Analytical solutions of non-scalar field equations, spinning particles in gravitationsl field.

I have wrote my first work "Mechanics of tangent rigid body" in which a rigid body is placed into tangent space that allowed me to represent it as a Hamiltonian system, in 1976. Since 1980 was working in the Space research Institute of the Acad. of Sci. of USSR, on problems of equilibrium of gravitating disk. In 1986 defended Ph.D. thesis on theoretical physics. In 1989 I have obtained a model of space-time containing rotating material disk. Built uniformly accelerated spherical coordinates in which I have solved Maxwell equations for the field of moving charge. I have generalized the isotrope complex tetrad method which was used then only in the oblate spheroidal coordinates, to the case of coordinates of prolate spheroid and succefully used this generalization for complete separating Maxwell equations in coordinate systems of oblate spheroid and round paraboloid. Later on, in 2009 I have completely separated Maxwell equations for waves in media whose electric permittivity and magnetic permeability are different functions in the space. This result is not published and its publication is not planned. Besides, I have obtained solution of Maxwell equations for the field of a particle, whose vector of dipole moment (electric or magnetic) is an arbitrary function of time. The published solutions, particularly, Deutsch solution of 1950 are erroneous). This result was presented on seminar of the Institute of Physics of Polish Academy of Siences in Warsaw (director -- Prof. J. Kijowski) in 1999, then in the Institute for Studies in Theoretical Physics and Mathematics (IPM) in Tehran in 2005 and was not published afterwards.

I have constructed the two-sheet Euclidean 3-space by putting that the ``radial'' coordinate in the oblate spheriodal system runs the whole of numerical line. Usually it does only non-negative values that produces two surfaces corresponding to its value equal to zero. These two surfaces are being identified that removes the boundary which would appear otherwise. If the lower limit of this coordinate is not put the boundary does not appear, but the space becomes two-sheet one with opposite sign of the ``radial'' coordinate on the sheets. This fact turned to be important when exploring my ``dual-to Kerr'' solution of the Einstein equation in which identifiction of the two surfaces turned to be impossible and the only way to avoid forming boundary is to introduce the second sheet. Thereby it was shown that my dual solution describes a stationary wormhole.

I have shown that in classical mechanics of mass point there exist two distinct objects called ``vector potential'', which have different geometric nature and define two distinct kind of dynamics. The object called so in classical electrodynamics is in fact co-vector, not vector which admits gradient (gauge) transformation, its square does not appear in Lagrangian of the mass point and does not appear in its Hamiltonian. Unlike it, the genuine vector potential does not admit gradient transformation, its square appears in the Lagrangian of the mass point and does not appear in its Hamiltonian. Ths difference reveal in terms of existence or non-existence of integrals of motion quadric on velocities.

Other new results, which are not being planned for publication, are related to development of analytical and geometric approaches to Navier-Stokes equation for planar and axially-symmetric flows.

I have built explicit representation of a fiber bundle with $SO(3)$ as the structure group that allowed me to propose a new geometric framework for the Yang-Mills theory. I have proved uniqueness of the Kerr-NUT solution among separating vacuum metrics (with N. Dadhich) and derived Papapetrou equation for photon from Lagrangian of electromagnetic field (with M. Safonova and A. T. Muminov).

1. Z. Ya. Turakulov, “Ravnovesie gravitiruyuschego zamagnichennogo gazovogo diska”, Astron. zh., 6:5 (1983), 866
2. Z. Y. Turakulov, “Relativistic rotating disk: the external field”, Int. J. Mod. Phys. A, 4:14 (1988), 3653      
3. Z. Ya. Turakulov, “Electromagnetic field of a charge moving with constant acceleration”, J. Geom. Phys., 14 (94), 305      
4. Z. Ya. Turakulov, “GROUP 21”, Physical Applications and Mathematical Aspects of Geometry, Groups and Algebras, Proc. of the XXI Int'l Colloquium on Group Theoretic Methods in Physics, 2, eds. H.-D. Doebner, W. Scherer, C. Schulte, World Scientific, Singapore, 1997

• Ulugh Beg Astronomical Institute of the Uzbek Academy of Sciences