Computer studies of polynomial solutions for gyrostat dynamics

Zyza A.V.
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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 equations whose right hand sides contain fifteen constant parameters. These parameters characterize the gyrostat mass distribution, as well as potential and non-potential forces acting on gyrostat. We consider polynomial solutions of Steklov–Kovalevski–Gorjachev and Doshkevich classes. The structure of invariant relations for polynomial solutions shows that, as a rule, on top of the fifteen parameters mentioned one should add no less than twenty five problem parameters. In the process of solving such a multi-parametric problem in this paper we (in addition to analytic approach) apply numeric methods based on CAS. We break our studies of polynomial solutions existence into two steps. During the first step, we estimate maximal degrees of polynomials considered and obtain a non-linear algebraic system for parameters of differential equations and polynomial solutions. In the second step (using the above CAS software) we study the solvability conditions of the system obtained and investigate 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 degree polynomials of the angular velocity vector component of the baracentric axis that is represented via hypereliptic function of time. The second solution is characterized by the following: the first component of velocity conditions is a second degree polynomial, the second component is a polynomial of the third degree, and the square of the third component is the sixth degree polynomial of the auxiliary variable that is an inversion of the elliptic Legendre 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 velocity vector are the second degree polynomials, and the square of the third component is a fourth degree polynomial of 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.

Keywords: polinomial solutions, the Kirchhoff–Poisson equation, gyrostat, potential and gyroscopics forces, Barnett–London effect, elliptical Legendre integrals
Citation in English: Zyza A.V. Computer studies of polynomial solutions for gyrostat dynamics // Computer Research and Modeling, 2018, vol. 10, no. 1, pp. 7-25
Citation in English: Zyza A.V. Computer studies of polynomial solutions for gyrostat dynamics // Computer Research and Modeling, 2018, vol. 10, no. 1, pp. 7-25
DOI: 10.20537/2076-7633-2018-10-1-7-25
URL: http://crm-en.ics.org.ru/journal/article/2653/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.
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