Linear orbital stability analysis of the pendulum-type motions of a Kovalevskaya top with a suspension point vibrating horizontally

The motion of a heavy rigid body is considered, with a point (suspension point) performing high-frequency horizontal vibration. The center of mass of the body is on the principal axis of inertia for this point. Within the framework of an approximate autonomous system, pendulum-type motions of the body are studied, they occur in a vertical plane containing the axis of vibration or in a plane perpendicular to it. For a body with a mass geometry corresponding to S. V. Kovalevskaya case, a linear orbital stability analysis of these motions is performed.


Introduction
The presence of high-frequency vibrations of the rigid body suspension has a significant influence on its motion. Vibrations can lead to the emergence of new types of body motions impossible for a body with a fixed point, or to a change in the stability of known particular cases of motion.
At the beginning of the 20th century, the phenomenon of dynamic stabilization of the inverted position of a mathematical pendulum due to vertical vibrations of the suspension point was first described [1]. Later, the mathematical pendulum and the pendulum systems were studied.
Recently, the influence of high-frequency vibrations on the dynamics of a heavy rigid body with a more complicated mass geometry, as well as more complex vibrations of its suspension point, have been investigated. In [2], an approximate autonomous system of Euler-Poisson type equations is obtained, which describes the motion of a heavy rigid body with an arbitrary mass geometry, with suspension point performing arbitrary periodic or conditionally periodic fast vibrations of small amplitude in three-dimensional space. Within the framework of this approximate system, a number of problems on dynamics of a body with a vibrating suspension has been solved. In particular, the orbital stability of the pendulum-type motions of the Lagrange top with a vibrating suspension has been studied [3].
Pendulum-type motions of a rigid body with a fixed point were first described in [4]. The orbital stability of the pendulum-type motions of a Kovalevskaya top with a fixed point was studied in [5] and [6].
This paper studies the influence of high-frequency horizontal periodic vibrations on the existence and stability of the pendulum-type motion of a rigid body with the center of mass on the principal axes of inertia for the suspension point. The problem of linear orbital stability of the pendulum-type motions for the Kovalevskaya top in the presence of such vibrations is considered in detail.
cos sin sin cos cos cos cos sin sin cos 2 Further research is carried out in the framework of the approximate system with Hamiltonian (1), (2). The errors of the solutions of this system with respect to ones of the complete system as well as the time interval, on which the approximate solutions are studied, are given in [2].
The system with the Hamiltonian (1), (2) admits particular solutions that correspond to pendulumlike body motions, where one of its principal axes of inertia not containing the center of mass G , takes a fixed horizontal position, and the radius vector OG moves in the vertical plane containing the axis of vibration (plane OXY ) or perpendicular to it (plane OYZ ).
In the first case 2 3 / 2, 0, 0, 0 P P θ π ϕ = = = = , and the change in the quantities ψ and 1 P is described by canonical equations with the Hamiltonian function To describe motions in the plane perpendicular to the axis of vibrations, we redirect the axes of the coordinate system OXYZ so that the vibrations of the suspension point occur along OZ . Part of the Hamiltonian (1) without the last term remains unchanged, and the vibration potential takes the form ( ) Note that in the case of the absence of vibrations of the body suspension point, the pendulum-like motions of the radius vector center OG can occur in any fixed vertical plane. However, there is a cyclic coordinate in the system if we choice the other coordinates axes and these motions are unstable with respect to spatial perturbations.
Further, we will assume that the body mass geometry corresponds to S. V. Kovalevskaya case, which is realized for 2, 1 µ η = = . The aim of the work is to describe and study the linear orbital stability of the pendulum-like motions of the Kovalevskaya top with a horizontally vibrating suspension point that correspond to the model systems with Hamiltonians (3) and (5).

Pendulum-type motions
The integral on the right side of (6) is expressed in terms of elliptic functions. Further, we use the notation sn for elliptic sine, ( ) and using some formulas from [7], we obtain The resulting solution is periodic, with frequency The motions in the region 2 Γ are oscillations that cover the lower and inclined equilibrium positions (if they exist), and In this case, the solution with the initial condition and its frequency is ( )   2  2  3  3  2  2  1  3  1  2  3  2  3  3  2  1  2   2 sn  ,  1  2  1  ,  1  1  ,  2sn , and its frequency is At  The study of orbital stability To study the orbital stability of the motions considered, we introduce action-angle variables I,w in the regions ΓΓ, Γ,,Γ ,Γ of oscillations and rotations of the model systems. The corresponding particular solutions of the complete three-degree-of-freedom systems, in which the "pendulum part" is written in variables , I w, will be taken as unperturbed motions. In the system with Hamiltonian (1), we introduce perturbations by the formulas Here 0 ω is a frequency of unperturbed motion. If pendulum-like motions of the system with Hamiltonian ( In expressions (8) and (9), the functions ( ) w ψ and ( ) 1 P w are calculated for the unperturbed motion. Further, we will consider the linearized equations of perturbed motion described by the Hamiltonian 2 H from (7) with regard to relations (8) and (9). At the energy level 2 0 H = that corresponds to the unperturbed motion we realize the isoenergetic reduction and, taking the quantity w as a new independent variable, we consider a non-autonomous reduced two-degree-of-freedom system with the Hamiltonian The stability criteria for the trivial equilibrium of a system with the Hamiltonian 2 H from (7) and the reduced system coincide [8]. The characteristic equation of the latter is α monotonically tend to 1 α = as the value N increases. Further, we will limit ourselves to the following points:     Note that in the limiting case where the suspension point of the body is fixed the results on the orbital stability of the pendulum-type motions of the Kovalevskaya top obtained in the paper are consistent with the results obtained in [5,6].