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 Mat. Sb., 2015, Volume 206, Number 12, Pages 29–54 (Mi msb8564)

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

Abstract: 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.

Keywords: first integrals, conformally Hamiltonian system, Poisson bracket, Kovalevskaya system, dynamical systems with quadratic right-hand sides.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant no. 14-50-00005.

DOI: https://doi.org/10.4213/sm8564

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English version:
Sbornik: Mathematics, 2015, 206:12, 1682–1706

Bibliographic databases:

Document Type: Article
UDC: 517.925
MSC: Primary 37J05; Secondary 37J30, 37J35, 70E45, 70H05, 70H06, 70H07

Citation: 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
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This publication is cited in the following articles:
1. I. A. Bizyaev, A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Topology and bifurcations in nonholonomic mechanics”, International Journal of Bifurcation and Chaos, 25:10 (2015), 15300–21
2. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Hojman Construction and Hamiltonization of Nonholonomic Systems”, SIGMA, 12 (2016), 012, 19 pp.
3. A. V. Borisov, P. E. Ryabov, S. V. Sokolov, “Bifurcation Analysis of the Motion of a Cylinder and a Point Vortex in an Ideal Fluid”, Math. Notes, 99:6 (2016), 834–839
4. V. V. Kozlov, “On the equations of the hydrodynamic type”, J. Appl. Math. Mech., 80:3 (2016), 209–214
5. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics”, Russian Math. Surveys, 72:5 (2017), 783–840
6. V. V. Kozlov, “Multigamiltonovost lineinoi sistemy s kvadratichnym invariantom”, Algebra i analiz, 30:5 (2018), 159–168
7. V. V. Kozlov, “Tensor invariants and integration of differential equations”, Russian Math. Surveys, 74:1 (2019), 111–140
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