On resonances on the Kovalevskaya exponents
type: Item
instance of: scholarly article
title: On resonances on the Kovalevskaya exponents
zbMATH Open document ID: 0820.58048
DOI: 10.1007/BF01210428
published in: Mathematical Notes
publication date: 1994-11-16T00:00:00Z
review text: There are obtained some results about the resonances on the Kovalevskaya exponents. The first concerns systems (1) \(\dot{x} = v(x)\), invariant with respect to the substitution \[ t \to \lambda^{-1}t,\;x_ 1 \to \lambda^{g_ 1} x_ 1,\dots, x_ n \to \lambda^{g_ n} x_ n,\;g_ i \in\mathbb{Q}. \] Kovalevskaya exponents (KE) are defined as the eigenvalues of the matrix \[ \left| {\partial v \over \partial x}(c) + \text{diag}(g_ 1,\dots, d_ n)\right| \] where \(c\) is from the Kovalevskaya solution \(x_ i = c_ i t^{-g_ i}\). Theorem. An additional rational integral of the system (1) leads to an additional integer-valued resonance on the KE. The second result is the following. Lemma. The KE of the Euler equations on a Lie algebra are paired: \(\rho_ i + \rho_{d+i} = g,\;1 \leq i \leq d\), in the restriction to the orbit \(O^* \ni c\) (\(2d = \dim O^*\), \(g + 1\) is the degree of the Hamiltonian of the Euler equations).
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 58E99
zbMATH DE Number: 687528
zbMATH Keywords: resonances
zbMATH Keywords: Kovalevskaya exponents
zbMATH Keywords: Euler equations
author: S. T. Sadètov
MaRDI profile type: Publication
cites work: Tensor invariants of quasihomogeneous systems of differential equations, and the Kovalevskaya-Lyapunov asymptotic method
cites work: Necessary condition for the existence of algebraic first integrals
cites work: A criterion for the nonexistence of an additional analytic integral in Hamiltonian systems with \(n\) degrees of freedom
cites work: Q4204849
cites work: Q4711785
full work available at URL: https://doi.org/10.1007/bf01210428
OpenAlex ID: W1567922417
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description: scientific article; zbMATH DE number 687528