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2013
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 Yehia H. M., El-Hadidy E. G. On the Orbital Stability of Pendulum-like Vibrations of a Rigid Body Carrying a Rotor 2013, vol. 18, no. 5, pp.  539-552 Abstract One of the most notable effects in mechanics is the stabilization of the unstable upper equilibrium position of a symmetric body fixed from one point on its axis of symmetry, either by giving the body a suitable angular velocity or by adding a suitably spinned rotor along its axis. This effect is widely used in technology and in space dynamics. The aim of the present article is to explore the effect of the presence of a rotor on a simple periodic motion of the rigid body and its motion as a physical pendulum. The equation in the variation for pendulum vibrations takes the form $$\frac{d^2 \gamma_3}{du^2}+\alpha [\alpha \nu^2+\frac{1}{2}+\rho^2- (\alpha + 1)\nu^2 sn^2u+2\nu \rho \sqrt{\alpha} cnu]\gamma_3=0,$$ in which α depends on the moments of inertia, $\rho$ on the gyrostatic momentum of the rotor and $\nu$ (the modulus of the elliptic function) depends on the total energy of the motion. This equation, which reduces to Lame’s equation when $\rho = 0$, has not been studied to any extent in the literature. The determination of the zones of stability and instability of plane motion reduces to finding conditions for the existence of primitive periodic solutions (with periods $4K(\nu)$, $8K(\nu)$) with those parameters. Complete analysis of primitive periodic solutions of this equation is performed analogously to that of Ince for Lame’s equation. Zones of stability and instability are determined analytically and illustrated in a graphical form by plotting surfaces separating them in the three-dimensional space of parameters. The problem is also solved numerically in certain regions of the parameter space, and results are compared to analytical ones. Keywords: stability, pendulum-like motions, planar motions, periodic differential equation, Hill’s equation, Lame’s equation Citation: Yehia H. M., El-Hadidy E. G.,  On the Orbital Stability of Pendulum-like Vibrations of a Rigid Body Carrying a Rotor, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 539-552 DOI:10.1134/S1560354713050067
 Yehia H. M., Elmandouh A. A. New Integrable Systems with a Quartic Integral and New Generalizations of Kovalevskaya’s and Goriatchev’s Cases 2008, vol. 13, no. 1, pp.  56-69 Abstract In his paper [1], one of us has introduced a method for constructing integrable conservative two-dimensional mechanical systems, on Riemannian 2D spaces, whose second integral is a polynomial in the velocities. This method was applied successfully in [2] to construction of systems admitting a cubic integral and in [3, 4] and [5] to cases of a quartic integral. The present work is devoted to construction of new integrable systems with a quartic integral. The potential is assumed to have the structure $V = u(y) + v(y)(a \cos x + b \sin x) + w(y)(c \cos 2x + d \sin 2x)$. This is inspired by the structure of potential in the famous generalization of Kovalevskaya’s case in rigid body dynamics introduced by Goriatchev. The resulting differential equations were completely solved only for time reversible systems. A 10-parameter family of systems of the searched type is obtained. Four parameters determine the structure of the line element of the configuration manifold and the others contribute only to the potential function. In the case of time-irreversible systems the governing equations were solved in the three cases when the metric is identical to that of reduced rigid body motion. Those lead to three new several-parameter generalizations of known cases, including the classical cases of Kovalevskaya, Chaplygin and Goriatchev. Keywords: integrability, quartic integral Citation: Yehia H. M., Elmandouh A. A.,  New Integrable Systems with a Quartic Integral and New Generalizations of Kovalevskaya’s and Goriatchev’s Cases, Regular and Chaotic Dynamics, 2008, vol. 13, no. 1, pp. 56-69 DOI:10.1134/S1560354708010073
 Yehia H. M. Kovalevskaya's integrable case: generalizations and related new results 2003, vol. 8, no. 3, pp.  337-348 Abstract More than 11 decades have elapsed since S. V. Kovalevskaya discovered her famous case of integrability of the equations of motion of a heavy rigid body about a fixed point [1]. Nevertheless, the last 17 years have witnessed the emergence of some amazing and even unexpected results generalizing this case or valid under the same condition $A = B = 2C$. In this paper we give a summary of the known integrable cases of Kovalevskaya's type and point out some of their generalizations. A total of four general and five conditional integrable cases is listed. Of those, two general and three conditional cases are introduced for the first time. Citation: Yehia H. M.,  Kovalevskaya's integrable case: generalizations and related new results , Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 337-348 DOI:10.1070/RD2003v008n03ABEH000250