Linear-and-fractional integral of Poisson equations in case of three linear invariant relations

G.V.Gorr

Donetsk National University

50, B. Khmelnitsky Prospect, apt. 3, Donetsk, 83050, Ukraine, ph. (062) 3359220

e-mail: math@ iamm.ac.donetsk.ua

E. л.Uzbek

Donetsk State University of Economics and Trade

after M.Tugan-Baranovsky

20, Vatutin Prospect, apt 18, Donetsk, 83050, Ukraine, ph. (062) 3379147

After the works by S.V.Kovalevskaya, J.Liouville, E.Husson, P.Burgatti, A.M.Lyapunov, P.Ya.Kochina appeared there was resolved the issue on conditions of the fourth algebraic integral existence under the problem of high-gravity solid motion having a fixed point; that integral does not depend explicitly on time dimension and contains an arbitrary constant. There is a simple enough demonstration of J.Liouville's theorem. The nonexistence of the first integral for this problem in general form is proved by V.V.Kozlov and S.L.Ziglin. The problem of existence of single-valued and analytical integrals was examined for generalized problems of dynamics as well. The given work calls attention to the approach based on the study of integral diversity of movement equations suggesting the existence of the first integral not in the whole phase expanse but in some diversities.

This paper examines conditions of existence of a linear-and-fractional primary integral of Poisson's equations under the problem of gyrostat movement in the field of potential and gyroscopic forces that is described with differential equations of G.Kirchhoff's type allowing three linear invariant relations. The new solution ofљ G.Kirchhoff - S.Poisson's equations which is expressed by means of elementary time functions is obtained.