Computer studies of polynomial solutions for gyrostat dynamics

We study polynomial solutions of gyrostat motion equations under potential and gyroscopic forces applied and of gyrostat motion equations in magnetic field taking into account Barnett – London effect. Mathematically, either of the above mentioned problems is described by a system of non-linear ordinary differential equationswhose right hand sides contain fifteen constant parameters. These parameters characterize the gyrostat massdistribution, as well as potential and non-potential forces acting on gyrostat. We consider polynomial solutionsof Steklov – Kovalevski – Gorjachev and Doshkevich classes. The structure of invariant relations for polynomialsolutions shows that, as a rule, on top of the fifteen parameters mentioned one should add no less than twentyfive problem parameters. In the process of solving such a multi-parametric problem in this paper we (in additionto analytic approach) apply numeric methods based on CAS. We break our studies of polynomial solutionsexistence into two steps. During the first step, we estimate maximal degrees of polynomials considered andobtain a non-linear algebraic system for parameters of differential equations and polynomial solutions. In thesecond step (using the above CAS software) we study the solvability conditions of the system obtained andinvestigate the conditions of the constructed solutions to be real.
We construct two new polynomial solutions for Kirchhoff – Poisson. The first one is described by the following property: the projection squares of angular velocity on the non-baracentric axes are the fifth degreepolynomials of the angular velocity vector component of the baracentric axis that is represented via hyperelipticfunction of time. The second solution is characterized by the following: the first component of velocity conditionsis a second degree polynomial, the second component is a polynomial of the third degree, and the square of thethird component is the sixth degree polynomial of the auxiliary variable that is an inversion of the ellipticLegendre integral.
The third new partial solution we construct for gyrostat motion equations in the magnetic field with Barnett – London effect. Its structure is the following: the first and the second components of the angular velocityvector are the second degree polynomials, and the square of the third component is a fourth degree polynomialof the auxiliary variable which is found via inversion of the elliptic Legendre integral of the third kind.
All the solutions constructed in this paper are new and do not have analogues in the fixed point dynamics of a rigid body.