The Stability of Certain Motion of a Charged Gyrostat in Newtonian Force Field

This work aims to study the stability of certain motions of a rigid body rotating about its fixed point and carrying a rotor that rotates with constant angular velocity about an axis parallel to one of the principal axes. This motion is presumed to take place due to the combined influence of the magnetic field and the Newtonian force field. The equations of motion are deduced, and moreover, they are expressed as a Lie–Poisson Hamilton system. The permanent rotations are calculated and interpreted mechanically. The sufficient conditions for instability are presented employing the linear approximation method. The energy-Casimir method is applied to gain sufficient conditions for stability. The regions of linear stability and Lyapunov stability are illustrated graphically for certain values of the parameters.

On motions of a conservative system on invariant manifolds

In [Irtegov and Burlakova, 2017], the algorithms for the qualitative analysis of conservative systems have been presented. These are based on the Routh-Lyapunov method [Lyapunov, 1954] and some its modifications [Irtegov and Titorenko, 2009] as well as computer algebra methods [Cox, Little, and O’Shea, 1997]. In the paper the application of the algorithms is demonstrated by analysing a conservative system, the study of which is also of interest. We conduct qualitative analysis for the differential equations describing the rotational motion of a rigid body with a fixed point in two constant force fields. Similar problems arise, e.g., in space dynamics [Sarychev and Gutnik, 2015], quantum mechanics [Adler, Marikhin, and Shabat, 2012], [Smirnov, 2008]. In the phase space of the problem, we isolate the invariant manifolds of maximal dimension and study the equations of motion on them. For these equations, solutions (and their families) corresponding in the original phase space of the problem to permanent rotations and pendulum-like oscillations of the body as well as the invariant manifolds of 2nd level, which these solutions belong to, have been found and their Lyapunov’s stability has been investigated. The possibility of stabilization for the motions of conservative systems, whose stability conditions have the form of some constraints on the constants of first integrals, is discussed.

On the Stability of Permanent Rotations of a Chain of Two Lagrange Gyrostats in a Central Gravitational Field

In this paper we study the problem of the motion of a two-gyrostat chain about a fixed point in a central gravitational field. We assume that the mass distribution of each gyrostat is analogous to the one of a Lagrange top, the gyrostatic moment of each gyrostat is constant relative to its carrier, and the center of a spherical joint connecting the gyrostats belongs to their dynamic symmetry axes. We establish and analyze sufficient conditions for stability of the chain’s permanent rotations about a vertical axis. Our findings extend corresponding results in the dynamics of a single gyrostat to a case of the two-gyrostat chain as well as generalize some of the known properties of permanent rotations in the many-body dynamics.