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    Ashraf Hussein

    Kafr El-Sheikh 33516, Egypt
    Department of Mathematics, Faculty of Science, Kafrelsheikh University


    Yehia H. M., Hussein A. M.
    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 half-plane. A byproduct of the process of solution is the construction of 12 Riemannian metrics whose geodesic flow is integrable with a quartic second integral.
    Keywords: integrable systems, quartic second integrals, Poincaré half-plane
    Citation: Yehia H. M., Hussein A. M.,  New Families of Integrable Two-Dimensional Systems with Quartic Second Integrals, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 2, pp.  211-242
    Hussein A. M.
    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.
    Keywords: rigid body, precessional motion, three irreducible fields
    Citation: Hussein A. M.,  Precessional Motion of a Rigid Body Acted upon by Three Irreducible Fields, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp.  285-292

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