This article is cited in 7 scientific papers (total in 7 papers)
Homogeneous systems with quadratic integrals, Lie-Poisson quasibrackets, and Kovalevskaya's method
I. A. Bizyaev*, V. V. Kozlov
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
We consider differential equations with quadratic right-hand sides that admit two quadratic first integrals, one of which is a positive-definite quadratic form. We indicate conditions of general nature under which a linear change of variables reduces this system to a certain ‘canonical’ form. Under these conditions, the system turns out to be divergenceless and can be reduced to a Hamiltonian form, but the corresponding linear Lie-Poisson bracket does not always satisfy the Jacobi identity. In the three-dimensional case, the equations can be reduced to the classical equations of the Euler top, and in four-dimensional space, the system turns out to be superintegrable and coincides with the Euler-Poincaré equations on some Lie algebra. In the five-dimensional case we find a reducing multiplier after multiplying by which the Poisson bracket satisfies the Jacobi identity. In the general case for $n>5$ we prove the absence of a reducing multiplier. As an example we consider a system of Lotka-Volterra type with quadratic right-hand sides that was studied by Kovalevskaya from the viewpoint of conditions of uniqueness of its solutions as functions of complex time.
Bibliography: 38 titles.
first integrals, conformally Hamiltonian system, Poisson bracket, Kovalevskaya system, dynamical systems with quadratic right-hand sides.
|Russian Science Foundation
|This work is supported by the Russian Science Foundation under grant no. 14-50-00005.
* Author to whom correspondence should be addressed
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Sbornik: Mathematics, 2015, 206:12, 1682–1706
MSC: Primary 37J05; Secondary 37J30, 37J35, 70E45, 70H05, 70H06, 70H07
I. A. Bizyaev, V. V. Kozlov, “Homogeneous systems with quadratic integrals, Lie-Poisson quasibrackets, and Kovalevskaya's method”, Mat. Sb., 206:12 (2015), 29–54; Sb. Math., 206:12 (2015), 1682–1706
Citation in format AMSBIB
\by I.~A.~Bizyaev, V.~V.~Kozlov
\paper Homogeneous systems with quadratic integrals, Lie-Poisson quasibrackets, and Kovalevskaya's method
\jour Mat. Sb.
\jour Sb. Math.
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