Institute of Systems Dynamics and Control Theory,

Siberian Branch of RAS,

134 Lermontov Str.,

664033 Irkutsk,

RUSSIA

E-mail: irteg@icc.ru

**Application of Computer Algebra Methods in Investigation
of Quite Integrable Systems**

**Abstract:**

Some results of qualitative analysis of the equations of rigid body's
motion in the ideal fluid, which has been conducted with the use of computer
algebra methods, are discussed. A case of Kirchhoff equations, which assume
four algebraic first integrals (Sokolov's case) - three quadratic ones
and one 4th degree integral - are considered. The Kirchhoff equations herein
are quite integrable. The phase space structure for such systems allows
one to apply algebra of first integrals and obtain rather detailed information
on qualitative properties of some classes of their solutions. In the present
work, the set of solutions, which attribute steady-state solutions to the
elements of first-integral algebra, has been investigated by Routh-Lyapunov's
method for the above type of differential equations. In particular, conditions
of stability and instability for the steady-state solutions have been obtained,
and parametric analysis of some of these conditions has been performed.
Questions of branching the families of steady-state solutions and the related
problems of transition from stability to instability are considered. Some
ways of application of computer algebra methods in solving above problems,
while including those of obtaining steady-state solutions themselves, are
discussed. Such computer algebra systems as "Maple" and "Mathematica" have
been used in the computations.