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Vladimir Dragović

Vladimir Dragović
Kneza Mihaila 36, 11001 Belgrade, p.p. 367, Serbia
Mathematical Institute SANU

Full Research Professor of the Mathematical Institute of the Serbian Academy of Sciences and Arts
Head of the Department of Mechanics
The founder and chairman of the Dynamical Systems group
The co-chairman of The Centre for Dynamical Systems, Geometry and Combinatorics of the Mathematical Institute of the Serbian Academy of Sciences and Arts

Born: July 26, 1967
1987: B.Sc. in Mathematics, University of Belgrade
1988-1992: aspirant at Moscow State University,Faculty of Mechanics and Mathematics,Department for Higher Geometry and Topology
1992: Dr. Sci. in Mathematics, University of Belgrade thesis:R-matrices and algebraic curves advisor: Boris Dubrovin, Moscow State University
1992-2000, 2004-2007: teacher in Mathematical High School, Belgrade
1993-1998: Department of Mathematics, University of Belgrade, courses: Differential Geometry,several graduate courses
1996-1999: Head of the Committee for mathematical competitions of high school students of Serbia
1996-1999: Department of Philosophy, University of Nis courses: Differential Geometry, Partial Differential Equations
2003-2008: Department of Sciences and Mathematics, University of Montenegro courses: Geometry, Analysis and Geometry on Reimann Surfaces, Integrable Dynamical Systems 1, Integrable Dynamical Systems 2
2004-2008: regualar associate member of ICTP Abdus Salam, Trieste, Italy
2004-2008: Director of the Mathematical High School, Belgrade
2008-2012 Researcher, the Mathematical Physics Group, University of Lisbon


Since 2012, Full Professor of Mathematics, University of Texas at Dallas
Since 1988, employed at Mathematical Institute SANU
Conducting of scientific work
Chairman of the Seminar Mathematical Methods of Mechanics, since its founding in 1993
Advisor for four M.Sc. theses and four Ph.D. theses at Department of Mathematics, University of Belgrade
2001-2007, 2010-: Member of the Committee for mathematics and mechanics of the Ministry for Sience of Serbia
2002-2005: Leader of Project 1643 of the Ministry for Sience of Serbia
since 2006: Leader of Project 144014 of the Ministry for Sience of Serbia
since 2005: Co-Leader of the Italian-Serbian project Geometry, topology and combinatorics of manyfolds and dynamical systems


1987: Award for the best graduated student of the Faculty for Sciences and Mathematics
2004: Award of the Union of mathematical societies of Serbia and Montenegro for achievements in mathematical sciences for at most 40 years old researchers
2010: City of Belgrade Annual Award for natural and technical sciences (jointly with Milena Radnovic)


