Hamad Yehia

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.

Four new integrable classes of mechanical systems on Riemannian 2D manifolds admitting a complementary quadratic invariant are introduced. Those systems have quite rich structure. They involve 11–12 arbitrary parameters that determine the metric of the configuration space and forces with scalar and vector potentials. Interpretations of special versions of them are pointed out as problems of motions of rigid body in a liquid or under action of potential and gyroscopic forces and as motions of a particle on the plane, sphere, ellipsoid, pseudo-sphere and other surfaces.

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.

The method introduced in [20] was applied in [21] and [22] for constructing integrable conservative two dimentional mechanical systems whose second integral of motion is polynomial up to third degree in the velocities. In this paper we apply the same method for systemaic construction of mechanical systems with a quartic integral. As in our previous works, the configuration space is not assumed an Euclidean plane. This widens the range of applicability of the results to diverse mechanical systems such as rigid body dynamics and motion on two dimensional surfaces of positive, negative and variable curvature. Two new several-parameter integrable systems are obtained, which unify and generalize several previously known ones by modifying the configuration manifold and the potential of the forces acting on the system. Those systems are shown to include as special cases, integrable problems of motion in the Euclidean plane, the hyperbolic plane and different types of curved two dimensional manifolds. The results are applied to problems of rigid body dynamics. New integrable cases are obtained as special versions of one of the new systems, corresponding to different choices of the parameters. Those cases include new generalizations of the classical cases of Kovalevskaya, Chaplygin and Goryachev.

In the present note we introduce a full isomorphism between the problems of motion of a triaxial rigid body and motion of a particle on a triaxial ellipsoid. Using this isomorphism, we can obtain a very special case of integrability in the last problem from an integrable case of the first problem. The new case is time irreversible and is not separable in any configurational variables for arbitrary initial conditions. It can be interpreted as a motion of an electrically charged particle under potential and Lorentz forces. The result is extended to cases of motion on one-sheeted and two-sheeted hyperboloid.

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.