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.