Dragović V., Gasiorek S., Radnović M.
Billiard Ordered Games and Books
2022, vol. 27, no. 2, pp.  132-150
The aim of this work is to put together two novel concepts from the theory of integrable billiards: billiard ordered games and confocal billiard books. Billiard books appeared recently in the work of Fomenko’s school, in particular, of V.Vedyushkina. These more complex billiard domains are obtained by gluing planar sets bounded by arcs of confocal conics along common edges. Such domains are used in this paper to construct the configuration space for billiard ordered games.We analyse dynamical and topological properties of the systems obtained in that way.
Keywords: integrable systems, topological billiards, billiard books, Fomenko graphs
Citation: Dragović V., Gasiorek S., Radnović M.,  Billiard Ordered Games and Books, Regular and Chaotic Dynamics, 2022, vol. 27, no. 2, pp. 132-150
Adabrah A. K., Dragović V., Radnović M.
Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials
2019, vol. 24, no. 5, pp.  464-501
We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with small periods. We observe a relationship between Cayley-type conditions and discriminantly separable and factorizable polynomials. Equivalent conditions for periodicity and elliptic periodicity are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal polynomials. In particular, the light-like periodic trajectories are related to the classical Chebyshev polynomials. Similarities and differences with respect to the previously studied Euclidean case are highlighted.
Keywords: Minkowski plane, relativistic ellipses and hyperbolas, elliptic billiards, periodic and elliptic periodic trajectories, extremal polynomials, Chebyshev polynomials, Akhiezer polynomials, discriminantly separable polynomials
Citation: Adabrah A. K., Dragović V., Radnović M.,  Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials, Regular and Chaotic Dynamics, 2019, vol. 24, no. 5, pp. 464-501
Dragović V., Radnović M.
Caustics of Poncelet Polygons and Classical Extremal Polynomials
2019, vol. 24, no. 1, pp.  1-35
A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean plane is presented. The novelty of the approach is based on a relationship recently established by the authors between periodic billiard trajectories and extremal polynomials on the systems of $d$ intervals on the real line and ellipsoidal billiards in $d$-dimensional space. Even in the planar case systematically studied in the present paper, it leads to new results in characterizing $n$ periodic trajectories vs. so-called $n$ elliptic periodic trajectories, which are $n$-periodic in elliptical coordinates. The characterizations are done both in terms of the underlying elliptic curve and divisors on it and in terms of polynomial functional equations, like Pell's equation. This new approach also sheds light on some classical results. In particular, we connect the search for caustics which generate periodic trajectories with three classical classes of extremal polynomials on two intervals, introduced by Zolotarev and Akhiezer. The main classifying tool are winding numbers, for which we provide several interpretations, including one in terms of numbers of points of alternance of extremal polynomials. The latter implies important inequality between the winding numbers, which, as a consequence, provides another proof of monotonicity of rotation numbers. A complete catalog of billiard trajectories with small periods is provided for $n=3, 4, 5, 6$ along with an effective search for caustics. As a byproduct, an intriguing connection between Cayley-type conditions and discriminantly separable polynomials has been observed for all those small periods.
Keywords: Poncelet polygons, elliptical billiards, Cayley conditions, extremal polynomials, elliptic curves, periodic trajectories, caustics, Pell’s equations, Chebyshev polynomials, Zolotarev polynomials, Akhiezer polynomials, discriminantly separable polynomials
Citation: Dragović V., Radnović M.,  Caustics of Poncelet Polygons and Classical Extremal Polynomials, Regular and Chaotic Dynamics, 2019, vol. 24, no. 1, pp. 1-35
Dragović V., Gajić B., Jovanović 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: Dragović V., Gajić B., Jovanović B.,  Note on Free Symmetric Rigid Body Motion, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 293-308
Dragović V., Gajić 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: Dragović V., Gajić B.,  Four-Dimensional Generalization of the Grioli Precession, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 656-662
Dragović V., Kukić K.
Systems of Kowalevski Type and Discriminantly Separable Polynomials
2014, vol. 19, no. 2, pp.  162-184
Starting from the notion of discriminantly separable polynomials of degree two in each of three variables, we construct a class of integrable dynamical systems. These systems can be integrated explicitly in genus two theta-functions in a procedure which is similar to the classical one for the Kowalevski top. The discriminantly separable polynomials play the role of the Kowalevski fundamental equation. Natural examples include the Sokolov systems and the Jurdjevic elasticae.
Keywords: integrable systems, Kowalevski top, discriminantly separable polynomials, systems of Kowalevski type
Citation: Dragović V., Kukić K.,  Systems of Kowalevski Type and Discriminantly Separable Polynomials, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 162-184
Dragović V., Gajić 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: Dragović V., Gajić B.,  On the Cases of Kirchhoff and Chaplygin of the Kirchhoff Equations, Regular and Chaotic Dynamics, 2012, vol. 17, no. 5, pp. 431-438
Dragović V., Kukić K.
New Examples of Systems of the Kowalevski Type
2011, vol. 16, no. 5, pp.  484-495
A new examples of integrable dynamical systems are constructed. An integration procedure leading to genus two theta-functions is presented. It is based on a recent notion of discriminantly separable polynomials. They have appeared in a recent reconsideration of the celebrated Kowalevski top, and their role here is analogue to the situation with the classical Kowalevski integration procedure.
Keywords: integrable dynamical system, Kowalevski top, discriminantly separable polynomials, systems of the Kowalevski type, invariant measure
Citation: Dragović V., Kukić K.,  New Examples of Systems of the Kowalevski Type, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 484-495
Dragović V., Gajić 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: Dragović V., Gajić B.,  Elliptic curves and a new construction of integrable systems, Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, pp. 466-478
Dragović V., Radnović M.
Bifurcations of Liouville tori in elliptical billiards
2009, vol. 14, no. 4-5, pp.  479-494
A detailed description of topology of integrable billiard systems is given. For elliptical billiards and geodesic billiards on ellipsoid, the corresponding Fomenko graphs are constructed.
Keywords: elliptical billiard, Liouville foliation, isoenergy manifold, Liouville equivalence, Fomenko graph
Citation: Dragović V., Radnović M.,  Bifurcations of Liouville tori in elliptical billiards, Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, pp. 479-494
Dragović V., Gajić 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: Dragović V., Gajić B.,  Hirota–Kimura Type Discretization of the Classical Nonholonomic Suslov Problem, Regular and Chaotic Dynamics, 2008, vol. 13, no. 4, pp. 250-256
Dragović V., Gajić 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: Dragović V., Gajić B.,  The Wagner Curvature Tenzor in Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 105-123

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