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.
Citation:
I. A. Bizyaev, V. V. Kozlov, “Homogeneous systems with quadratic integrals, Lie-Poisson quasibrackets, and Kovalevskaya's method”, Sb. Math., 206:12 (2015), 1682–1706
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\paper Homogeneous systems with quadratic integrals, Lie-Poisson quasibrackets, and Kovalevskaya's method
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\vol 206
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\pages 1682--1706
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This publication is cited in the following 11 articles:
A. V. Tsyganov, “O tenzornykh invariantakh dlya integriruemykh sluchaev dvizheniya tverdogo tela Eilera, Lagranzha i Kovalevskoi”, Izv. RAN. Ser. matem., 89:2 (2025), 161–188
Peter H. van der Kamp, David I. McLaren, G. R. W. Quispel, “On a Quadratic Poisson Algebra and Integrable Lotka – Volterra Systems with Solutions in Terms of Lambert's $W$ Function”, Regul. Chaot. Dyn., 2024
Valery V. Kozlov, “Integrals of Circulatory Systems Which are Quadratic
in Momenta”, Regul. Chaotic Dyn., 26:6 (2021), 647–657
Kozlov V.V., “On the Ergodic Theory of Equations of Mathematical Physics”, Russ. J. Math. Phys., 28:1 (2021), 73–83
V. V. Kozlov, “Tensor invariants and integration of differential equations”, Russian Math. Surveys, 74:1 (2019), 111–140
V. V. Kozlov, “Multi-Hamiltonian property of a linear system with quadratic invariant”, St. Petersburg Mathematical Journal, 30:5 (2019), 877–883
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
Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Hojman Construction and Hamiltonization of Nonholonomic Systems”, SIGMA, 12 (2016), 012, 19 pp.
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
V. V. Kozlov, “On the equations of the hydrodynamic type”, J. Appl. Math. Mech., 80:3 (2016), 209–214
I. A. Bizyaev, A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Topologiya i bifurkatsii v negolonomnoi mekhanike”, Nelineinaya dinam., 11:4 (2015), 735–762
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