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Borislav Gajic

Kneza Mihaila 36, Belgrade
Mathematical Institute SANU


Dragovic V., Gajic B., Jovanovic B.
Note on Free Symmetric Rigid Body Motion
2015, vol. 20, no. 3, pp.  293-308
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.
Keywords: Euler equations, Manakov integrals, spectral curve, reduced Poisson space
Citation: Dragovic V., Gajic B., Jovanovic B.,  Note on Free Symmetric Rigid Body Motion, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 293-308
Dragovic V., Gajic B.
Four-Dimensional Generalization of the Grioli Precession
2014, vol. 19, no. 6, pp.  656-662
A particular solution of the four-dimensional Lagrange top on $e(4)$ representing a four-dimensional regular precession is constructed. Using it, a four-dimensional analogue of the Grioli nonvertical regular precession of an asymmetric heavy rigid body is constructed.
Keywords: rigid body dynamics, Grioli precession, four-dimensional Lagrange case
Citation: Dragovic V., Gajic B.,  Four-Dimensional Generalization of the Grioli Precession, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 656-662
Dragovic V., Gajic B.
On the Cases of Kirchhoff and Chaplygin of the Kirchhoff Equations
2012, vol. 17, no. 5, pp.  431-438
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. Four-dimensional analogues of the Kirchhoff and the Chaplygin cases are defined on $e(4)$ with the standard Lie–Poisson bracket.
Keywords: Kirchhoff equations, Kirchhoff case, Chaplygin case, algebraic integrable systems
Citation: Dragovic V., Gajic B.,  On the Cases of Kirchhoff and Chaplygin of the Kirchhoff Equations, Regular and Chaotic Dynamics, 2012, vol. 17, no. 5, pp. 431-438
Dragovic V., Gajic B.
Elliptic curves and a new construction of integrable systems
2009, vol. 14, no. 4-5, pp.  466-478
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 Hess-Appel’rot system is established. A sort of separation of variables is suggested for the Hess-Appel’rot system.
Keywords: elliptic curves, $L-A$ pair, integrability, Hess-Appel’rot system, separation of variables
Citation: Dragovic V., Gajic B.,  Elliptic curves and a new construction of integrable systems, Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, pp. 466-478
Dragovic V., Gajic B.
Hirota–Kimura Type Discretization of the Classical Nonholonomic Suslov Problem
2008, vol. 13, no. 4, pp.  250-256
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 so-called steady-state rotations. The last property completely corresponds to well-known property of the continuous Suslov case. The explicite formulae for solutions are given. In $n$-dimensional case we give discrete equations.
Keywords: Hirota–Kimura type discretization, nonholonomic mechanics, Suslov problem, rigid body
Citation: Dragovic V., Gajic B.,  Hirota–Kimura Type Discretization of the Classical Nonholonomic Suslov Problem, Regular and Chaotic Dynamics, 2008, vol. 13, no. 4, pp. 250-256
Dragovic V., Gajic B.
The Wagner Curvature Tenzor in Nonholonomic Mechanics
2003, vol. 8, no. 1, pp.  105-123
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.
Citation: Dragovic V., Gajic B.,  The Wagner Curvature Tenzor in Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 105-123

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