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2013
Impact Factor

Alexander Savin

Moscow Aviation Institute

Publications:

Bardin B. S., Rudenko T. V., Savin A. A.
On the Orbital Stability of Planar Periodic Motions of a Rigid Body in the Bobylev–Steklov Case
2012, vol. 17, no. 6, pp.  533-546
Abstract
We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that a mass geometry corresponds to the Bobylev–Steklov case. The stability problem is solved in nonlinear setting.
In the case of small amplitude oscillations and rotations with large angular velocities the small parameter can be introduced and the problem can be investigated analytically.
In the case of unspecified oscillation amplitude or rotational angular velocity the problem is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients.
Keywords: Hamiltonian system, periodic orbits, normal form, resonance, action-angel variables, orbital stability
Citation: Bardin B. S., Rudenko T. V., Savin A. A.,  On the Orbital Stability of Planar Periodic Motions of a Rigid Body in the Bobylev–Steklov Case, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 533-546
DOI:10.1134/S1560354712060056
Bardin B. S., Savin A. A.
On the Orbital Stability of Pendulum-like Oscillations and Rotations of a Symmetric Rigid Body with a Fixed Point
2012, vol. 17, no. 3-4, pp.  243-257
Abstract
We deal with the problem of orbital stability of planar periodic motions of a dynamically symmetric heavy rigid body with a fixed point. We suppose that the center of mass of the body lies in the equatorial plane of the ellipsoid of inertia. Unperturbed periodic motions are planar pendulum-like oscillations or rotations of the body around a principal axis keeping a fixed horizontal position.
Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of the perturbed motion are obtained in Hamiltonian form. Regions of orbital instability are established by means of linear analysis. Outside the above-mentioned regions, nonlinear analysis is performed taking into account terms up to degree 4 in the expansion of the Hamiltonian in a neighborhood of unperturbed motion. The nonlinear problem of orbital stability is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients. The orbital stability is investigated analytically in two limiting cases: small amplitude oscillations and rotations with large angular velocities when a small parameter can be introduced.
Keywords: Hamiltonian system, periodic motions, normal form, resonance, action–angle variables, orbital stability
Citation: Bardin B. S., Savin A. A.,  On the Orbital Stability of Pendulum-like Oscillations and Rotations of a Symmetric Rigid Body with a Fixed Point, Regular and Chaotic Dynamics, 2012, vol. 17, no. 3-4, pp. 243-257
DOI:10.1134/S1560354712030033

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