Olga Kholostova

This paper examines the motion of a time-periodic Hamiltonian system with two degrees of freedom in a neighborhood of trivial equilibrium. It is assumed that the system depends on three parameters, one of which is small; when it has zero value, the system is autonomous. Consideration is given to a set of values of the other two parameters for which, in the autonomous case, two frequencies of small oscillations of the linearized equations of perturbed motion are identical and are integer or half-integer numbers (the case of multiple parametric resonance). It is assumed that the normal form of the quadratic part of the Hamiltonian does not reduce to the sum of squares, i.e., the trivial equilibrium of the system is linearly unstable. Using a number of canonical transformations, the perturbed Hamiltonian of the system is reduced to normal form in terms through degree four in perturbations and up to various degrees in a small parameter (systems of first, second and third approximations). The structure of the regions of stability and instability of trivial equilibrium is investigated, and solutions are obtained to the problems of the existence, number, as well as (linear and nonlinear) stability of the system’s periodic motions analytic in fractional or integer powers of the small parameter. For some cases, conditionally periodic motions of the system are described. As an application, resonant periodic motions of a dynamically symmetric satellite modeled by a rigid body are constructed in a neighborhood of its steady rotation (cylindrical precession) on a weakly elliptic orbit and the problem of their stability is solved.

The motion of a heavy rigid body with a mass geometry corresponding to the Hess case is considered. The suspension point of the body is assumed to perform high-frequency periodic vibrations of small amplitude in the three-dimensional space. It is proved that for any law of vibrations of this type, the approximate autonomous equations of the body motion admit an invariant relation (the first integral at the zero level), which coincides with a similar relation that exists in the Hess case of the motion of a body with a fixed point. In the approximate equations of motion written in Hamiltonian form, the cyclic coordinate is introduced and the corresponding reduction is performed. For the laws of vibration of the suspension point corresponding to the integrable cases (when there is another cyclic coordinate in the system), a detailed study of the model one-degree-of-freedom system is given. For the nonintegrable cases, an analogy with the approximate problem of the motion of a Lagrange top with a vibrating suspension point is drawn, and the results obtained earlier for the top are used. Some properties of the body motion at the nonzero level of the above invariant relation are also discussed.

The motion of a solid (satellite) carrying a moving point mass in the central Newtonian gravitational field in an elliptical orbit of arbitrary eccentricity is considered. The law of motion of a point mass is assumed to allow for the existence of relative equilibria of the “body-point” system in the orbital coordinate system. A nonlinear stability analysis of these equilibria is carried out, based on the construction and normalization of the area-preserving mapping generated by the motions of the system.

The motion of a nonautonomous time-periodic two-degree-of-freedom Hamiltonian system in a neighborhood of an equilibrium point is considered. The Hamiltonian function of the system is supposed to depend on two parameters $\varepsilon$ and $\alpha$, with $\varepsilon$ being small and the system being autonomous at $\varepsilon=0$. It is also supposed that for $\varepsilon=0$ and some values of $\alpha$ one of the frequencies of small linear oscillations of the system in the neighborhood of the equilibrium point is an integer or half-integer and the other is equal to zero, that is, the system exhibits a multiple parametric resonance. The case is considered where the rank of the matrix of equations of perturbed motion that are linearized at $\varepsilon=0$ in the neighborhood of the equilibrium point is equal to three. For sufficiently small but nonzero values of $\varepsilon$ and for values of $\alpha$ close to the resonant ones, the question of existence, bifurcations, and stability (in the linear approximation) of the periodic motions of the system is solved. As an application, periodic motions of a symmetrical satellite in the neighborhood of its cylindrical precession in an orbit with small eccentricity are constructed for cases of the multiple resonances considered.

