Ashraf Hussein
Kafr ElSheikh 33516, Egypt
Department of Mathematics, Faculty of Science, Kafrelsheikh University
Publications:
Yehia H. M., Hussein A. M.
New Families of Integrable TwoDimensional Systems with Quartic Second Integrals
2020, Vol. 16, no. 2, pp. 211242
Abstract
The method introduced in [11] and [12] is extended to construct new families of severalparameter
integrable systems, which admit a complementary integral quartic in the velocities.
A list of 14 systems is obtained, of which 12 are new. Each of the new systems involves a number
of parameters ranging from 7 up to 16 parameters entering into its structures. A detailed
preliminary analysis of certain special cases of one of the new systems is performed, aimed at
obtaining some global results. We point out twelve combinations of conditions on the parameters
which characterize integrable dynamics on Riemannian manifolds as configuration spaces. Very
special 7 versions of the 12 cases are interpreted as new integrable motions with a quartic
integral in the Poincaré halfplane. A byproduct of the process of solution is the construction
of 12 Riemannian metrics whose geodesic flow is integrable with a quartic second integral.

Hussein A. M.
Precessional Motion of a Rigid Body Acted upon by Three Irreducible Fields
2019, Vol. 15, no. 3, pp. 285292
Abstract
We consider a quit general problem of motion of an asymmetric rigid body about a fixed
point, acted upon by an irreducible skew combination of gravitational, electric and magnetic
fields. Two of those three fields are uniform and the third has a more complicated structure.
The existence of precessional motions about a nonvertical axis is established. Conditions on the
parameters of the system are obtained. An alternative physical interpretation is given in the
framework of the problem of motion of a rigid body immersed in an incompressible perfect fluid,
acted upon by torques due to two uniform fields.
