Borislav Gajic
Publications:
Dragovic V., Gajic B., Jovanovic B.
Note on Free Symmetric Rigid Body Motion
2015, vol. 20, no. 3, pp. 293308
Abstract
We consider the Euler equations of motion of a free symmetric rigid body around a fixed point, restricted to the invariant subspace given by the zero values of the corresponding linear Noether integrals. In the case of the $SO(n − 2)$symmetry, we show that almost all trajectories are periodic and that the motion can be expressed in terms of elliptic functions. In the case of the $SO(n − 3)$symmetry, we prove the solvability of the problem by using a recent Kozlov’s result on the Euler–Jacobi–Lie theorem.

Dragovic V., Gajic B.
FourDimensional Generalization of the Grioli Precession
2014, vol. 19, no. 6, pp. 656662
Abstract
A particular solution of the fourdimensional Lagrange top on $e(4)$ representing a fourdimensional regular precession is constructed. Using it, a fourdimensional analogue of the Grioli nonvertical regular precession of an asymmetric heavy rigid body is constructed.

Dragovic V., Gajic B.
On the Cases of Kirchhoff and Chaplygin of the Kirchhoff Equations
2012, vol. 17, no. 5, pp. 431438
Abstract
It is proven that the completely integrable general Kirchhoff case of the Kirchhoff equations for $B \ne 0$ is not an algebraic complete integrable system. Similar analytic behavior of the general solution of the Chaplygin case is detected. Fourdimensional analogues of the Kirchhoff and the Chaplygin cases are defined on $e(4)$ with the standard Lie–Poisson bracket.

Dragovic V., Gajic B.
Elliptic curves and a new construction of integrable systems
2009, vol. 14, no. 45, pp. 466478
Abstract
A class of elliptic curves with associated Lax matrices is considered. A family of dynamical systems on $e(3)$ parametrized by polynomial a with the above Lax matrices are constructed. Five cases from the family are selected by the condition of preserving the standard measure. Three of them are Hamiltonian. It is proved that two other cases are not Hamiltonian in the standard Poisson structure on $e(3)$. Integrability of all five cases is proven. Integration procedures are performed in all five cases. Separation of variables in Sklyanin sense is also given. A connection with HessAppel’rot system is established. A sort of separation of variables is suggested for the HessAppel’rot system.

Dragovic V., Gajic B.
Hirota–Kimura Type Discretization of the Classical Nonholonomic Suslov Problem
2008, vol. 13, no. 4, pp. 250256
Abstract
We constructed Hirota–Kimura type discretization of the classical nonholonomic Suslov problem of motion of rigid body fixed at a point. We found a first integral proving integrability. Also, we have shown that discrete trajectories asymptotically tend to a line of discrete analogies of socalled steadystate rotations. The last property completely corresponds to wellknown property of the continuous Suslov case. The explicite formulae for solutions are given. In $n$dimensional case we give discrete equations.

Dragovic V., Gajic B.
The Wagner Curvature Tenzor in Nonholonomic Mechanics
2003, vol. 8, no. 1, pp. 105123
Abstract
We present the classical Wagner construction from 1935 of the curvature tensor for the completely nonholonomic manifolds in both invariant and coordinate way. The starting point is the Shouten curvature tensor for the nonholonomic connection introduced by Vranceanu and Shouten. We illustrate the construction by two mechanical examples: the case of a homogeneous disc rolling without sliding on a horizontal plane and the case of a homogeneous ball rolling without sliding on a fixed sphere. In the second case we study the conditions imposed on the ratio of diameters of the ball and the sphere to obtain a flat space — with the Wagner curvature tensor equal to zero.
