability of Euler equations on Lie algebra \(\mathrm{so}(4, \mathbb C)\)
zbMATH Open document ID: 1143.37047
author: Sasho Ivanov Popov
author: Jean-Marie Strelcyn
published in: Israel Journal of Mathematics
publication date: 2008-05-27T00:00:00Z
review text: The authors of this very interesting paper study a class of Euler equations on the complex Lie algebra \(L\), that is \(\text{so}(4,\mathbb C)\) with a diagonal quadratic Hamiltonian \(H\). The system of these equations has the form \[ {dx_i\over dt}=\{x_i,H\}, \quad 1\leq i\leq n, \] where the element \(x=\sum_{i=1}^{n}x_ie_i^{\ast }\in L^{\ast }\) (\(L^{\ast }\) is dual to \(L\)), \(x_i\in C^{\infty }(L^{\ast })\) (\(1\leq i\leq n\)), \(\{e_1^{\ast },e_2^{\ast },\dots,e_n^{\ast }\}\) is the basis dual to a fixed basis \(\{e_1,e_2,\dots,e_n\}\) of \(L\), \(\{x_i,H\}\) is the Lie-Poisson bracket, and \(H\in C^{\infty }(L^{\ast })\) is a given function, that is a diagonal quadratic Hamiltonian. It is well known that the above stated system always admits three functionally independent first integrals in polynomial form. The authors prove that if the system has a rational first integral functionally independent of the known three ones so called fourth integral, then it has a polynomial first integral that is also functionally independent of them. The fourth integral is obtained in explicit form.
reviewed by: Dimitar A. Kolev
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 17B80
Mathematics Subject Classification ID: 37K30
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 5279161
zbMATH Keywords: Euler equation
zbMATH Keywords: integrability
zbMATH Keywords: Lie algebra
zbMATH Keywords: \(\text{so}(4,\mathbb C)\)
zbMATH Keywords: Lie-Poisson bracket
zbMATH Keywords: Casimir function
zbMATH Keywords: Hamiltonian
MaRDI profile type: Publication
full work available at URL: s11856-008-0012-7
OpenAlex ID: W1968323996
Wikidata QID: Q115377714
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DOI: 10.1007/S11856-008-0012-7
description: scientific article; zbMATH DE number 5279161