We consider the motion of Lagrange’s top with a suspension point performing the specified highfrequency periodic motion with small amplitude in three-dimensional space. The approximate autonomous system of equations of motion written in the form of canonical Hamiltonian equations is investigated. The problem of the existence and number of stationary rotations of the top about its dynamical symmetry axis is solved. The study of stability of the corresponding equilibrium positions of the reduced two-degree-of-freedom system for fixed values of the cyclic integral constant depending on the angular velocity of rotation is carried out. For suspension points’ motions allowing for stationary rotations about the vertical, a detailed linear and nonlinear stability analysis of these rotations and rotations about inclined axes is carried out. For a number of other cases of the suspension point motions a linear stability analysis is carried out.

The motion of a time-periodic two-degree-of-freedom Hamiltonian system in the neighborhood of the equilibrium being stable in the linear approximation is considered. The weak Raman thirdorder resonance and the strong fourth-order resonance are assumed to occur simultaneously in the system. The behavior of the approximated (model) system is studied in the stability domain of the fourth-order resonance. Areas of the parameters (coefficients of the normalized Hamiltonian) are found for which all motions of the system are bounded if they begin in a sufficiently small neighborhood of the equilibrium. Boundedness domain estimate is obtained. А disturbing effect of the double resonance on the motion of the system within the boundedness domain is described.

We consider the motion of a heavy rigid body with one point performing the specified highfrequency harmonic oscillations along the vertical. In the framework of an approximate autonomous system of differential equations of motion two new types of permanent rotations of the body about the vertical are found. These motions are affected by presence of fast vibrations and do not exist in the case of a body with a fixed point. The problem of stability of the motions is investigated.

Motions of a time-periodic, two-degree-of-freedom Hamiltonian system in a neighborhood of a linearly stable equilibrium are considered. It is assumed that there are several resonant thirdorder relations between the frequencies of linear oscillations of the system. It is shown that in the presence of two third-order resonances the equilibrium is unstable at any ratio between resonant coefficients. Approximate (model) Hamiltonians are obtained which are characteristic of the resonant cases under consideration. A detailed analysis is made of nonlinear oscillations of systems corresponding to them.

Stability of permanent rotations around the vertical of a heavy rigid body with the immovable point (Staude’s rotations) is investigated in assumption of a general mass distribution in the body and an arbitrary position of the point of support. In admissible domains of the five-dimensional space of parameters of the problem the detailed linear analysis of stability is carried out. For each set of admissible values of parameters the necessary conditions of stability are received. In a number of cases the sufficient conditions of stability are found.

Motions of a non-autonomous time-periodic Hamiltonian system with one degree of freedom are considered. The Hamiltonian of the system contains a small parameter. The origin of the phase space is a linearly stable equilibrium of the unperturbed or complete system. It is supposed that the degeneration takes place in the unperturbed system with regard for terms of order less than five (the frequency of small nonlinear oscillations does not depend on the amplitude), and a resonance (up to the sixth order inclusively) occurs. For each resonance case a model Hamiltonian is constructed, and a qualitative investigation of motion of the model system is carried out. Using Poincare’s theory of periodic motions and KAM-theory we solve rigorously the problem of existence, bifurcations and stability of periodic motions of the initial system. The motions we study are analytical with respect to fractional (for resonances up to the forth order inclusively) or integer (resonances of fifth and sixth orders) degrees of the small parameter. As an illustration, we analyze resonance periodic motions of a spherical pendulum and a Lagrange top with a vibrating point of suspension in the presence of the degeneration considered.

We study the motion of a satellite (a rigid body) in a circular orbit about its centre of mass. The satellite is subject to the central Newtonian gravitational field. The satellite’s principal central moments of inertia $A$, $B$ and $C$ are assumed to satisfy the equation $B=A+C$. This equation holds for thin plates. Particular motions occur when the plate executes pendulum-like oscillations of an arbitrary amplitude in the plane of the orbit. A linear analysis of the orbital stability of this motion is carried out. In the plane of parameters of the problem (an amplitude of oscillations and an inertial parameter) domains of orbital linear stability and instability of oscillations of the satellite are obtained both numerically and analytically.