74, R.Luxemburg, 83114, Donetsk, Ukraine
Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine
Kovalev A. M.
Invariant and integral manifolds of dynamical systems and the problem of integration of the Euler–Poisson equations
2004, vol. 9, no. 1, pp. 59-72
The problem on inclusion of an invariant manifold of a dynamical system into a set of integral manifolds is considered. It is shown that this inclusion is always possible unless the invariant manifold is a singular one (consisting of singular points of the system) of codimension one. It enables us to study invariant manifolds using the equation for integrals instead of the Levi-Civita equations containing uncertain factors. The defining role of the singular manifolds in shaping a phase portrait of a dynamical system is established and the following from this integrals properties are obtained. The results are applied to the analysis of solutions of the Euler–Poisson equations. A new treatment of the fourth integrals in Euler's, Lagrange's, Kovalevskaya's cases is proposed. It is proved that under the Hess conditions there is a fourth integral of which special cases are Euler's and Lagrange's integrals as well as the Hess and the Dokshevich solutions.