On resonances on the Kovalevskaya exponents
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).
scientific article; zbMATH DE number 687528
scientific article